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Dynamics on quantum graphs as constrained systems
Vol. 59 (2007)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
DYNAMICS ON QUANTUM G R A P H S AS C O N S T R A I N E D SYSTEMS G. E DELL'ANTONIO Department of Mathematics, University of Rome "La Sapienza", 00185 Rome, Italy, (e-maih [email protected]) (Received September 8, 2006 - Revised January l l, 2007)
We study the possibility of regarding the dynamics on a quantum graph as limit, as a small parameter ~ --~ O, of a dynamics with a strong confining potential. We define a projection operator along the first eigenfunction of a transversal operator and, under suitable assumptions, we prove that the projection of the solution strongly converges along subsequences to a function that satisfies the Schr6dinger equation on each open edge of the graph. Moreover the limit dynamics is unitary. If the limit is independent of the subsequence, one has a limit one-parameter group, generated by one of the self-adjoint extensions of a symmetric operator defined on the open graph (with the vertices deleted). The crucial role of the shape of the confining potential at the vertices is pointed out. Keywords: limit dynamics, constrained systems.
Introduction In applications one often encounters quantum systems which have a quasi onedimensional structure, in the sense that the corresponding wave function is localized for all times near a one-dimensional structure, typically a graph in IK3 or in I1~2. These systems include, e.g. aromatic molecules, in which the conduction electrons are confined by molecular forces to move in a small neighbourhood of a graph defined by the valence electrons, optical fibers and miniature electronic devices such as nanotubes, in which the electrons are compelled by external forces to move in a region that may be regarded as a fattened graph. These systems can be described by the Schr6dinger equation with some confining potential, and therefore by a formalism which contains a small parameter ~, the ratio between the linear size of the support of the wave function in the direction(s) transversal to the graph and its size in the direction of the graph. An alternative description introduces boundary conditions (mostly Dirichet b.c.) at the boundary of a small tubular neighbourhood of the graph; in this case the small parameter is the aspect ratio, namely the relative size of the diameter of this neighbourhood versus the typical length of the edges. The presence of a small parameter suggests to analyze the problem by adiabatic (or multiscale) methods [1], with some care because the limit set is not a smooth manifold but rather a metric graph. [267]
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G. E DELL'ANTONIO
On the other hand, the theory of quantum graphs, i.e. of quantum mechanical theoretical structures associated to one-dimensional metric graphs, has been much developed in the past few years [2-4]. It is based on Krein's theory of self-adjoint extensions of second-order differential operators defined on open metric graphs, and it is now a fairly complete theory, including properties of the scattering matrix (for connected graphs that have one or more semi-infinite edges); in the latter context, a rather accurate analysis has been made of inverse scattering problems. If one could prove that the physical problem of interest is well approximated, for small values of the parameter E, by quantum mechanics on a metric graph, one could take over the mathematics developed for quantum graphs. Moreover, since the limit problem can be treated exactly, one could estimate the error as a function of E and perhaps even set up a recurrence scheme. In the physical literature of the past few years such possibility has been given for granted, with some degree of success. More specifically, it is commonly assumed that the limit dynamics for E ~ 0 exists and is characterized by giving at each vertex Kirchhoff's conditions, namely continuity of the wave function and vanishing of sum of the directional derivatives at each vertex along oriented links. As we shall see if one does not choose properly the structure of the confining potential in a neighbourhood of the vertices of the graph (or the shape of the boundary) it is not evident that there is a simple limit dynamics, i.e. a dynamics which has as generator a self-adjoint operator specified by some boundary conditions at the vertices. It is fair to say that the mathematical analysis of this problem is still at a primitive stage. One should notice that most of the mathematical research on this problem has been devoted in the past