Integral representations of the schrödinger propagator

Vol. 62 (2008)

REPORTS ON MATHEMATICAL PHYSICS

No. I

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR LORENZO ZANELLI, PAOLO GUlOTTO

and FRANCO CARDIN

Dipartimento di Matematica Pura ed Applicata, Universita di Padova, Via Trieste 63 - 35121 Padova, Italy (e-mails:[email protected]@[email protected]) (Received October 24, 2007 - Revised January 22, 2008)

We consider the Schrodinger equation for the Hamiltonian operator H = - ~ !:1 + V (x), where V is a potential function modeling one-particle scattering problems. By means of a strongly converging regularization of the Schrodinger propagator U (t), we introduce a new class of integral representations for the relaxed kernel in terms of oscillatory integrals. They are constructed with complex amplitudes and real phase functions that belong to the class of global weakly quadratic generating functions of the Lagrangian submanifolds At C T*jRn x T*jRn related to the group of classical canonical transformations cf>k. Moreover, as a particular generating function, we consider the action functional A[y] = J~ !m IJi(s)1 2 - V(y(s»ds evaluated on a suitable finite-dimensional space of curves y Ere H1([o, t], R"). As a matter of fact we obtain a finite-dimensional path integral representation for the relaxed kernel. Keywords: Schrodinger equation, semigroups of linear operators, oscillatory integrals, symplectic geometry, path integrals.

1. Introduction In this paper we consider the Schrodinger equation: ihato/(t, x) = {

-~l:1o/(t, x) + V(x)o/(t, x), 2m

(1)

0/(0, x) = qJ(x) E H 2 (]Rn),

where V E CJ(]Rn) is a compact supported energy potential function, e.g. modeling one-particle scattering problems. Our aim is to construct a class of integral representations for the Schrodinger propagator o/(t) = e-kt H tp, More precisely, we introduce the following s-dissipative regularization li 2

V

HE := --1:1 - - - 2m (i + £)2 of the Hamiltonian H. Setting

[19]

20

L. ZANELLI, P. GUIOTIO and F. CARDIN

by standard arguments it follows that o/(t,')

L 2 ( jRn )

=

lim o/£(t, .).

(2)

£---+0+

Notice that 0/£ (t, .) is obtained by solving - i ~£ Ot 0/£ = H£ 0/£. The main result of our work is to prove that 0/£ admits an integral representation o/£(t, x) =

[

U£(t, x, y)q;(y) dy,

JjRn

with the kernel U£(t, x, y)

=

~

-'-2-S(t,x,v,u) Ii e(l+E)II P£ (t,

(3)

x, y, u) duo

jRk

In (3), the phase function S is an arbitrary weakly quadratic at infinity generating function for the Lagrangian submanifold n At:={cy,~;x,p) E T*JR x T*JRn: (x,p) =¢~(y,~)} = {(y,~; x, p) E T*JRn x T*JRn : p = VxS, ~ = - VyS,

0

= VuS},

denoting the graph of the canonical transformation, the classical flow

¢~,

(4)

related

2

to H(x, p) = fm + Vex). The complex amplitude p; is depending on the choice of the generating function S. Its construction is based on the following argument. We first derive an appropriate strongly convergent operator series expansion for HE tH

e -71

E

=

Ii

Ii

Lj B£,/t), whose terms admit kernels constructed by the use of P£,j'

Then we show that P; = Lj P:,j is L 1 convergent and fulfills (3). These last two properties are precisely guaranteed by the above introduced s-regularization. A particular application of (3) is a path integral representation of 0/£. Indeed in [8] it has been proved that the classical mechanical action functional t I A[y] = -m Ir(s)1 2 - V(y(s))ds,

i

o 2

evaluated on a suitable space of curves ret, x, y) C H1([0, t]; JRn), that is in fact a finite-dimensional manifold, corresponds to a weakly quadratic global generating function for At. As a consequence we can derive a finite-dimensional path integral representation, [

o/E(t, x)

= Jm jRn

[

U£(t, x, y)q;(y) dy ,

U£(t, x, y)

= Jr

- , - '-A[y] E2l!i e(l+

P:(dy), (5)

['(t,x,y)

where P:(dy) := y*[p;(u)du] is well defined as the image measure of the map yet, x, y, ·)(s) : JRk -+ ret, x, y) on the complex measure P;(t, x, y, u)du; in this way we obtain a complex measure on I' (t , x, y). It is very important to point out that the construction (3) is global in time, overcoming the problem related to the occurrence of caustics. This is the well-known

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

21

problem of nontransversality of the Lagrangian submanifolds At that occurs after a critical time, and this behaviour does not allow the existence of a unique global generating function S (t, x, y) for an arbitrary large time. We overtake this problem by using the direct generalizations of this function, namely the class of global generating functions S (t, x, y, u) involving auxiliary parameters u E JRk that permit, as shown in (4), to generate the whole Lagrangian submanifold for all times, even in the presence of non-transversality. Different techniques have been used to find representations of the Schrodinger propagator. First of all we mention the WKB methods [19], [20], where the obstruction of caustics gets across naturally. This difficulty has been solved, for example, with the theory of tunnel canonical operator, see Maslov [20]. Other results have been obtained by the use of Wigner functions and Wigner measures, see for example [27]. Several results have been obtained following the theory of Global Fourier Integral Operators, developed by Hormander and Duistermaat, see [9, 12]; in this framework we mention the recent important results [6, 7, 11, 13, 16, 17, 25, 26, 29]. In particular, the results of L. Kapitansky and y. Safarov [13], A. Laptev and 1. M. Sigal [16], and moreover of A. Hassel and J. Wunsch [11], represent the nearest references to the present study of the Schrodinger propagator. In comparison with these works, one of the main different features of this article is the class of phase functions involved. Indeed, here we construct oscillatory integrals of type (3) with global, in time and space, real phase functions, giving for them an explicit representation. More precisely, they constitute the whole large class of weakly quadratic generating functions for the Lagrangian submanifold At. Another new character of our work corresponds to the construction of integral operators intrinsically related to canonical transformations. These are not necessarily belong to L 1 and they may not can be E1.O. because in general the functions smooth. An advantage with this type of oscillatory integrals is the simple proof of a theorem for the change of phase functions, with the determination of a linear operator defined on L 1 acting on the functions P:. It is important to mention other techniques that can be found in the literature providing different representations of the propagator, namely the path integral formulations, see for example [2, 14, 24]. In particular, Albeverio and Mazzucchi [2] proved a rigorous formulation of Feynman path integral by means of infinitedimensional Fresnel integrals for polynomially growing potentials, while Robbin and Salomon [24] obtained a finite-dimensional phase space path integral in connection with metaplectic representation, but only for exactly quadratic Hamiltonians. With respect to this framework we underline the innovative character of our path integral formulation (5); indeed, in spite of the s-regularization introduced for the propagator, we show the determination of a well-defined integral over a space of curves y Ere H 1([0, t], JRn) (independent of c). This set exhibits the structure of finite-dimensional manifold, with a complex measure P:(dy) absolutely continuous w.r.t. the Lebesgue measure.

p:

22

L ZANELLI. P. GUIOTIO and F. CARDIN

Finally, our present treatment involves scattering-like potential energies V (i.e. compactly supported): this allowed us to catch rather quickly the above global setting for the cited generating functions weakly quadratic at infinity. Removal of this restrictive hypothesis on V will be the subject of a next paper. The plan of our paper is the following. After some preliminaries in Section 2, Section 3 develops some general results about the series expansion for Co-semigroup of linear operators. These results do not depend on the particular choice of the generator and will be applied in Section 6 to the central case of the Schrodinger group. Before doing this we have to develop some machinery about generating functions, that are the basic ingredient in order to build the integral representation. To this purpose, in Section 4 we first recall the definition of generating function weakly quadratic at infinity (GFWQI). Then, following a result by Viterbo [31] about the equivalence of two GFWQI, we prove that, under suitable assumption on the potential V, there exists a GFWQI for the Lagrangian submanifold At. The next step, Section 5, is to introduce integral operators of the type we will use for our representation, that is, integral operators with kernel b(t,x,y) =

f

JF.k

eiASCt.x,y,u)p(t,x,y,u)du.

In particular, we study some operations among such operators, proving that this class is closed under these operations. Finally, in Section 6, we apply the previous machinery to the semigroup e-L¥tH£ and we derive the existence of by following the plane drawn above. The last remark goes to the special case in which we apply the integral representation to the functional given by the classical mechanical action, that is A [y]. This allows us to derive a finite-dimensional path integral representation of 1frE;'

p:

2.

Preliminaries and settings

In the following X will denote a Banach space, 'H a Hilbert space. If I c ]R is an interval, C (I; X) and C k (I; X) will be, respectively, the sets of all continuous and continuously k times differentiable functions 1fr : I c ]R -+ ,1'. If I is closed and bounded such spaces will be endowed with the standard norms k

1/1frlle(l;x)

:= sup 1/1fr(t)llx, tEl

111frll e

k(l;x)

:=

L 1/1fr(j)ll eo(l;x) j=O

(here 1fr(j) denotes the strong derivative of order j). Shortly, we will write 111frll eo, 1I1frll el , . . . for such norms. We will denote also by Cb(I; X) the set of functions 1fr E C(I; X) such that 1/1frll eo < +00. Such notation is kept also in the case of classical spaces Ci([a, b] x ]Rd; ]Rm), that is the spaces of functions F E Ck([a, b] x ]Rd; ]Rm) such that F with all his derivatives up to order k are bounded on [a, b] x ]Rd.

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

23

The space of bounded linear operators on X will be denoted, as usual, by L:(X). This is a Banach space endowed with the operator norm that will be denoted by IILII.c(x) (or shortly IILII if it is clear what do we intend) for an operator L E L:(X). If 1fr E C([O, T]; X) then the Riemann integral JOT 1fr(s) ds is well defined. Such integral is also well defined in the case of a 1fr belonging to Cb(]O, T]; X), a case which we will need in what follows. We will need also to define convolutions. To this end let G : [0, T] ---+ L:(X) and assume that G is strongly continuous (namely, G(-)~ E C([O, T]; X) for any ~ E X). In this case, if 1fr E CO([O, T]; X) is well defined the convolution G * 1fr(t) := i t G(t - s)1fr(s) ds,

'it E [0, T]

and, in fact, this defines a map of C ([0, T]; X) into itself. If, moreover, we assume that G is strongly differentiable and G' : [0, T] ----+ L:(X) we have that the defined above mapping takes values in C l ([0, T]; X). All these facts hold true also in the case of G :]0, T] ----+ L:(X) under the assumption G(-)qJ E CbOO, T]; X) and G'(')qJ E CbOO, T]; X), respectively. The notation {etL}t>o stands for the Co-semigroup generated by a (generally unbounded) linear operator L : D(L) C X ----+ X which satisfies the hypotheses of Hille-Yosida theorem. For such matters we will refer to Pazy [22]. We recall that if qJ E D(L), the function 1fr(t) = etLqJ is a strong solution of the Cauchy problem 1fr'(t) = L1fr(t),

t

~ 0,

(6)

{ 1fr(0) =qJ,

that is, 1fr E CIOO, +00[; D(L)) n C([O, +00[; D(L)) and satisfies (6). Now, let us briefly recall some well-known facts about Schrodinger equation. First, if the potential V E LP(]Rn) with p > nl2 and p ~ 2 then the self-adjoint operator H := -;~tlx + V(x) is well defined on the domain H 2(]Rn) (here H 2(]Rn) stands for the usual Sobolev space W 2,2(]Rn)) . By Stone theorem it follows that the unitary group of linear operators e- ktHis well defined, and for any the function i tH 1fr(t, x) := e- 7i

qJ E

H 2(]Rn)

qJ(x)

is a strong solution for the Schrodinger equation

I

ih8t1fr(t, x) = ( -

1fr(0, x)

3.

;~ tl + V(X)) 1fr(t, x),

= qJ(x),

The series expansions for semigroups of linear operators In this section we assume that L : D(L) C X ----+ X is an infinitesimal generator of a Co-semigroup {et L }t>o. In particular, L is a densely defined linear operator

24

L. ZANELLI, P. GUIOTIO and F. CARDIN

satisfying the hypotheses of Hille-Yoshida theorem. It is well known that if L E £(X) the series expansion e' L = '£':'=0 (t L)j / j! holds true (and, indeed, this is the way to define the semigroup). However, this series does not make sense, generally, if L is an unbounded operator generating a Co-semigroup. In this framework, the main aim of this section is to prove that the semigroup etL admits a general class of series expansions of bounded linear operators, +00

etLq; =

L Bj(t)q;,

sp

E

X.

j=O

To this end, let us introduce the vector space

9

:= {G :]0, T] ---+ L:(X) : G(·)q; E Cb(]O, T); X)

Vep E X},

and the mapping QL :

9

---+

g,

[QL(G)(t)]q;:= etLq; - i t G(s)e(t-s)Lq;ds,

q; E X.

(7)

c g. We denote W L := QL(Q), and observe that it is an infinite-dimensional affine subspace of g. The set W L contains lots of elements and in particular etL belongs to W L (indeed etL = QdO)). We provide here a characterization of W L which will be useful in what follows.

It is easy to check that QL(Q)

THEOREM 3.1. Let X be a Banach space, the family of operators W E 9 belongs to W L if and only if there exists D C D(L), linear subspace dense in X, such that (i) W(·)ep E C([O, T); X), Vq; E X, (ii) W (O)q; = tp, Vq; E X, (iii) W(·)ep E CI(]O, T]; X), Vq; E D, (iv) there exists a constant C T :::: 0 depending only on T such that

II (W(t)L - W'(t))q;11 :'S CTIlq;II,

Vep E D, Vt E]O, T].

Proof: Suppose first that W(·) E W L • Then there exists G E

9 such that

W(t)q; = etLq; - i t G(s)e(t-s)Lq;ds.

The first three properties are easily verified with D := D(L). For the last, notice thm W(t)Lq; - W'(t)q; = etL Lip

-1

t

G(s)e(t-s)L Lip ds - LetLq;

o

+

G(t)q;

= G(t)q;,

1 t

+

Vep

G(s)Le(t-s)Lq; ds

E

D(L)

From the Banach-Steinhaus theorem it follows that C T := Therefore II(W(t)L - W'(t))epll

=

IIG(t)q;1I :'S CTIIq;II,

SUPtE)O.T)

IIG(t) II <

Vq; E D(L), t E ]0, T].

+00.

25

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

Conversely, suppose that the map W (t) : [0 , T] -+ £ (X ) satisfies (i) , . . . , (iv). By (iv) it turn s out that the operator W(t)L - W ' (t) can be extended from D to X. Let G (t ) be such linear exten sion. Clearly, by definition, if q; E D we have G ( ·)q; E Cb(]O, T]; X ). For a general q; E X there exists a l/f E D such that 11q; -l/f ll :::: e (here e > 0). Therefore II G (t)q; - G (to) q;II :::: IIG (t )( qJ -l/f )1I

:::: 2Cr llqJ -l/fll :::: 2C r c

+ IIG(t )l/f -

+ IIG(t )l/f -

+ IIG(t)l/f -

G (to) l/fll

+ IIG(to)(l/f -

qJ) 1I

G (to)l/fll

G (to)l/fll .

Hence lim suPt---+1o IIG(t) q; - G (t o)qJll :::: 2C r s . Because e is arbitrary, from this it follow s that GOqJ E COO, T]; X) . By the same argument it follows that IIG O qJ llco :::: CrllqJll, so G(')qJ E CbOO, Tl ; X) for any ip E X. Now, take ip E D C D(L) and e > O. We have el Lq;

-1

1

G(s) e(l - s)LqJ d s = el Lq;

-1

1

[W (s)L - W ' (s)]e(t-s)Lq; d s

d

= el LqJ

1 +

= etLqJ

+ W (t) qJ -

= el LqJ

1

+

e

-

[W (s ) e(I-S)LqJ] ds

ds

[W (s )e (t- s)LqJ] ;:: W (c ) e(l - li)LqJ .

(8)

From the Banach-Steinhaus theorem it follows that SUPSE[O.T] IIW (s ) II < calling that W (s )etLqJ --+ W (O)eILqJ = etLqJ as e -+ 0+

+00.

Re-

and because II W(s)elt- E)LqJ - W (s )etLqJll::::

sup IIW (s )lIll e(t- Ii)LqJ - el LqJ ll --+ 0, SEIO. T]

e -+ 0+,

we deduce that W (s )e lt- li)LqJ --+ elL qJ as e -+ 0+. Finally, by letting e -+ 0+ in (8) we obtain that QL(G ) = W . 0 We notice here that the cla ss W L does not depend on "small" perturbations of the operator L, namely on perturbations by a bounded linear operator. This concept will be defined in the following. PROPOSITION 3.1. L et X be a Banach space and l et L I : D(L) ~ X -+ X and L 2 : D (L 2) ~ X -+ X be two infinitesimal gen erato rs of Ce-semigroups such that D (L d D (L 2) is dense in X. L et Ll qJ - L 2qJ = LqJ, VqJ E D (L I ) D (L 2) with L E £ ( X ). Then we have W L , = W Lz.

n

n

Proof : We prove onl y that W L1 c W Lz' the oppo site inclu sion is the same. Let WI E W L1. From Theorem 3.1 we know that there exists a certain D I C D (Ld linear subspace den se in X on which WI satisfies the properties (i)-(iv). From the BanachSteinhaus theorem K r := SUPIE[O.T] IIW (t )11 is finite. In order that WI E WL z we

26

L. ZANELLI, P. GUIOTIO and F. CARDIN

have just to check (iv) with L2 in place of Li, We define D 2 := D, nD(L 2). Clearly D 2 is a dense linear subspace of X. Moreover, by taking cp E D2 C D(Lj) n D(L 2) we have II(Wj(t)L2 - W{(t))cpll

+ L 1 ) - W{(t))cpll :::: IIWj(t)(L2 - Lj)cpll + II(W j(t)L j - W{(t))cpli :::: IIWj(t)ll·II(L 2 - Lj)cpll + CTllcpl1 :::: (K TC1,2 + CT)IIcpll.

= II(W j(t)(L 2 -

Lj

0

In particular we have the straightforward COROLLARY E W L2.

3.1. With the hypothesis of the previous proposition, we have

exp{tLd

Consider now the Cauchy problem 1/f'(t) = L1/f(t), i : 0,

(9)

{ 1/f(0) = tp,

where L is an infinitesimal generator of a Co-sernigroup e'!', The next theorem will show that any pair (G, W := QL
1/I(t)

= W(t)cp +

1

G(t - s)1/f(s) ds.

(10)

Proof: If 1/I(t) = elLcp is a strong solution of (9) with tp E D(L), the conclusion follows immediately from the definition of W = QL
The last proposition allows us to consider series expansion for 1/1. Indeed, by the successive approximations, setting Bo(t)cp := W(t)cp,

1 1

Bj+j(t)cp:=

G(t - s)Bj(s)cp ds,

we have

00

1/I(t) =

L Bj(t)cp, j=O

vi

>: 0,

(11)

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

and this series is strongly convergent for any cp the followin result .

E

27

X fixed. In other words we have

THEOREM 3.2. Let X be a Banach space and L : D(L ) ~ X --* X be a densely defined linear operator generating a Ce-semigroup, G E 9 and W = ~h(G) E W L. Then

+00

elLcp =

L Bj(t)cp,

(12)

j=O

where the operators Bj(t), j ::: 0, are defined by (11). 3.1. Notice that in the case L = La + K with arbitrary bounded operator K (for instance, L = i 2~ /:). - V in the Schrodinger equation (1)), choosing

*

REMARK

W(t)

= elLo

(that is W(t)

= ei~ItJ.),

G(t) = W(t)L - W'(t)

the corresponding G is

= eILO(Lo + K)

- LoelLO = eILOK,

and the series expansion becomes

elLcp = elLocp +

+

1

11ill

1

e(I -I\ )Lo K el\Locp dt,

e(l-t\ )Lo K e(t\ -t2)Lo K et2LOcpdt2dt,

+ ...

(13)

This is one of the more classical perturbative methods to construct the sernigroup However, the choice of W is not necessarily linked to semigroups. Indeed, we will use a different kind of operators to represent the series of e' L in a suitable way. This permits us to discover a Feynman-like representation of the propagator for the solution.

e' (Lo+K ).

REMARK 3.2. It is important to point out that for each W E W L it is necessary to have different series expans ion (12).

Finally, expansion (13) allows us to prove a stability result for "lower order" perturbations of the generator L of the semigroup. THEOREM 3.3. Let X be a Banach space and Lo : D(L o) ~ X --* X be an infinitesimal generator of a Cc-semigroup etLo and a family {KeJe>O C £(X) is such that lime--+o liKe II = O. Finally, let L; := Lo + K, and etLt be the Ce-semigroup generated by L e • Then, if CT := SUPtE[o,TjlletLolI,

(14)

Proof: Indeed, recalling that there exists M > 0, by (13) we have that

wE

lR such that lIe tLo l1 ~ Me M

,

28

L. ZANELLI, P. GUlOTTa and F. CARDIN

IjetL Ecp - elLOcpll:s:

111

1

e(t-I\)LOKeel\Locpdtlll

111 1
+

=

(t

1

\

II K ,II JC{

= (eIIKEIICTI -

e(t-Ij )LoKee(t\-IZ)Lo Keel2Locp

~) CTII~II

1) CTllcpll,

o

and now the conclusion is straightforward.