to the study of the existence of the limit of the spectrum. In fact, in some applications, mostly in chemistry, one is interested in energy estimates, and therefore in properties of the spectrum. This analysis makes use of compactness (to ensure existence of the limit) and mini-max techniques and is in general limited to the lowest lying eigenvalues. On the other hand, if one is interested e.g. in charge conduction or in scattering problems, one would have to prove that the dynamics has a limit and that this limit coincides with the dynamics defined on the quantum graph by a well-defined self-adjoint operator. Altematively one may set up a scattering theory for the system and prove that in some sense the scattering matrix can be reduced to a transition matrix on the graph. Here we are interested in the problem of existence of a limit dynamics. Some preliminary results about the possibility of defining the dynamics on a quantum graph as limit of the (regular) dynamics defined by the Schrrdinger equation with a constraining potential have been obtained in collaboration with L. Tenuta in [5]. We are aware of a few recent papers in which the problem of limit dynamics is posed. The related parabolic problem of diffusion processes on graphs is analyzed in [6]. For quantum mechanics, if the operator is the Laplacian and if the constraint is given in terms of Neumann boundary conditions at the border of a fattened tube Saito [7] establishes in some cases weak resolvent convergence, which is not enough
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to guarantee a limit dynamics. In the case of Dirichlet b.c. Post [8] proves norm resolvent convergence, and therefore existence of limit dynamics with Dirichlet b.c. at the vertices, in some cases of fattened graphs shrinking at the vertices. In [9] Molchanov and Veinberg study the same problem from the point of view of comparison between scattering matrices. Although the model these authors study is somewhat remote from the physical situation, the paper is very inspiring. A relevant analysis of the limit problem is given in [10] where a simplified problem is considered. We also mention a recent publication [11] where the problem of a limit dynamics is treated. Following [5] we prove that for any given initial condition (subject to natural condition, e.g. to be of finite energy) the projection (in a sense that will be specified) of the flow on the graph defines (not necessarily in a unique way) on the graph a function of time which away from the vertices satisfies the Schrrdinger equation. But each function in this collection depends on two times (the time in which the initial condition has been set and the running time) and therefore it may not define a flow. The existence of a limit dynamics depends in an essential (and so far largely unknown) way on the shape of the constraining potential at the vertices. We also comment briefly on the possibility of introducing wave operators W~ and scattering operators S, for small E ~ 0, and the corresponding reductions ff'E and SE on the graph (it will be essential for that purpose to have at least weak decoupling for channels corresponding to different eigenvalues of an operator which acts on the degrees of freedom transversal to the graph). If under suitable conditions the family S, converges along subsequences when E ~ 0 to an isometry So one could find conditions on the constraining potential at the vertices under which the limit flow can be described by a Schrrdinger's operator on the graph since the S matrix for the SchrSdinger evolution on the graph for a self-adjoint extension of the Laplacian under some boundary condition on the vertices are known [2, 3] (in particular their dependence on the energy is completely specified). 1.
Generalities
Consider a graph F with a finite number of vertices Ui, i = 1. . . . . V, each vertex has a finite number nvi of (open) edges ei attached to it; the graph is metric i.e. each edge is diffeomorphic to a segment (0, a) of the real line (a = + ~ is allowed). This defines the length of an edge and allows the introduction on each edge of differential operators, in particular second-order elliptic. The couple made of a metric graph and a differential (or pseudodifferential) operator defined on it is usually called quantum graph (a detailed review on this topic is given in [2]). By imposing suitable boundary conditions one associates to a quantum graph a self-adjoint operator, and therefore a dynamics. For a general procedure we refer to [3, 4]. This gives unitary evolutions on the graph and scattering matrices if at least one of the edges has infinite length.