4. Generating functions for Lagrangian submanifolds in T*IRn and T*IRn x T*IRn In this section we review some topics from the theory of generating functions for canonical transformations. Our main goal will be to prove the existence of a representation for the class of global generating functions weakly quadratic at infinity for the Lagrangian manifolds generated by Hamiltonian flows. Symplectic geometry - symplectic manifolds and their Lagrangian submanifolds - is the natural framework to treat this subject. Adopting standard notations, as in [2, 4], we denote by w = dp r.dx = 2::7=1 dp, /\ dx' the natural symplectic 2-form on T*IRn. The notion of Lagrangian submanifold of T*lR. n can be introduced by thinking of it as a multivalued generalization of the graph of the differential of a function / on IRn, graphtzij') = {(x, p) E T*IR n : p = Vx/(x), x E IR n}. It is easy to see that: WlgraPh(f)

= a;ixj/

i

dx /\ dx!

== 0,

dimgraph(f)

= n,

and graph(f) is globally transverse to the fibers of n : T*IRn ---+ IRn. Thus, we and dim L = n = say that L C T*IRn is a Lagrangian submanifold if: wlL =

°

~dim(T*IRn).

The Maslov [21] and H6rmander[l2] theories give draws of the local (with respect to the base manifold IRn = {x}) description of the Lagrangian submanifold L by means of the generating functions Sex, u). Here we give is the definition of global generating function: DEFINITION 4.1. A generating function for a Lagrangian submanifold L is a smooth function S = S (x, u) : IRn x IRk ---+ IR such that (i) L = {(x, p) E T*IRn : p = VxS, = VuS }, (ii) zero (in IRk) is a regular value of the map (x, u) f------+ VuS(x, u).

°

In many questions of symplectic topology the generating functions quadratic at infinity and weakly quadratic are of particular interest:

29

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

4.2. A generating function S: IRn x IRk -+ IR is called • quadratic at infinity (GFQI) if there exists a compact set K C IRk and a nondegenerate quadratic function a(x, u) = (a(x)u, u) such that Sex, u) = a(x, u)

DEFINITION

Vu

E IRk \

K.

• weakly quadratic at infinity (GFWQI) if there exists a nondegenerate quadratic function a (x, u) such that IIS(x,') - a(x, ')lI c I (]Rk ) <

n

"Ix E IR

00,

.

Of course, a GFQI is a GFWQI. GFQI are very important in the description of Lagrangian manifolds. The basic reason is that there is a fundamental theorem due to Viterbo stating that any generating function S of a Lagrangian manifold is obtained from a generating function quadratic at infinity S by a mixture of three basic operations that now we will introduce. . DEFINITION 4.3. Let S = Sex, u) be a generating function for a Lagrangian submanifold L, S: IRn x IRk ~ R We say that S is obtained by • A stabilization from S, if there exists a regular non degenerate quadratic function a = a(x, v) == (a(x)v, v), v E IRh and Sex, u, v) == sex, u) + (a(x)v, v). • A fibered diffeomorphism from S, if there exists a regular u : IRn x IRk ~ IRk such that u (x, .) is a diffeomorphism such that S(x, v) == S (x, u (x, u)). • An addition of constant from S, if there exists a constant C E IR such that

Sex, u)

==

sex, u)

+ C.

REMARK 4.1. Although it is trivial to check that each of these operations on S gives still a generating function S for the same Lagrangian submanifold, it is a rather intriguing fact that by means of these three operations we can exactly classify all the generating functions of a large enough class - see Theorem (4.1) below - of Lagrangian submanifolds. REMARK 4.2. Consider the stationary condition for a generating function Sex, v, ii) of L, with respect to the set of the parameters v,

0= V'vS(x, v, ii).

(15)

Suppose that there exists a smooth function g : IRn x

IRk

-+ IRk such that

v=g(x,v)

realizes the relation (15); then it is possible to determine a reduced new generating function S, Sex, v) = sex, g(x, v), v).

Indeed, {(x, p) E T*IR n

:

p = V'xS(x, v), 0 =

= {(x, p) E T*IR

n

= {(x, p) E T*IR

n

:

p = V'xS

v.s«, v)}

+ V'vS(x, v, v)l(x,g(x,V),v)

. V'xg(x, v, v),

0= V'vS(x, v, v)!
p

= V'xS, 0 = V'vS(x, v, v)

0

= V'vS(x, v, v)} = L

30

L.

ZANELLI, P. GUlOTTa

and F.

CARDIN

REMARK 4.3. The Hamiltonian vector field X H, related to the Hamiltonian function = (x, p), the

H : T*JRn ---+ JR, is defined by ixHw = -dH. Denoting curves by y related Hamilton system y = XH(y) reads:

y

=

JV H(y), {}

{~= VpH(x, p), p

where

= -VxH(x, p),

J

=

-II (II)]) .

(((])

We denote by
=

t

JVH(
It is well known that Hamiltonian flows send Lagrangian submanifolds into Lagrangian submanifolds. Furthermore, for any Lagrangian submanifold L belonging to a regular fiber of a Hamiltonian H, L C H-I(e), we have XH(x, p) E T(x,p)L.

The following theorem (see [31]) establishes the equivalence between the generating functions quadratic at infinity for Lagrangian manifolds in T*JRn related to Hamiltonian flows, with compact supported Hamiltonian: THEOREM 4.1 (Viterbo). Let Po E JRn, L o := {(x, Po) E T*JRn : x E JRn} and L, =
5\

and a constant C, such that diagram commutes,

JRn X JRk

and 52 +C are equivalent, that is the following R

!

) JRn X JRk

!

s] (r,«)

JR

(16)

S2(t,.j+C

id

~

JR

REMARK 4.4. Theorem 4.1 has been generalized by Theret (see [30]) for weakly quadratic generating functions. The assumption about the compact support of H (x, p) in Theorem 4.1 is not natural for the case of mechanical Hamiltonians of type H(p, x) = 2~p2 + Vex). However, with some work we can extend the result also to this case under a suitable assumption on V. First, we begin to consider in some detail the product manifold T*JRn x T*JRn, and we will equip it in a suitable symplectic structure w in order to deal with

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

31

canonical transformations as Lagrangian submanifolds in such a 'graph structure'. Let us consider the standard projections T*]Rn ~ T*]Rn x T*]Rn ~ T*]Rn

and we carry out a symplectic structure io on T*]Rn x T*]Rn ~ T* (]Rn x ]Rn) by the following twofold pull-back of the standard symplectic 2-form on T*]Rn, W := pr; w - pri

to

= dp2

1\

dX2 - dp,

1\

dXI.

We point out that for any fixed time t E [0, T] the set At := (cy,~, x, p)

T*]Rn x T*]Rn : (x, p) = ¢k(y,~)}

E

(17)

is a Lagrangian submanifold of (T*]Rn x T*]Rn, w), that is

wlA t =

0,

1

"2 dim(T*]Rn x

dim At = 2n =

T*]Rn).

Therefore we can try to determine a global generating function At = (cy,~; x, p) E T*]Rn x T*]Rn : p = "\IxS, ~ = -"\IyS, 0= "\IuS}.

Now we prove the validity of Theorem 4.1 for a different class of Hamiltonians. COROLLARY 4.1. Theorem 4.1, for (weakly) GFQI, does work also for the Lagrangian submanifolds At C T*]Rn x T*]Rn, related to the noncompact supported

(in T*]Rn) mechanical Hamiltonian H (x, p) := V is compact supported (in ]Rn).

fm2 + V (x),

where the potential energy

Proof: The Hamilton equations for H, with the initial conditions x(o) = Xo and = reads . 1 ~: m P' x(O) = Xo, { p - -"\Ix Vex), p(O) = 0.

pea)

°

First of all we observe that if Xo E ]Rn/ supp V then the solution is x(t) = Xo and pet) = 0, namely the identity operator on (xo, 0) for each t ~ 0. While in the case Xo E supp V we enquire about the behaviour of the solution by means of inequalities for the norms Ilx(t)1I and IIp(t)II. Indeed, simply we find: Ilx(t) {

xoll :::

~

sup IIp(t)llt,

m tE[O,Tj

IIp(t)1I ::: II v, Vlloot.

In this way, for the time interval t

E

[0, T], we find the uniform boundness:

xoll ::: ~I1"\1x VllooT, { IIp(t)11 ::: II "\Ix VllooT. Ilx(t) -

32

L. ZANELLI, P. GUIOITO and F. CARDIN

In view of the first inequality we can say that if Xo is contained in the compact set supp V then the solution x (r) is contained in a greater compact set K T C lRn , for all t E [0, T]. Moreover, for the second inequality we have that p(t) is contained in the closure of the ball B(O, IIVx VllooT) C lRn , centered in with radius R = IIVxVllooT. As a consequence we have that (x(t), p(t» E K T x B(O, IIVx VllooT) =: Q T C T*lRn x T*lRn.

°

Consider the characterstic function X for the set above, namely: XnT(x, p) =

I,

V(x, p) E QT, V(x, p) E lR2nj Q T .

{ 0,

Now we can define the Hamiltonian H := H . X, which it is compact supported, and due to the previous observations it is easy to see that t; := ¢kLo = ¢j;L o,

"It E [0, T].

This is an important equivalence because we can say that~Lt is also generated by a Hamiltonian flow with compact supported Hamiltonian H. Define the map, see Viterbo [31], h : T*lRn x T*lRn ---+ T*(lRn x lRn), (y,~;

x, p)

A

A

t---+ (X; P):=

(X-2-' + Y -2-; ~ +p p -~, y -

)

x ,

and observe that it is a symplectic isomorphism, sending the diagonal into the zero section. Indeed, the invertibility is straightforward, while

h*w(2n) = ~(dp - d~) /\ (dy

+ dx) + ~(dy -

dx) /\ (d~

+ dp)

= dp /\ dx - d~ /\ dy = win) - w~n) = W, tells us that the pull back of the 2-form w2n on T*(lRn x lRn) is exactly w on T*lRn x T*lRn, so it is a symplectic map. We consider the set At := h(A t) which is a Lagrangian submanifold of T*(lRn x }Rn) because the map h is symplectic and At is Lagrangian. In view of this relation, we can find a generating function for At if we know S for At,

s«. Xl, X2,~) := S(t, a.B, u) + (a +,8 - 2X l )y + X 2(a -

,8),

~ = (a,,8, y, u).

(18)

For the same reason, if we compute h -1,

h -1

(Xl, X2; PI, PZ)

t---+

:

T*(}Rn x }Rn) ---+ T*}Rn x T*}Rn,

(y,~; x, p) := (Xl

we can determine S if we know S(t, x, y, X) :=

+ ~P2, X2 -

~Pl; Xl - ~P2' X2 + ~Pl)'

S,

s«, a,,8, u)-y(,8-~jl)+x(,8+~jl)-jla,

X = (a,,8, u; u). (19)

33

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

These two transformations preserve the property of quadraticity at infinity, so we are going to deduce that the existence and uniqueness (modulo stabilizations and fibered diffeomorphism) that we are going to prove for generating function S is also proved also for S. Now we see that At is generated by a Hamiltonian flow with a compact supported Hamiltonian function, and consequently this permits us to apply Theorem 4.1 and conclude. Indeed, we easily observe that At = qir/A o), namely it is generated by a Hamiltonian flow, where ~

~

~

~

~

~

l~

H(XI, X 2; PI, P2):= H(X I - 2P2,

~

X2

l~

+ 2PI)

. Xh(QTxQ-r)'

Note that this Hamiltonian is compact supported because we have used the characteristic function X on the set h(Q T x QT), where the map h and the set QT have been previously defined. In view of the compactness of h(QT x QT) we can now apply Theorem 4.1 to have the existence and uniqueness, after the action of a stabilization, a fibered diffeomorphism and the sum of a constant, for the generating functions S of At weakly quadratic at infinity, but the transformations (18) and (19) preserve the property of quadraticity at infinity, so we conclude the validity of the theorem also 0 for At. Now we are going to present an explicit construction of a global weakly quadratic generating function for the above defined Lagrangian submanifold At. Before doing this, we borrow a result by [8], where a global generating function is provided but which, in general, is not weakly quadratic. LEMMA 4.1. Suppose that V E H (x, p) :=

2

fm + V (x).

c 2(]Rn)

with compact support,

and define

There exists a finite-dimensional space of curves (yX,

Y P)

E

['(t,x,y)CH I([0,T];]R2n) such that

1 t

Set, x, y, v)

= A[y(t, x, y,

v)]

=

[yP(s)yX(s) - H(yX(s), yP(s))] ds

(20)

corresponds to a global generating function for the Lagrangian submanifold At := (Ao), for every t E [0, T].

c/Jk

Proof: We have first to introduce some notation. Let N E Nand u. := 2n(2N + 1). In the space L 2([0, t]; ]R2n), given v E ]RJL == (]R2N+I)2n, V = (V-N, .. " VN) with v J E JR 2n , let PN(V):=

L

vJeJCr),

IJI:sN

be the trigonometric polynomial with coefficients v and lP'NL2([0, t]; ]R2n) the linear subspace of L 2([0, t]; JR2n) whose elements are {PN(V)}VElR.k. We will use the notation PN(V) = (VX, vP), where v X, v P E L 2([0, t]; ]Rn). Finally, define QN L 2( [0 , t]; ]R2n) := lP'NL 2([0, t]; JR2n)-l.

34

L. ZANELLI, P. GUIOTIO and F. CARDIN

In [8] it has been proved that, if N is big enough', namely if N

II V 2 H II L'X)

. (l

IS

such that

+2:;:) T < 1,

(21)

then it is possible to construct a global generating function for any Lagrangian submanifold At, t E [0, T], in the following way:

s«. x, y, v) = A[y(t,x, y, v)] =

l

t

[yP(s)yX(s) - H(yX(s), yP(s))] ds,

(22)

where A[·] is the Hamilton-Helmholtz functional evaluated on a suitable set of curves2 y = (yX, yP) E r(t,x, y) c H!([O, t]; ~2n) depending on (t, x, y, v) E [0, T] X ~n X ~n X ~JI (where J.L = J.L(T, H, n) = + 1)). The curves (yX, yP) have the following structure:

2n(2N(T, H)

y X yX(s):= y + - . - s +

ll

t

{

yP(s):= -

t

lflX(r)dr

10t

s lflX(r)dr - -

0l

+

too

s

lflP(r) dr,

t

10r lflX(r)dr,

lfl

x:= X V +

i', (23)

0

lfl P:= v P + f P,

where (VX , vP) == v while (f", fP)

==
E QNL 2([0, t]; ~2n)

must solve the fixed point equation (r, fP) = QNJV H (h(t, x, y, (VX, vP)

More precisely, this equation for (r,

+ tJ',

f P) reads:

1 1 r(-) = -QNhP(t, x, y, v P + fP)(.) = -QN m m

{ fP(.) =

1-

fP))).

fP(r) dr,

0

(24)

(25)

-QNVV(hx(t, x, y, V X + rK)).

We remark that the solution of this equation corresponds exactly to the fixed point of a contraction if the condition (21) is fulfilled, and of course as, a consequence, such a solution exists and is unique. Generally, rand f P are nonlinear functionals", on v, and [8] provides some good information on which that will be useful in what follows. In particular, there exist constants C, C' ::: such that

°

Ilf(t, x, y, v)(.)II L 2 ~ Ct,

IIVvf(t, x, y, v)(-)II L 2 ~ C't.

(26)

1In order that Eq. (24) be generated by a contraction map. 2Clearly HI ([0, t]; IR 2n) is the usual Sobolev space. 3We stress the fact that the upper indexes x and p are just labels that recall us with which component we are working. On the other side the lower indexes are true variables for the various functions involved. In particular, v X , v P are elements of L 2([0, t]; IR n ) that depend only on v E IRJL whereas f~'P(t,x, y, v) is again in L 2([0, t]; IR n) and it depends on x, y, v and t explicitly.

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

35

o REMARK 4.5. In the previous lemma we have briefly resumed the construction of a global generating function for All and it turned out that the space of curves F'(z, x, y) played a fundamental role. We now enquire in detail about such a set. First of all we underline that, if we fix the variables (t, x, y) E [0, T] x ~n X ~n , then the set of curves

I'(r, x, y) := {(yX(t, x, y, v)

I yP(t, x,

y, v)) E H\([O, t]; ~2n) : v E ~Ji}

(27)

has the structure of finite-dimensional manifold induced by the map y (t, x, y, .) : ~IL -----+ H \ ([0, t]; ~2n). Indeed, the inverse of this injective map, together with I' (t, x, y), represents exactly a unique local chart needed to define the manifold structure. Now it is important to observe that r(t, x, y, v)(s), as well as yX(t, x, y, v)(s), depend only on ._ ( x x x x) u.V_N"",V_\,V\"",V N

m2N·n _, 1TJlk

Ell'\,.

-.ll'\,.

mIL

ell'\,.

so they do not depend on vP and v~. Indeed, by (25) we deduce that fP does not depend on vP and v~, and in the light of the linear relation between rand f P, the same holds true also for r . Finally, in view of the definition of yX(t, x, y, v)(s), it is straightforward to recognize that the same conclusion holds for it. In view of these observations, we define a reduced space of curves, where now we consider only the curves yX lying in the configuration space ~n (instead of T*~n) and with the reduced effective parameters u E ~k: I'(r, x, y) := {yx(t, x, y, u)

E H\([O,

t]; ~n) : u E ~k}.

(28)

As we have previously observed for the other set of curves, I' has the structure of a finite-dimensional manifold, but now the dimension is k := 2N . n. The importance of the reduced space of curves I' (t , X, y) will be more clear in the next proposition where we will show that it can be used, together with the action functional A[yX] =

r ~mlyX(s)12 2

Jo

- V(yX(s))ds,

to construct a global weakly quadratic generating function for At. On the other hand, by using I'(r, x, y) and the Hamilton-Helmholtz functional A[y] =

it

[yP(s)yX(s) - H(yX(s), yP(s))] ds,

it has been obtained only a global generating function. Exactly for this reason, this reduced set F'(z, x, y) and the functional A [.] are the constituents which should be considered in building our type of path integral representation for the Schrodinger propagator, and this will be the subject of the last theorem of the paper.

36

L. ZANELLI, P. GUIOnO and F. CARDIN

PROPOSITION 4.1. Suppose that V E c2 (~n ) with comp act supp o r t, and consider

r,;; + 2

H(x, p) := VeX). Then there exists a generating function S(t, x, y, u ) w eakly quadratic at infinity f or the Lagrangian submanifold At such that in the interval t E [0, T] it assumes th e form S(t , x , y ,u )=m

Ix - Yl2

2t

m

2

+-Iu l 2

(29)

+c(t,x,y, u),

where the auxiliary parameters u E ~k wi th k = k (T , V ) E N, and th e remaining term c(t , ·) E ct (~n x ~n X ~k; ~), c( ·, x, y, u ) E Cl([O, T] ; ~) .

Proof: By Lemma 4.1, it is possible to construct a global generating function for any Lagrangian submanifold A t , t E [0, T], as

1 1

Set , x, y, v )

= A[y (t , x, y , v)] =

More explicitly, S et, x, y, v) =

= =

yP(s ) yx (s ) - H (y X(s ) , y P(s ) d s

r

10

it

y P(s )yx (s ) _ _ 1 ly P(s )12 - V(yX(s)) ds

2m WN

1

2m

+ Q N )Y P(S)' (JP'N + Q N)YX(S)

IW N + Q N )yP (s )l 2 - V (y X(s ) )ds

1/

JP'Ny P(s ) · JP'N YX(S)

1 2m

+ QN yP(s)· QNYX(S)

1 2m

- - / JP'N y P(s )/ 2 - - I Q N y P(S) /2 - V(y X(s) ds .

We remember the structure of the curves y X(s) := y

1 {

yP (s ) := -

r

X y +- s + 10 (px(r )

1

to

1

(30)

it - -

=

[y P(s)YX (s ) - H(y X(s ) , y P(s))] ds.

t

(j>X(r ) dr

0

+

is o

s

dr - -

(j> P(r )

t dr ,

r (j>X(r ) 10

dr ,

(j> X := v

0

(j> P :=

X

(3 1)

+ r,

u!' + fP ,

(32)

and the fixed point equation that determines (r , fP): (33)

37

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

But now it is easy to check that (33) can be rewritten in the form QNyX(t, x, y, v) = -.!..QNyP(t, x, y, v),

m

{ QNyP(t, x, y, v)

(34)

= -QNVV(yx(t, X, y, v)).

From the first of (34) we have that (31) becomes

s«. x, y, v) =

1 t

lP'NyP(s) ·lP'NYX(S)

1 m. - -2 IlP' NyP(s)12 + -IQ NyX(s)1 2 - V(yX(s)) ds. (35) m 2 We recall now that the function S so constructed generates the Lagrangian submanifold At, At := {(y,~; x, p)

E

T*~n x T*~n : (x, p) = k(y,~)}

{(y,~; x, p)

E

T*~n x T*~n : p

=

= VxS,

~

= -VyS,

0

= VvS}.