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G. E DELL'ANTONIO
We consider the case in which the quantum graph is viewed as immersed in R d, d > 1, and we are interested in the possibility of recovering some of these evolutions as limit, in a suitable sense, of the solutions of the Schr6dinger equation in j~d in presence of a positive potential V which is zero on the graph and in the limit takes the value +c~ elsewhere. Whether this is possible depends crucially on the shape of the level sets of the potential (or the shape of the boundary of the fattened tube) near the vertices; this is not surprising, as the dynamics we expect to obtain in the limit depends on the boundary conditions for the differential operator on the graph. Call F ° the open set obtained from F by deleting the vertices. We shall prove under fairly general assumptions that the function obtained by projection from the solution of Schr6dinger's equation with the constraining potential via convergent subsequences satisfies Schrrdinger's equation on F°; the matching of this solution at the vertices to form a flow (a one-parameter group of unitaries) is much more difficult to achieve. The same result would be obtained in the case of Dirichlet boundary conditions. As in [5] we consider only the case in which n = 2, each edge is a (straight) open segment and the differential operator along each edge is - d 2 / d ~ 2 where ~ parameterizes the edge. We consider a constraining potential of the form V(e, x) - - D~r(x) - - ,
x 6 ~2,
(1.1)
E2
where E is a small parameter (we shall consider the limit E ~ 0) and the function D~ coincides with the distance function Dr from the graph at least in the complement of a neighbourhood of the vertices of linear size ~ where ot is a fixed number, 0 < ot < 1/2. Notice that the distance function is continuous but not differentiable at the vertices. Let O, be defined by (1.2)
OE -- {x c R 2 : D r ( x ) > ~} tO N~,
where N, is a small neighbourhood of the vertices of linear size E~, 0 < ot < 1/2. We introduce in O~ channels, corresponding for each point on q E F ° to the decomposition with respect to the eigenfunctions of a "transversal" operator Hq, q ~ r "°. For each edge ei denote by e~ the open set ei A O~ and denote by ,E~ = e~ ® ~ the cylinder in ]t~2 of base e~. We can choose the size of the open neighbourhoods in such a way that ~E tAi ,Ei is a covering of O~. In each E~ one can introduce orthogonal coordinates; we shall denote by xi the coordinate along ei and with Yi the coordinate along the direction perpendicular t o ei. We denote by Yi the subset of E~ defined by Yi =- {x : x 6 ~ ,
D(x, ei) < D ( x , ek)},
Yk#i.
(1.3)
Notice that the closure of tAiY,, covers ~,' and that by construction Yi N Yk = 13 if
i#k.
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271
In Y/ the hamiltonian takes the explicit form 02 ax 2 + I-Ii',
H --
d2 y{ . 2. +. E2. H iE =-- dy
1
I
(1.4)
(H/E the transverse operator ). A term - 7l I has been added so that the spectrum of H is bounded below by 0. We shall denote by ~n,i(Yi) the restriction to ~i~Eof the n-th eigenvector of H/~. We take initial data of the form OPO = ~ ( ~ i ( X i ) i
® ~,i(Yi),
~i ~ Cm(ei),
(1.5)
where the functions q~i have support in e~. We shall use the notation dPt -~- eitHop O. Using rearrangement inequalities to control the L 2 norm in terms of the energy, it is easy to prove that there is a positive constant Cl such that the L 2 norm of restriction of dot to OE is bounded by Cl E~ (notice that this may not be true for the H 1 norm). We c a l l (gen,i(xi, t) the projection of OPt on the edge ei along the n-th channel and shall denote by ~bn~(t) the collection of the ~b~,i. With these notation, we have the following result. THEOREM 1.1. Given any real T, for each t E [0, T] and for each value of the indices i the family (parameterized by E) q)~(xi, t) strongly converges along subsquences to an element ~i(t) C L2(ei) with ~ i ]tPi(t)[2 = E i Iq~i(0)l2. Moreover qb(t) E Hi(l-'0) uniformly in t and therefore ~b(x; t) -- {~bi(t)} is continuous up to the boundary in each edge. For every t there is a unitary map Utl,t 2 in L2(F)U i L2(ei) which is such that 4~(t2) = Ut 1,t2(p (tl).