Therefore, the stationary conditions, 0= VvS(v) = (VvxS(v), VvpS(v)),

(36)

establish the following relations: lP'Nyx(t, x, y, v) = -.!..lP'NyP(t, x, y, v),

m

(37)

{ lP'NyP(t, x, y, v) = -lP'NVV(yx(t, x, y, v)).

Recalling parametrizations (32), the previous relations read X -

y

V

X

1

VX(s)+-t-- J=mlP'N {

(

X ;+ Jo(S V

)

vP(r)+fP(t,x,y,vX)(r)dr,

(38)

vP(s) = -lP'NVV(yX(t,x, y, v))(s).

The first equation can be rewritten in the form

x -y VX(s) + t

-

X

X

~ yt V

1 ( V~ -lP'N

-

m

yt

+

is

fP(t, x, y, vX)(r)dr

0

)

1 = -lP'N m

is

vP(r)dr.

0

(39)

In view of Remark 4.2, we can insert this equation, with its previous form (37), in (35) to obtain a new reduced generating function,

1 t

S=

mlP'NYX(s) ·lP'NYX(S)

-

~1lP'NyX(S)12

-

+ mQNYX(s) . QNYX(S)

~IQNyX(S)12

- V(yX(s))ds

38

L. ZANELLI, P. GUIOTTO and F. CARDIN

=

=

=

t

m IlPNYX(s)!2

10

2

r2 r 10 2

m IlPNyX(s)12

10

+m 2

IQ N yX(s)1 2 _ V(yX(s)) ds

+ m Ir(s)1 2 _ 2

V(yX(s)) ds.

m l yX(s)1 2 _ V(yX(s)) ds.

(40)

This is an important step because now we see that the generating function S is obtained by using the action functional A[yX]

=

i

t

1

-mIYX(s)1 2 - V(yX(s))ds

o 2

(instead of the Hamilton-Helmholtz functional), and evaluating it only on the curves yX E F'( t , x, y). So we realize that the new parameter space associated to S is u := (V=-N' "V=-l' vf, ... v~) E lR 2N .n = jRk C jRJL because, in view of the observation made above, yX has dependence only on u and not on all the other parameters (v~;

vP).

Now we write down a more detailed representation for this function:

r [I

m s«. x, y, u) = 10 "2

2

VX(s)

=m Ix - Yl2

2t

x t - y - Ji Vo m ++ "2lr(s)12 1

+ -mit 2

VX(s) - -Vo

1

+ -mit Ir(s)1 2ds 2

=m

lx

0

- Yl2

2t

2 1

.Jt

0

V(yX(s)) ] ds

ds

i t V(yX(s))ds 0

+m 2

L

!Uj12

O
m t' +"2 10 (Ir(t, x, y,

u)(s)!2 - V(yX(t, x, y, u)(s)))ds.

(41)

Finally, by defining c(t, x, y, u) := i t

[~ Ir(t, x, y, u)(s)1 2 -

V(yX(t, x, y, u)(s)) ]dS'

we conclude that S(t, x, y, u) = m

Ix - Yl2

2t

m

2

+ -lui + c(t, x, y, u) 2

generates the same Lagrangian submanifold At as previous S, but it is the expected generating function weakly quadratic at infinity. In order to check this final sentence,

39

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

it is enough to consider the nondegenerate quadratic form P(u) := ¥luI 2, for which

liS -

Pll c!(JRk)

+ II'VuS -

=

liS -

=

Ix - Yl2 2t + Ilc(t, x, y , ·)IIcO(JRk ) + lI'Vu c(t , x, y, ') lI c o(JRk) < +00.

Pllco(JRk)

'Vu P

II CO(JRk)

Moreover, by using the Cl-regularity property of f(t , x, y , u)(r) and yX(t , x , y, u)(r) in the variables (x, y, u) and the C~ -regularity property with respect to t, we obtain the same for c(t ,x , y ,u). 0 As a consequence of Corollary 4.1 and Proposition 4.1, now we prove a general result about all weakly quadratic generating functions of At. THEOREM

4.2. Suppose that V E C 2(JRn with compact support, and let

2

H(x, p) := f;; + V(x) . Then any generating function S(t, x, y, for the Lagrangian submanifold At satisfies the relation

Stt , x , y , v[) =m

VI)

weakly quadratic

+ Q(t , x , y)wI . WI Ix - Yl2 m 2 2t + "2IV21 + c(t, x, y , V2) + Q(t, x , y)W2' W2 + C,

(42)

where (V2 , W2) = <1>(t , x, y, VI , w[) is a suitable fibered diffeomorphism on the parameter space ]RtL = ]Rk! X ]Rk! = ]Rk2 X ]Rk2, Q and Q are suitable non degenerate matrices, C is a constant. Proof : We consider the generating function of Proposition 4.1: [r - Yl2

m 2 +-Iul +c(t ,x,y,u) , uE]Rk. (43) 2t 2 Applying Corollary 4.1 we deduce that any other generating function is connected with this one by combinations of three operations: stabilization by suitable nondegenerate matrices Q(t, x , y ) and Q (t, x , y) on some ]Rk( and ]Rk2, composition with a fibered diffeomorphism (t , x , y , .) : ]RtL ---+ ]RIot, sum of a constant C (see Definition 4.3). From the composition of S with these operations we obtain the expected result. 0

S(t ,x, y ,u)=m

S. Integral operators related to generating functions In this section we construct integral operators whose representations are related to the class of generating functions studied in the previous section. More precisely we make a direct link between the representations of the kernel of this operators and weakly quadratic at infinity generating functions for the Lagrangian submanifolds At. We show some important properties, that will be useful in the next section. Consider a generating function S : [0, T] x ]Rn x ]Rn x ]Rk ---+ ]R weakly quadratic at infinity for At. The existence of such functions was proved in Theorem 4.1.

40

L.

ZANELLI, P. GUIOTIO and

F.

CARDIN

We are interested here to study integral operators of type

[B(t)cp] (x):= ( b(t,x, y)cp(y) dy,

I~n

where

b(t,x,y):= ( eiJ...s(t,X,y,u)p(t,x,y,u) du, }[f.k

(44)

where A E JR and p is some complex amplitude function. Among all the possible hypotheses on p that make B(t) a bounded linear operator on LZ(JRn) we will introduce the following class of densities. DEFINITION 5.1. Let Sk be the class of all complex functions p : [0, T] x JRn X JRk ----+ C, P = p(t, x, y, u) such that (i) SUPxE[f.n IIp(t, x, " ·)IIL1([f.nx[f.k) .:::: C T < +00, "It E [0, T]; (ii) SUpYE[f.n IIp(t, " y, ·)IIL1([f.n x[f.k) .:::: CT , "It E [0, T].

X

JRn

It is easy to check that if p

E

Sk then B(t) is well defined and B(t)

E

£(LZ(JRn».

LEMMA 5.1. The family of operators {B(t)}tE[O,T] is uniformly bounded on LZ(JRn), that is IIB(t)11 .:::: C T for all t E [0, T].

Proof: By Schur lemma and the properties of p

E

Sk we have

IIB(t)ll z,:::: (sup ( Ib(t, x, y)ldy) . (sup ( Ib(t, x, Y)ldx) XE[f.n }[f.n yE[f.n }[f.n .:::: ( sup { ( Ip(t, x, y, u)ldudy) . ( sup { ( Ip(t, x, y, u)ldudx) XE[f.n }[f.n }[f.k yE[f.n }[f.n }[f.k

= sup

XE[f.n

IIp(t,x,', ·)IIO([f.nx[f.k)· sup IIp(t,', y, ·)IIL1([f.nx[f.k). yE[f.n

o

.:::: C~

REMARK 5.1. Operators of type (44) are the base in constructing the integral representations (3). The next theorem is a crucial step because it allows us to prove that the kernels admits the same type of representation for each generating function S. THEOREM 5.1. Assume that S\ and Sz are two generating functions for At weakly quadratic at infinity with the space of parameters JRkl and JR k2, respectively. Then, for any (t, x, y) E [0, T] x JRn X JRn and A E JR, there exists a bounded linear operator (45)

such that (

i;

eiJ...Sl(t,x,y,U)p(u) du = (

».

eiJ.. S2(t,x,y,v) [Np] (v)dv,

YAER

Moreover, (46)

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

In particular, if p E S kI' then p = N p, that is belongs to Sk2 for all )., E R

p(1 , x , y,

41

v) := [NI ,x,y,}.p(t , x, y, .) ] (v )

Proof: By Viterbo's theorem 4.1 there exist stabilizations

+ (QI(1 , x, y) u , u) , 52(t , x, y, v) + (Q 2(t , x , y) ii, v ),

51 = 51(1 , x , y , u ) 52 =

of the generating function s 51 and 52 on the same space of parameters (u , i:i ), (v, ii) E ]Rk = ]Rk I X ]RkI = ]Rk2 X ]Rk2 , We will call w the new parameters, that is u = TIkI W, U = TIkI w. Moreover, there exists a fibered diffeomorphism RI,x,y : ]Rk ----+ ]Rk

such that 51 and 52 are equivalent, that means 51 (1 , X , y , w )

=

52(t , x, y, RI ,x,y(w )) ,

Now, define a

lulZ " ( Q ( 1 (u) '= e- -y- t" I I,x,y)-u,u- ) I,X ,y,}. , (2rr) k/2

This function clearly fulfills the identity

~

k

eiA(Ql (l,x,y)U.U)a

IRI

I (u) du = 1. I,X,y,).