(1.6)
In addition, the estimates we give are sufficient to prove that the limit gPi of the projection on the open edge ei satisfies in a weak sense the equation i~7(gi(Xi, t) --
02 O x i ~ i ( x i ' t).
REMARK 1. For more general initial data we have only a weaker result [5]. Indeed, if the initial data are of the form *o = ~
dPn,i(xi) ® ~n,i(Yi), i
~ [~bn,i[H1 < OO, i,n
given any real T, for each t e [0, T] and for each value of the indices i and n the family (parameterized by E) ~b/'/,l E .(xi, t) weakly converges along subsquences to an element qbn,i(t) E L2(ei). Moreover ~ ( t ) = {~bn~i} ~ Hi(F0) uniformly in 0 < t < T and therefore ~b(x; t) is continuous up to the boundary in each edge. Also in this
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G. E DELL'ANTONIO
case the estimates we give are sufficient to prove thet for each n and ei the limit ~bi,n of the projection on the open edge ei satisfies in a weak sense the equation 0 02 i - ~ n , i (Xi, t) = -- Ox2 ~n,i (Xi, t ). We sketch briefly the proof of Theorem 1.1. One can easily prove that on overlapping patches the transformation between different sets of coordinates along different edges is a diffeomorphism. On each edge define the "projection operator" I-I~,n from C ~ ( R 2) to L2(ei) by
(Fli,n~)(x) -- dpn,i =
dyi~n.i(yi)dp(xi, yi)dyi.
(1.7)
ei Explicit computations give the estimate, for each value of the index k, f dxllOxl-I.,iep(x)l[LZ(ei) k 2
2 k 2 <_ ][qbillL2(ei) IlOx~)l[L2(~2)
and therefore 1-I~n,i extends to a unique operator (of norm _< 1) from Hk(o~) to Hk(1R). From this it follows that sup ~bn ~ i(Xi)(t)llHlei [0,T]
<
(1.8)
We now make use of the following two properties: in Of the square of the distance from the graph is a polynomial of second order in the variables xi, Yi and ~,~ is a polynomial in yi/.C/-E times exp(-y/2/2e). Taking into account the structure of H~ and using the explicit expression of Fl,~,i, by lengthy estimates [5] one proves that there is a constant C such that, for sufficiently small value of e, one has on each edge ej 1
~
1
1
C
- - d2 rq~t~ _< -'E ~llOxj~bt I12 + ~llOyj4~ll 2 + +2E
(1.9)
Let 6 be a positive number, and let f8 be the characteristic function of the set of points which are at a distance greater that D ' from I'. From (1.9) one has, for a positive constant C1, 1 sup Ilfa4'~llL2(~2) -< C ~ = . (1.10) [0,T] On the other hand, if ~Pk 6 C2 has support in ek one has, exploiting (1.10), =
1 z
+
1
(clrz -- y2)¢k k,n + E"¢kE
k,n,
(1.11)
where En is the difference between the energy of the n-th excited state of the harmonic oscillator and its ground state (which corresponds to n = 1). From this,
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273
it is rather easy to deduce [5] exploiting (1.10) that for a dense set of functions ~t limc(q~, lp ~ n ) = 0,
E 0
n > 2,
(1.12)
i.e. if the initial data is in the first "channel" in all edges, the scalar product of the solution with functions supported on any edge ei and orthogonal to Xln, .E n >__2, vanishes with e. Since the number of links is assumed to be finite, this proves that in a weak sense there is asymptotic decoupling between the solution with initial data of the form (1.5) and each of the channels that correspond to n > 2. Notice that by itself this result does not imply strong convergence of t~t since the weak convergence is not uniform in j and m > 2 and in the set of functions in the scalar product in (1.12). In order to prove strong convergence of the projected flow in the (open) links ei we now exploit the fact that ~i,1 correspond to ground states. Consider the energy form QE associated to the operator HE. It is left invariant by the flow of HE, and is is bounded uniformly in • on the vectors we chose as initial states. We have seen that, for each t and with an error in L 2 n o r m of the order at most ~ we can write the eitHEqbO =--(a t in the form
(at = ZqSn,i(t, Xi) ®~,i(Yi), i
~
II~i,nll 2 < CX),
i,n
where q)n,i(t, Xi) are supported in the (open) sets ei. One has O(c~t) = O(q)o) ~ C. But 0 E En Q(~b~) = xi~k,t( t, xi)[I 2 + --I[qb~n(Xi)ll 2] + O(~/~). (1.13) • k.n
~[11
Since all terms on the right-hand side are positive, it follows for each value of i,
Z
ll(9~,n(t'xi)ll2 < C3~/-~.