Then, for any pEL I (]RkI ; C), we have

1

ei},SI (t,X,y,u)p (u ) du

k

IR I

= =

=

=

1 _ k

k

IR I xJR I

1 1

k k IR I x JR I

k IRl xIRI k

ei},(SJ{t,x.y.U)+(Q I (l ,x,y)u,U» p(u) al , (u ) du dll I,X.y,}.

ei}.SI (I,x,y,W)P (TI k w )a l (n - w) d w I t .x .y .); kI ei}.SZ(I .X ,y,Rt,x,y(W» p( TIk w )a l ( TI- w) d w I I,X ,y,}. k( i}.S2 (I,X,.v,W)p(TI R- I (w))a l (TI- R- 1 (w )) IJR- I (w ) ! dw kI I,x,y I,x,y,). kl I,x,y I,x,y

[ e JIR k2xJRk2

-1 -

IRkZ

ei }.S2(I,X,y,v) [NI,X,y,}. p] (v) d v ,

where , of course, [NI,x,y,}. p] (v ) I I I (v,v) l dii . (47) l : = l k_ei }.(QZ(I,X,Y)V,V)P (TI k( R- RI,x,y (v ,ii))a I,X,y,).(TIkl I,X ,y(v, v)) IJ Rtix , » IR Z

It remains to prove that N

==

NI,x.y,}. E £(L I(]Rk1); L1(]RkZ)), This is straightforward

42

L. ZANELLI, P. GUIOTIO and F. CARDIN

because INp(v)1 ::: L

k2

Ip(n kl R;';,y(v, v))IO"/X,y,A(n kl R;';,y(v,

v))l] R;';,y(v, v)1 dv,

hence IINpIILI(lRh) =

r INp(v)1 i-.

dv

::: L k2 (Lk2IP(nkl R;';jv, v))\a/X,y,A (n kl R;';jv, v))I] R;';jv, v)! dV)dv =

r _

J~hxlRh

jp(nkl w)la/X,y,A (n kl w) dw

= (Lkl jp(u)j dU) (Lkl

at~X'Y'A(U)dU)

= IIp11£l(lRh)'

For the last part of the statement, just the last inequality shows that IIp(t, x, y, ')II LI(ffih) ::: IIp(t, x, y, ')II£l(ffikl)' and by this and the fact that p E Ski it follows immediately that AER

p E Sk2' for any D

We now consider two operators of type (44) based upon two different p and different S, the generating functions for At weakly quadratic at infinity. By the previous theorem, modulo a linear transformation, we can assume that the two kernels are of type b1,2 (z, x , y) =

~ eiAS(t,x,y,u)p 1,2 (t , x " y u) du • ffik

The next proposition, that indeed is an immediate consequence of Theorem 5.1, shows that convolution of the two operators generated in this way is still of the same type. COROLLARY 5.1. Consider A E JR, S(t, x, y, u) GFQI of At, PI,2 E Sk and let b 1,2 be the corresponding kernels for the operators B1,2 defined as in (44)4. Moreover, suppose that IIPI (t, x, y, ')IILI(JRk) is uniformly bounded in t E [0, T] and y E lR.n , that is: sup sup IIPI(t, x, y, ')IIL1(ffik) < +00. tE[O,t] yEJRn

Finally, the operator

4This is strictly more than PI

E

Sko but enough for our purpose.

43

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

is well

defined and is of type (44) with the corresponding P given by p(t, x, Z, u) = it Nt,X,Z,AbS(t, x, z. .) ds ,

where bs(t, x, z. w) = PI(t - s, x, y, U)P2(S, y, Z, v),

w = (y, u, v).

Proof: First notice that, by definition, B(t)cp = i t B I (t - s)B2(s)cp ds

(in bl (t - x, y) in b2(s, y, z)cp(z) dz d Y) ds = in (it in bl(t -s,x, y)b2(s, Y,Z)dYdS) cp(z)dz. = it

s,

Now, by definition of bi.i.

r

J~n =

=

b l(t-s,x,y)b2(s,y,z)dy

in (ik eiAS(t-s,x,y,ul PI (t - s, x, y, u) dU) (ik eiAS(s,y,z,vl P2(S, y, z. v) dV) dy

r (r

J~kx~k J~n

eiA[S(t-s,x,y,u)+S(s,y,Z,vl]PI (t - s, x, y, U)P2(S, y,

z. v) d Y) dudv,

all the exchanges in the integrals being justified by the application of the FubiniTonelli theorem, that holds true in our hypotheses. and Define now the space of parameters w := (y, u, v) E IRn x IRk X IRk = notice that the function

IRn+2k,

s,«. x, z, w) := St: -

s, x, Y, u)

+ Sis, y, z. v)

is still a generating function of At, with the parameters wand for 0 < s < t. Introducing the auxiliary kernel bs(t, x, z, w) := PI (t - S, x, y, u)pz(s, y, Z, v), and applying Theorem 5.1 we can write

r b l(t-s,x,y)b2(s,y,z)dy= J~n+2k r eiASS(t,X,z,wlbs(t,x,z,w)dw

J~n

=

r

J~k

eiAS(t,x,z,ul Nbs(u) du,

where N was defined in (47). We have to say that Nbs stands for Nt,X,Z,AbS(t, x, z, '),

44

L. ZANELLI. P. GUIOTIO and F. CARDIN

and moreover that N depends on s. We can finally write the form for the kernel of B(t) , being

b(t , x , z)

= it l

k

eiAS(t,x,z,u )Nb s(u) du = l k eiAS(r.x,z,ul(it Nb s(u) dS)

du,

so that the dependence on S disappears in p(t , x, Z, u) := J~ Nb.tu) ds. Finally we have just to check that p E Sk . To this end, by (46), we have that !Ip(t, x,

Z,

')IIL1(JRk) :::

it !INbsOIILl (JRk)

ds :::

it IlbsII Ll (JRn+2k) ds.

Now { {IPI(t-s,X,y,U)P2(S,y,z,v)ldvdudy JJRn JJRk JJRk

IlbsIILI(JRn+2k)::: {

<

sup r E[O, Tj, YE!R!'

and since P2 E

Sk

Ilpl (r, x, y, ')II LI( JRk)llp2(S, y , ' , ') IILI (JRnxJRk)

o

the conclusion is immediate.

6. The Schrodinger propagator This section is the core of this work. It is well known that if V E L P(~n; lR.+ ), ~ and p ~ 2, then the differential operator H = - ~~ 6. + V (x) is self-adjoint on the domain H 2 (]Rn) . By Stone theorem, it defines the unitary group of linear p >

*

operators on L 2 (R"); we will denote such unitary group by e- tH. Also in the case of VEL 00 (]Rn;~) the same conclusions hold; in particular, as we have done in the previous sections, there will be considered V E c 2 (]Rn; ]R ) with compact support . This class of potentials arises in many problems of quantum mechanics, e.g. in modeling of the scattering or molecular dynamics. For example, we can provide a simplified model for the ammonia molecule N H3, which is usually described by a double-well potential with a rapidly asymptotic condition like limlx!---+oo V (x) = 0, see [10]. For a typical single particle scattering problem (see K. Yajima [32]) the potentials satisfy lV(x)1 ::: C(x)-/3 where (x) = (l + IxI 2) 1/ 2 and f3 > 5/2. Compactly supported potential energies are also involved in many statistical models, see e.g. [28]. The starting point for a derivation of Feynman path integral representation formula is the following well known perturbative series expansion, (48)

Unfortunately, the Feynman path integral that arises in this expansion is purely formal. In order to have a rigorous Feynman-like formula, together with a more general class of integral representations, as we said in Section 3 we want to write

45

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

a different series

.

00

L

e-ktHcp =

Bj(t)cp.

j=O This will be performed in such a way because we want to show the connection between this set of series operators and the class of generating functions weakly quadratic S for the Lagrangian submanifold At, introduced and studied in Section 4, by an appropriate integral representation. In order to do this, we introduce a suitable regularization He of the Hamiltonian H, and we prove the following integral representation:

r (r

(i+e)tH e- I l ecp(x) = J~n

~S(t.x,y.u)

J~k e(l+<)/I

h

)

p£(t,x,y,u)du cp(y)dy.

(49)

We point out that this representation is obtained for any generating weakly quadratic function S. In particular, the generating function constructed in Proposition 4.1 corresponds to the use of the action functional A[y] = J~(!m ly(s)1 2 - V(y(s)))ds evaluated on a suitable finite-dimensional space of curves r (t, x, y) ~ ]Rk. This permits us to obtain a finite-dimensional path integral formula Ue(t)cp(x) =

~

~n

1

ru,x,y)

e(l+~2)/lA[Y] P:(dy)cp(y)dy.

This will be the subject of the last theorem of this section. We begin by proving that a suitable s-regularization of the Hamiltonian allows to approximate the unitary group eTHEOREM

6.1. Assume that V

htH .

E Loo(]Rn),

and set

2

li V He =--~. 2m (i+8)2

Then e- ktH is the strong limit of e-4ftH£, that is i+e H e

, e-Tt Irm e--+O+

cp

L 2 (TIll ""n )

=

i H

e -,;t tp,

(50)

Proof: By triangular inequality we have lim sup e--+O+

lie -L¥tH£ cp - e -ktHcp I

2::

L

lim sup I e-4ftH£ tp e--+O+

-

e-4ftH tp I

2

L

+ lim sup lie -4ftHcp -

e-k

e--+O+

tH

cpll

2'

L

For the first term in the right-hand side we observe that 2

2

) li V (li K£ := He - H = - 2m ~ - (i + 8)2 - 2m ~ + V = -

[

1]

1 + (i + 8)2 V

46

L. ZANELLI, P. GUIOTTO and F. CARDIN

is a bounded operator on L 2(lR.n ) , "1£ > 0, and satisfies lime->o+ IIKel1 = 0. Therefore, applying Theorem 3.3 we conclude that Ile-f.¥tHecp - e-f.¥tHCPt2::: (eiIKeIICTt -

1) Cr llcp IIL2 e~ 0,

where

for any t

E

[0, T]. For the second term

I e-f.¥tH cp-e -k

tH

cp

I L2 = I e-*tH cp-cp I L2

°

e--+O+ -----+,

by the strong continuity of the semigroup e- ~ t H .

D

In the following definition we will assume that S is the generating function constructed in Proposition 4.1, where the decomposition S=m

Ix - YI 2 2t

m 2+c(t,x,y,u) +-luI

2

is proved. DEFINITION

6.1. We introduce the set

I:;;:= {a E L1(lR.k ;

q :

lk

a(u)e-U";e)/i¥IUI2du

= I}.

(51)

Then, given a: E I:;:, we define the integral operators with parameters t E]O, T] and e > 0, We(t)cp(x):= [

1

([

JHf.n lHf.k

(2rrt(i

+ c)lit/ 2

e-U";e)/iS(t,X,y,U)a:(u) dU) cp(y) dy.

(52)

We will denote by We(t, x, y) the kernel of We(t), that is, We(t, x, y) :=

~

JRk (2rrt(i

1

+ £)Ii/m)n/2

e

__I_S(txvu) n U+e)/i " . ' a (u) duo e

(53)

The operators We(t) so defined are not directly related to the particular function S because, in view of Theorem 5.1, we can always represent its kernel with the use of another phase function S but as a consequence we have to transform with a suitable linear operator. Notice that

a:

I:;:

-l/(i + s) = -£/0 + £2) + i/O + £2),

this shows that contains also functions a1 can be a:(u) = Ce ue with 0< a < £~.

a: growing at infinity; for example, it

With this settings, we have the following theorem.

47

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

E

°

b:,

6.2. Let a: E {We(t)}tE[O,Tj, e > as defined by (52). Then £(L 2(]Rn)) and satisfies the properties of Theorem 3.1, that is:

THEOREM

We(t)

CO([O, T); L 2(]Rn)), Vep E L 2(]Rn); 2 (ii) limHO+ We(t)ep = tp, Vep E L (]Rn); (iii) We(-)ep E CI(]O, T); L 2(]Rn)), Vep E H 2(]Rn); (iv) there exists a constant Ce,T ~ depending on e and T such that (i) We(-)ep

E

°

"It E)O, TJ, where L := _i

+ £H h

e

= _i

+£ (_~/:). _ h

e

2m

V

(i

+ £)2

).

Proof: We first prove that We(t) E £(L 2(]Rn)) for all t E]O, T) (We(O) = I). Due to the Schur lemma, we prove that

sup { IWe(t, x, Y)I dy < +00, XEIRn

JIRn

sup { IWe(t, x, y)1 dx < +00. yEIRn

JIRn

Because these two estimates are similar, we will limit the proof only to the first one. To do this, first notice that (

JIRn

IWe(t, x, y)! dy

.:::

iii IRn

IRk

(2m(i

1

+ £)hjmt/ 2

exp

(1+ -. h(l

s)

)

S(t, x, y, u) ah(u) e

I dudy.

Now, we recall that a particular generating function S, constructed in Proposition 4.1, has the decomposition Set, x, y, u) = m

Ix - Yl2 2t

m

2

+ -lui + c(t, x, 2

(54)

y, u).

But all the others are obtained by fibered diffeomorphism, stabilizations and sum of constants, so the proof in what follows does not change so much in the general case. However, in this case we can obtain the estimate

1 ) I .:::exp( -h(I+£2) e (IX -2t y!2 +2IuI2+c(t,X,y,u) m )) m ,

exp ( - h(i+£/

I

"It

E

[0, T),

(55)

48

L. ZANELLI, P. GUIOTIO and F. CARDIN

Therefore,

Indeed, in view of the definition (51) of a: and by using the property c(t, .) with respect to all variables, we have: M::=

r exp(-

sup

(t,x,Y)E[O,TlxlR2n llRk

8

/i (l

2

+ 8)

(mIUI2+C(t,X,Y,U)))la:(U)ldU <

L oo

E

+00.

2 (56)

Henceforth, noticing moreover that

Ln (2Jr/it~/m)n/2 1

mlx-Yl 2 ~ exp dy = ( ) ( / i ( l + 8 2) 2t) 8 I

n/2

8

,

we finally conclude that, y

°

<

8

< 1,

t

E

[0, T],

and by this inequality the conclusion is immediate. For the remaining statements of the lemma, the unique serious part is the proof of point (iv). Let q; E H2(~n). By direct computation and integration by parts, we have We(t)Leq; - W:(t)q;

=

LJw,«,

x, y)(- i :

8 ( -

;~ ~yq;(y) - (iV:~)2 q;(y)) ) - at w,«, x, y)q;(y) ]dY

r [(i +2m8)/i ~y We(t, x, y) + We(t, x, y) (iV Y)] + 8)/i( - at We(t, x, y) q;(y) dy

=

llR

=:

s:r o,«.

n

x, y)q;(y) dy.

(57)

49

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

By computing all the derivatives in We(t , x, y) we deduce

~

V (y ) C!. yS . - [ Rk (I + e)1i 2m

G o{t , x, y) =

x a h( u) ex p ( e

=:

r

+ . (I

e

+ e2) 1i

(1

1

+ e)1i

IV y Sj2

n

2m

2t

-- + - + . (I

1

+ e)1i

ats

]

seiJ'£,hS)dU

ge(t, x, y, u )eiA£.hS du ,

(5 8)

i Rk

for

1 = (1 + e 2 )1i .

A o.h

By using (54) in (58) we obtain

go(t, x , y, u)

= [ (i +e)1i n + 2t

1

n

C!.yc

V(y)

2M - 2t

+

(i

+

(i +e)1i 2m

I

+

1

IV ySol 2

(i +e)1i

2m

IVycl2

l

]

+ e)1i at So + (i + e)1i atc

a h(u )

+

1

VySo . V yc

(i+ e)1i

m

( +e)

(2 rrt(i 0+ e) li) nj2 exp - (1

e2)1i S

V (y ) C!.yc 1 IVycl2 1 V ySo ' V yc 1 ] = [ -- + + + atC (i + e)1i 2m (i + e)1i 2m (i + e) h m (i + e)1i

( e)

a: (u ) x (2rr t(i + e) n)n j2 exp - (1 + e2) /i S

because , if we denote as So := mi x - y!2 j 2t, then we have

IV SOl 2 - - + arso=o. 2m

We can now proceed with the estimates: it is ea sy to check that for e that

Igc(t,

x, y, u)! ~

C2 [ C1 + T x

C3 + T1x -

la:(u)1

(2rr /i ~t)nj2

~

1 we have

]

y!

Iexp ( -

e

)\. S

( 1 + e 2 ) /i

(59)

The con stants C 1,2.3 are also depending on T: 1

C\ :=- sup

2m tElO.T)

IIC!. yc(t,·)lloo '

(60) 2

C2:= sup t E[O. T ]

II

V( ·) + IVyc(t, ') 1 + V yC(t , .) + atC(t, .) II 2m

00

,

(61)

50

L. ZANELLI, P. GUIOITO and F. CARDIN

C3:=

SUp tE[O,T]

v :«, .) I II

t

where ",1100 is evaluated with respect to (x, y, u) also (55),

(62)

,

00 E jRn x jRn X

jRk.

Therefore, recalling

sup { IGE(t, x, y)ldy

JJRn

XEJRn

::: sup { XEJRn

I(

JJRn JJRk

::: sup { XEJRn

x

+ C2 + C31x Ii

la:(u)1

(2nlit~/m)n/2

XEJRn

X

{ [C I

JJRn JJRk

_ YI]

Ii

lexp (-

e (1 + e 2)1i

s) I

du dy

[CI + C + C31x _ YI] la:(u)/ JJRn JJRk Ii Ii (2n lit ~/ m)n/2

< sup { -

{ IgE(t, x, y, u)1 du dy

JJRn JJRk

::: sup { XEJRn

gE(t, x, y, u)eiAE,hS(t,X.y,u) dul dy

ex p ( -

2

(

(l

€ 2

+e

)Ii

(m Ix - Yl2 2t

+ m lul 2 + ctt , x, 2

y, U»))dU dy

: : M: sup JJRn{ [CI + C + C3 1x _ YI] Ii n 2

xEJRn

x where

1

(2nlit~/m)n/2

exp (

m Ix - YI (l + e 2)1i 2t €

2)d Y,

M: is defined in (56). Integrating the others terms we have vt E ]0, T],

(63)

vt E ]0, T].

(64)

because

In the same way

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

51

Finally, by (63) and (64),

II We(t)Lecp -

r IGe(t, X, y)1 dX) Ilcplliz i; IGe(t, X, y)1 d Y) (sup YElRn]lRn

W;(t)cplliz :::: (sup ( xElRn

: : [M:(

C1 +

r

~2 + ~~) (~)n =: Ce,T 'it

E

which is the conclusion.

]0, T], D

Applying Theorems 6.2 and 3.2 we have immediately theorem.

0': '2;:

'2;:

THEOREM 6.3. Let E be fixed with defined by (51), We(t) defined by (52) and let Ge(t) := We(t)L e - W;(t) with L; := _ite He. Then, the semigroup of bounded operators generated by the operator He

=

liz

--~

2m

-

V --""7

(i+c)Z

admits the strongly convergent series expansion ' e) (H

e -flHe cP

00

= 'L" ' Be,j(t)cp,

(65)

j=O

where Be,o(t) := We(t), and

1o,« 1

Be,j+I(t)CP =

s)Be,j(s)cp ds,

'ij

~ 0.

REMARK 6.1. Notice that there is not a unique series expansion (65), because E On the other hand it is still an open problem this depends on the choice of to clarify if there exists a particular choice of such that the corresponding series expansion becomes semiclassical, namely if

0': '2;:.

IIBe,j(t)1I :::: lij Ke,j,T,

0':

Vj ~ 1,

'it

E

[0, T].

This aspect needs more accurate estimates for the norm of these operators, so in the light of its relevance, will be enquired in a future work. The next step is to study the form of series expansion (65). In particular, we

(ite)

want to show that there is an appropriate integral representation of eI He tp, In order to do this we have to study the convergence of the kernels of the operators Bj(t).

0': '2;:

THEOREM 6.4. Let e, Ii > 0, E be fixed. Choose a generating function S weakly quadratic at infinity for the submanifold AI' There exists then a function p: = p:(t, x, y, u) such that exp(- (i : £)

in,)cp(X) = in u,«, x, y)cp(y) dy,

'icp

E

n),

H 2(lR

(66)

52

L. ZANELLI, P. GUlOTTa and F. CARDIN

where

Ue(t, x, y):= [exp(

JJRk

(l

i

+e

2

)Ii

Set, x, y, U))P:(t, x, y, u) duo

(67)

Proof: By (65) we have

+ s) tHe)ep =

exp ( - (i

Ii

f:

Bj(t)ep,

Vep

E

H 2 (JR n).

j=O

We recall that Bg(t)ep(x) = [

[ J.il?n (JJRk

. 1

/2

(2rrt(i

+ s)lit

(2rrt(1

+ s)li)n

exp(- . 1 (1

+ s)1i

Set, x, y, u))a:(u) dU) ep(y) dy.

Defining

h

1

Pe oCt, x, y, u) := ,

/2

(S

exp -

(l

+e

2

) h

)Ii

Set, x, y, u) a e (u),

we have that Be,o(t)ep(x) =

Ln (Lk exp(nl.e,hS(t, x, y, u) )p;,o(t, x, y, u) dU) ep(y)dy,

where

1

A

e,

h---::-

-

(l

+ s2)1i .

It is easy to check that P;,o

E Sk. where the class Sk is defined by Definition 5.1. Moreover, by definition of the operator Ge(t) and (58),

Ge(t)ep(x) =

Ln (Lk eiAE,hSge(t, x, y, u)

dU) ep(y) dy.

It is easily verified that, by (59), ge E Sk. Furthermore, by (63), Ilge(t, x, y, ·)IILI is uniformly bounded in (t, y) E [0, T] x JRn. Therefore, applying Corollary 5.1 j-times

we obtain that Be,j(t)ep(x)

= Ln (Lk eiAE.IlS(t,x,y,u) P;,j(t, x, y, u) dU) ep(y)dy,

where P;,j(t,x,y,u) =

it t,X,y,A N

e,1l [ge(t-s,x,b,U)p;,j_l(S,b,y,

~)] ds.