n>l
On the other hand, the evolution is unitary. It follows Ill~bt - t~°ll2 ~ Y ~ II~,l (t' xi)H2 - Z i i
Ilqbi,l(O' xi)ll2l < C4~r-E"
For each k the sequence q~" k,t is equibounded in H 1 and therefore for each value of t one can extract from ~bk,t E a subsequence that converges in LZ(eg) to a function ~bk(t) which for each value of t belongs to H 1, and is therefore continuous up to the boundary of each link e~. If one defines q~(t) ~ {~bk(t)},
II~b(t)ll2 = ~
IIq~kll2, k
one has II~(t)ll2 = 11~(O)ll2.
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G. E DELL'ANTONIO
In a separable Hilbert space, weak convergence and convergence of the norm implies strong convergence; we have therefore proven strong convergence along subsequences in L2(F) of ~b~(t) to ~b(t) and that Ut : ~b(0) --+ ~b(t) is for each t a unitary map in L2(F). Notice that since we have only proven convergence along subsequences there is no reason why this map should possess the group property U~ .Us = Ut+s. To prove that the limit ~b(t) satisfies Schrtdinger's equation in each (open) link, remark that from our previous estimates it follows that q~" 1,k E C 1 ([0, T]), L2(R) is equicontinuous from [0, T[ to LZ(~). Taking the weak limit as • --> 0 of both sides of the equation for • > 0 one has on each edge e~ iOtq~l,k
:- --102~l k 2
in D'([0, T]) x ek.
'
(1.14)
It is not difficult to see that Oxqb~,k converges strongly to 0xtPl,k in the open edge ek. This concludes the brief sketch of the proof of Theorem 1.1. [] REMARK 2. Eq. (1.6) is not sufficient to guarantee the existence of a limit (Hamiltonian) flow. Indeed, without strong hypothesis on the shape of the confining potential in a neighhourhood of the vertices, the group property need not hold, Ui
tl,t2
= Ui
tl,r
Ui
r,t2"
(1.15)
Our methods, which rely on embedding and compactness, are not suited to prove uniqueness. It may be possible to prove it if one makes strong assumptions on the shape of the level sets of the potential V, (or on the shape of the boundaries of a fattened tube at the 1 scale). Uniqueness implies that the group property holds; by Theorem 1 the one-parameter group U ( t ) is weakly continuous and therefore has a generator, which by (1.14) must be a self-adjoint extension of the free Laplacian defined on I'0. In general, it is difficult to control the properties (and even the existence) of the limit of the derivative of the solution along different directions at a vertex. We remark that some of the statements one finds in the literature should be taken with care, due to the singular nature of the solution in ~2 in the limit when • --+ 0. Indeed, the projection on the graph of the solution tends to a function that is at least continuous up to the vertices, but the t r a n s v e r s a l part of the solution becomes very singular. This makes very difficult a correct use e.g. of Gauss' law at the vertices.
2.
Particular shapes and some remarks
Consider first the graph composed of a single edge e of length 1, and consider the problem of deriving, in the limit E ~ O, the dynamics induced on e by
DYNAMICS ON QUANTUM GRAPHS AS CONSTRAINED SYSTEMS
275
Schr6dinger's equation in S---[0, 1] x IR (with coordinates x, y) with Hamiltonian HE =-- Ha + KEx '
d2
KEx =
dy 2
+ y2er2(x )
-
-1I ,
(2.1)
Ixl < 8(E),
(2.2)
E
where erE(x) -- 1,
1>
Ixl > ~(~),
Ixl
erE(x) -- 2 - 8(~---)'
and Hd = - 0 x 2 with Neumann boundary conditions at 0 and 1. The function 8(E) will be chosen later, with limE_+0S(E)= 0. For each value of x the operator /~x has discrete spectrum and the lowest eigenvalue is erE(x). We denote by ~x",E(y) the n-th eigenfunction of the operator KE(x), depending parametrically on x. If the harmonic potential did not depend on x, the problem would be solvable by separation of variables, and the free flow with Neumann boundary conditions along the x-line would be recovered trivially. Since the potential depends on x only in a neighbourhood of the origin which vanishes when E--+ 0, we expect that in our case the limit motion be on (0, 1] (with Neumann b.c. at 1) and that at the origin different boundary conditions may appear. We study only the projection onto the first eigenfunction of KE(x) and choose an initial datum of the form lP0(x, y) = 4~(X)~xI'E (y),
(2.2)
where the support of 4,(x) is contained in Ix[ > 8(e). Let QE be the quadratic form associated to the operator HE,
if
QE0P) = (Vx~P, Vx~P) + -
IqSxl2(erE(lxl)- 1)dx.
(2.3)
E
[xl_<6E
We choose 4~(x) with support in IxL > 8E and such that Iq~lff = C. Then Q0P0) = C and, since the energy form is conserved one will have Q ( ~ t ) = C, Vt. Let ~tE(t, x) =--eitHElpO be the solution of Schr6dinger's equation with initial datum ~P0. For each value of x we decompose ¢tE(t,x) in the eigenfunctions of HE,x, ~t(x, y, t) = Z
~n(X' t ) ) ~ ( y ) .
(2.4)
tl
We want to prove the lemma. LEMMA 1. For each value of n > 2 one has, in the strong sense, lim ~b~(x, t) = 0
E--+0
(2.5)
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G. E D E L L ' A N T O N I O
Moreover the function ~l(X, t) - limE~0~b~(x, t) defined on (0, 1) has L 2 norm equal to one, is continuous at the origin, is twice differentiable away from the origin, and it solves d 1 d2 i--;-u(X,at t) -- 2 dx 2u(x' t), u(0, t) = 0, 3u(1, t) = 0. (2.6) Proof: From (2.3) one has C = (Vx~t, -Vx~t) + (Vy~Pt, VyTh) + f f dxdy(Ttt, ~[y2cre(x)2 - 1 ] ~ t ) .
(2.7)
According to (2.4) we expand the second and third terms in (2.7). We remark that all terms are positive and that the sum is convergent since the flow is unitary. One has 1
C = IVx~tl 2 + y ~ 3(dPt", (na,(x)- 1)q~tX). n
(2.8)
K_
With our choice of ~r~(x) one has
n~r,(x) > 1,
Vx,
Vn >_2.
From (2.8) Z n k b t ' ~ 1 2 < caE.
(2.9)
n>l
We conclude that, in the limit E --+ 0, if one starts with a wave function that is in the lowest transversal channel, then there is strong decoupling of the other channels. Moreover, (2.8) provides an explicit estimate of the coupling when E is small. Remark also that (2.8) implies II~b°(.,t)ll2 = 1, Vt. We prove now that q~0(., t) c H 1. From (2.9) we have C > rlVx~Pt]l2. Making use of the explicit form of the eigenvectors of the harmonic oscillator one has from (2.9) C _> IV~b'(x, t)lz+Z(Vx logcr~(x))+ ~[Vx 3 logcrE(x)[qbE(x)[Zdx > l lVq~E(x, t)122.
(2.10)
This proves that ~b~(., t) is in H 1 uniformly in e and in t and therefore its limit (which we have proved to exist) belongs to H 1. The fact that it is a weak solution of the free Schr6dinger equation outside any neighbourhood of the origin follows from weak* convergence; further estimates show that the function is actually C 2 outside the origin. We now analyze the behaviour of the solution in a neighbourhood of the origin. By (2.8)
1(9~(x)12(crE(x)- 1)dx < --.