(68)

Therefore exp(- (i

+ s) tHe)ep(X) Ii

=

f: JJRn[ JJRk

[exp(iAe,hS(t, x, y, u))p;,/t, x, y, u)ep(y) du dy.

j=O

(69)

53

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

Our aim is now to exchange the sum with the two integrals in (69). In order to do this it is enough to prove the total convergence of the series L~oexp(iA£,hS{t, x, y, U))P:,/t, x, y, u)cp(y), that is the convergence in the norm L 1 (lRn x IRk). With this respect, note that 00

L

II

exp(iA£,hS(t, x, -, e))p:.j(t, x, ., e)cp(')IILI(JRnxJRk)

j=O 00

sL

IIp:,j(t, x, ., e)cp(.) IILI(JRn xJRk)

j=O

=

f

f f

/P:,/t, x, y, u)ldu /cp(y)/ dy.

j=O JJRn JJRk

Now, by (46) and (68) we have that

lip:,/t, x,

y, ')IILI(JRk)::::

t

Jo

Ilg£{t - s, x, 0,

~)p:' j-l (s, 0, y, q) IIIL (JRnxJRkxJRk)

ds.

Recalling (59), IIg£{t - s, x, 0, ~)P:,j-l (s, 0, y, q)1I0(JRn xJRk xJRk)

therefore h

IIp£,/t, x, y, ')II L'(JRk)

s t(f f

Jo JJRn JJRk

Ig£{t-s,x,z,v)IIIP:j_l(S,Z,y")II LI(JRk) dv dz)dS, '

and iterating this inequality we obtain h

IIp£,/t, x, y, ')IIL'(JRk)

:::: f(f , f

,nlg£{tk-l-tbXbXk+l,Vk)IIIP:,o{tj,Xj+l,y")IILI(JRk)dVdX)dt, J 1l j J(JRn)J+I J(JRk)J k=1

= dv, :::: ti-i ::::

dv], dx = dx, .. , dXj+lo dt = dt, ... dt], D..j = {(tl, ... , tj) : :::: tl :::: t} is the j-dimensional simplex and with Xo = x and to = t. Now, by definition of P:,o' by (55) applied in a similar way of (63) we

where dv

o :::: tj

obtain

54

L. ZANELLI, P. GUIOTIO and F. CARDIN

sup ~ [ Ige(tj-l - tj, Xj, Xj+l, vj)l, II P:,o(tj, XJ+l, y, XjEJRn JRn JJRk

')IILl(JRk)

dXj+l dv]

: : K: < +00,

K:

where the constant depends (as shown) on 8 and Ii but also on t, that we consider fixed. In order to conclude we observe that iterating the previous estimate we get fl fl [ fl t j I/PE,j(t, x, y, ')IILl(JRk) ::: KE,j l: dt = KE,j j!' J

and now the conclusion follows easily. THEOREM 6.5.

Let H

0

= - ~~ ~ + V (x)

where V E C 2(lRn) with compact support,

and 1/f(t) := e -kt H tp, ip E H 2(lRn). Choose a generating function S weakly quadratic at infinity for the submanifold At. Then, for any 8 > there exists p:(t, x, y, u) such that

°

[

"/JAt, x) = JJRn

([

(70)

~s(t.x,y,u) fl

JJRk e(l+E

)

PE(t, x, y, u)du cp(y)dy.

)Il

Proof: In the Theorem 6.1 we proved

Now, applying Theorem 6.4 we conclude immediately. 6.6. Take the operator H = -~~~x + V(x) where V compact support. Then the solution of the Schrodinger equation THEOREM

I

iliad/(t, x) 1fr(0, x)

= (-

= cp(x)

E

;~ ~x + V(X)) 1fr(t, x),

0 E

C 2(lRn) with

(71)

H 2(lRn),

admits the path integral representation: 1/f(t,')

L2(JRn) =

lim 1frE(t, '), E-+O+

1frE(t, x) = [ UE(t, x, y)cp(y) dy, JJRn UE(t, x, y) =

1

r(t,x,y)

where y

E

exp (

(l

i

+8

2

)Ii

(72)

(73) A[Y]) PEfl(dy),

r(t,x,y) C H1([0,t];lR n), and r(t,x,y) is a finite-dimensional manifold.

INTEGRAL REPRESENTATIONS OF THE SCHRODINGER PROPAGATOR

55

Proof: As a corollary of the previous result we choose the global weakly quadratic generating function described in Proposition 4.1, where t

Set, x, y, u)

= A[y XCt, x, y, u)] =

1

1

-mlv(s)!2 - V(y(s)) ds.

o 2

(74)

The action functional A[·] is evaluated on a suitable set of curves y(.) == y X(t, x, y, u)(·) E H1([0, t]; ~n) with parameters (t , x, y, u) E [0, T] x jRn X ~n X jRk, that are constructed as yX(t,

X,

y, u)(s) := y

(V(s) := UX(s)

x- s y + +-

1 s

(V(r) dr - -S t o t

1 t

(V(r) dr,

0

+ rCt, x, y, u)(s).

(75)

By definition we have UX ( . ) E JID N L 2 ( [0, r]; jRn), while for the determination of the 2 function E QN L ( [0, r]; R") see Lemma 4.1. The space of curves r(t,x,y) := {yXCt,x,y,u)Olu E jRk} C HI([O,t];~n) exhibits the structure of finite-dimensional manifold, I' (t, x, y) ~ jRk, as we have seen in Remark 4.5, and consequently it is a well-defined measurable space. We can define a complex measure on it as

ro

Psh(dY) := y*[p:(u)du],

namely the image measure of the map yX (t, X, y, .) (s) : ~k -+ I' (t, x, y) on the complex measure p:(t, x , y , u)du . To conclude, we consider the kernel of integral representation (70) and for the particular construction of Psh(dy) we can apply the theorem of change of variables from ~k to I'(r , x, y), Ue(t,x,y)=

=

=

r r J~k r J~k

exp ( exp (

JrCt.x,y)

(l

+e

i

2

(l

+e

i

2

(l

+e

exp (

)Ii

)Ii

i

S(t,X,y ,U))P:Ct, X,y,U) du

A[yX(t, x, y, U)])P:Ct, x, y, u) du

A[Y]) PeIlCdy).

2

(76)

)Ii

o REFERENCES [I J B. Aebischer, M. Borer, M. Klin, Ch. Leuenberger and H. M. Reimann: Symplectic Geometry. An introduction based on the seminar in Bern (1992). Progress in Mathematics, 124. Birkhauser, Basel 1994. [2] S. Albeverio and S. Mazzucchi : Feynman path integrals for polynomially growing potentials, J. Funct. Anal. 221, I (2005), 83-1 21. [3] H. Amann and E. Zehnder: Periodic solutions of asymptotically linear Hamiltonian systems, Manus. Math. 32 (1980), 149-189. [4] V. I. Arnold: Mathematical Methods of Classical Mechanics, Springer, Berlin 1978.

56

L. ZANELLI, P. GUIOTTO and F. CARDIN

[5] D. Bambusi, M. Degli Esposti and S. Graffi: Uniform convergence of the Lie-Dyson expansion with respect to the Planck constant, Atti Accad. Naz: Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007), no. 2, 153-162. [6] D. Bambusi, S. Graffi and T. Paul: Long time semiclassical approximation 0 f quantum flows: a proof of the Ehrenfest time, Asymptot. Anal. 21 (1999), 2, 149-160. [7] J. Butler: Global h Fourier Integral Operators with complex-valued phase functions, Bull. Lond. Math. Soc. 34 (2002), 479-489. [8] F. Cardin and A. Marigonda: Global World Functions, J. Geom. Symm. Phys. 2 (2004), 1-17. [9] J. J. Duistermaat: Fourier Integral Operators, Birkhauser, 1996. [10] V. Grecchi and A. Sacchetti: Critical metastability and destruction of the splitting in non-autonomous systems, J. Statist. Phys. 103 (2001), no. 1-2, 339-368. [11] A. Hassel and J. Wunsch: The Schrodinger propagator for scattering metrics, Annals of Mathematics 162 (2005), 487-523. [12] L. Hormander: Fourier integral operators, Acta Math. 127 (1971), 1-2, 79-183. [13] L. Kapitansky and Y. Safarov: A parametrix for the nonstationary Schrodinger equation. Differential operators and spectral theory, 139-148, Amer. Math. Soc. Trans!' Ser. 2, 189, Amer. Math. Soc., Providence 1999. [14] V. Kolokoltsov: A new path integral representation for the solution of Schrodinget; heat and stochastic Schrodinger equations, Mathematical Proceedings of Cambridge Phylosophical Society, 2000. [15] V. Kolokoltsov and V. Maslov: Idempotent Analysis and Its Applications, Kluwer, Dordrecht 1997. [16] A. Laptev and I.M. Sigal: Global Fourier integral operators and semiclassical asymptotics, Rev. Math. Phys. 12 (2000), no. 5, 749-766. [17] A. Martinez and K. Yajima: On the Fundamental Solution of Semiclassical Schrodinger Equations at Resonant Times, Commun. Math. Phys. 216 (2001), 357-373. [18] A. Martinez: An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer, New York 2002. [19] V.P. Maslov: The Complex WKB Method for Nonlinear Equations I, Linear Theory, Birkhauser, 1994. [20] V. P. Maslov: Semi-classical Approximation in Quantum Mechanics, Dordrecht 1981. [21] V. P. Maslov: Perturbation Theory and Asymptotic Methods (Russian), Izdat. Moskov. Univ., Moscow 1965. [22] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer. [23] M. Reed and B. Simon: Functional Analysis, vol 1-4. Academic Press, 1980. [24] J. Robbin and D. Salomon: Feynman path integral on phase space and metaplectic representation, Mathematische Zeitschrift 221 (1996), 307-355. [25] M. Ruzhansky and M. Sugimoto: Global L 2-boundedness theorems for a class of Fourier integral operators, Commun. Partial Differential Equations 31 (2006), no. 4-6, 547-569. [26] M. Ruzhansky and M. Sugimoto: Global boundedness theorems for Fourier integral operators associated with canonical transformations. Harmonic analysis and its applications, 65-75, Yokohama Publ., Yokohama 2006. [27] C. Sparber, P. A. Markowich and N. J. Mauser: Wigner functions versus WKB-methods in multivalued geometrical optics, Asymptot. Anal. 33 (2003), 2, 153-187. [28] H. Spohn: Lorge Scale Dynamics of Interacting Particles, TMP, Springer 1991. [29] D. Tataru: Outgoing parametrices and global Strichartz estimates for Schrodinger equations with variable coefficients. Phase space analysis of partial differential equations, 291-313, Progr. Nonlinear Differential Equations Appl., 69, Birkhauser Boston, Boston 2006. [30] D. Theret: A complete proof of Viterbo's uniqueness theorem on generating functions, Topology and its Applications 96 (1996), 249-266. [31] C. Viterbo: Symplectic topology as the geometry of generating functions, Math. Annalen 292 (1992), 685-710. [32] K. Yajima: Dispersive estimates for Schrodinger equations with threshold resonance and eigenvalue, Commun. Math. Phys. 259 (2005), no. 2, 475-509.