(2.11)
E
Let 8 be chosen so that l i r n ~ 0 ( e / 6 ( e ) ) - - + 0 . From (2.8) we conclude
f lVx(O~(x,t))ldx <_C.
(2.12)
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277
Consider now 4~1(x2) - 4~l(xl) for 0 < xl < x2 < BE. In view of (2.12) sup
I~b(x2)-4~(xl)1 < C1/21xl -x211/2 < C1/2 2 v / ~ E ) .
~(e)>_lx21>lxll>_0
Suppose now that lim. SUpx~0 lim. super0 ~b~ = K ~= 0 Then for some C1 > 0 one has lim. sup ~p~(x) > C1, 0 ~ x ~ BE, for some 3, limE~0S(e)= 0. This contradicts (2.11). We conclude that the projection of the solution converges pointwise to a function which is zero at the origin, and it has a subsequence that converges to a function in H 1. Since we have already proven that away from the origin the sequence converges to a solution of the free Schr6dinger equation, we can conclude that the sequence converges to a solution of the Schr6dinger equation in (0, 1) with Dirichlet b.c. at the origin and Neumann b.c. at 1. This ends the proof of Lemma 1. [] From Lemma 1 we can derive another lemma. LEMMA 2. Let F be any graph with a (locally)finite number of vertices, and consider the problem of finding a dynamics on I" as limit of the dynamics on ~ 2 given by the Hamiltonian HE, HE = --A + Ve,
V(~, x) -- D~(x) ~2 '
X E ]t~2,
where E is a small parameter (we shall consider the limit ~ -+ 0), D~ coincides with the distance function from the graph Dr at least in the complement of a neighbourhood of the vertices of linear size E~ where ot is a fixed number 0 < o r < 1/2. Assume moreover on one of the edges, e, ending in a vertex v, the constraining potential has the form given in Lemma 1, i.e. 1
2 2y2cr2(x)~re(x)
~
1,
1 > Ixl > 8(e),
Ixl ~rE(x) ~- 2 - 8(e~'
Ixl < 8(e),
(2.2)
where we have denoted by x the coordinate along the edge e, with origin at the vertex v, and by y the coordinate in the transversal direction. The function cr is as in Lemma 1 (i.e. the strength of the potential is increasing towards 0). The limit solution on the edge e is zero (i.e. it satisfies Dirichlet b.c.) at the vertex v. The proof is obtained from Lemma 2 by an easy localization argument making use of the estimates one has obtained on the fast decay of the solution away from the graph I'. [] We now briefly comment on the possibility of making use of scattering theory to compare the limit process with the Hamiltonian flow generated by some self-adjoint extension at the vertices.
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G. E DELL'ANTONIO
We shall consider only a graph with a single vertex, which we take to be the origin of a system of coordinates in the plane, and n > 2 semi-infinite edges ei. Suppose that uniqueness of the limit flow holds. It is reasonable to consider the scattering limit, which of course will depend on the shape of the potential inside the ball of radius one, and to expect that for E very small the corresponding scattering operator be close to the scattering matrix for one of the self-adjoint extensions (in 13] these scattering matrices have been analyzed in detail). The equation satisfied by the limit flow on F may be derived by the inverse scattering method. To define the wave operator (and therefore the scattering operator) in our setting one must resort to multichannel scattering theory and make use of a suitable reference operator H~. Consider a graph with a single vertex and N semi-infinite edges attached to it. A reasonable choice of asymptotic space is the direct sum 7"/as = @~=1(L2(~+ x R)). One can take as a reference Hamiltonian 1
H~ = .
-~ -
N
A x -
2
Oy --~ E 2
12 I,
where AN is the one-dimensional Laplacian on (e, +cx~) with Neumann boundary conditions at the origin. In order to construct a suitable map Y' between 7"/as and L2(~ 2) (or rather with the subspace in L2(]~3) of absolute continuity for H) one has to take into account the fact that the support of the region O, becomes vanishingly small when --+ 0 and that the support of the ground state of the transversal Hamiltonian is exponentially small with E. This map Y' can be chosen in such a way that lim,__,0 IIY'q~ll - IIq~ll = 0 for a dense set of functions 4~. In view of the strong asymptotic decoupling of the higher transversal channels of the transversal hamiltonian from the one corresponding to the ground state, it is natural to consider the "wave operator" W =- lim
lim eitHI"IE~,Y'e -itHO,
E---~0t---~
where FIE is the projection operator on the first channel and ~, is the characteristic function of the F,, (notice that it converges to 1 as E --+ 0). If the limit flow is unique, this sequence of operators should converge, for E --+ 0 to the one of wave operators described in [3]. Acknowledgements
Part of this work has been done at Gakushuin University, Tokio; I am grateful to prof. K. Yajima for the very warm hospitality. REFERENCES [1] G. E Dell'Antonio and L. Tenuta: Dynamics on Quantum Graphs as constrained systems, J. Phys. A37 (2004), 5605-5624. [2] P. Kuchement: Quantum Graphs I: some basic structures, Waves in Random Media 12 (2002), 1-24. [3] V. Kostrigin and R. Schrader: Kirchhoff's rule for quantum wires, J Phys. A32 (1999), 595-630.
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[4] S. Novikov: Schr6dinger operators on graphs and symplectic structures, Fields Inst. Commun. (Providence) 24 (1999), 397413. [5] G. E Dell'Antonio and L. Tenuta: Quantum graphs as limits of constrained systems, J. Math. Phys. 47 (2006), 072102-072129. [6] H. Wentzell and M. Freidlin: Diffusion processes on graphs and the averaging principle, Ann. Probab. 21 (1993), 2215-2245. [7] Y. Saito: Convergence of the Neumann Laplacian on shrinking domains, Electronic J. Diff. Equations 2000 (2000), 1-25. [8] O. Post: Branched quantum wires with Dirichlet b.c., J. Phys. A38 (2005), 4917-4932. [9] S. Molchanov and B. Vainberg: Transition from a network of thin fibers to a metric graph, to appear in Contemporary Mathematics 2006. [10] U. Smilansky and M. Solomyak: The quantum graph as limit of a network of physical wires, to appear in Contemporary Mathematics 2006 [11] V. Belov, S. Dobrokhotov and T. Tudorowskii: Asymptotic solution of nonrelativistic equations of quantum mechanics in curved nanotubes, Theor. Math. Physics 141 (2004), 1562-1592.
REPORTS ON MATHEMATICAL PHYSICS
No. 3
DYNAMICS ON QUANTUM G R A P H S AS C O N S T R A I N E D SYSTEMS G. E DELL'ANTONIO Department of Mathematics, University of Rome "La Sapienza", 00185 Rome, Italy, (e-maih [email protected]) (Received September 8, 2006 - Revised January l l, 2007)
We study the possibility of regarding the dynamics on a quantum graph as limit, as a small parameter ~ --~ O, of a dynamics with a strong confining potential. We define a projection operator along the first eigenfunction of a transversal operator and, under suitable assumptions, we prove that the projection of the solution strongly converges along subsequences to a function that satisfies the Schr6dinger equation on each open edge of the graph. Moreover the limit dynamics is unitary. If the limit is independent of the subsequence, one has a limit one-parameter group, generated by one of the self-adjoint extensions of a symmetric operator defined on the open graph (with the vertices deleted). The crucial role of the shape of the confining potential at the vertices is pointed out. Keywords: limit dynamics, constrained systems.
Introduction In applications one often encounters quantum systems which have a quasi onedimensional structure, in the sense that the corresponding wave function is localized for all times near a one-dimensional structure, typically a graph in IK3 or in I1~2. These systems include, e.g. aromatic molecules, in which the conduction electrons are confined by molecular forces to move in a small neighbourhood of a graph defined by the valence electrons, optical fibers and miniature electronic devices such as nanotubes, in which the electrons are compelled by external forces to move in a region that may be regarded as a fattened graph. These systems can be described by the Schr6dinger equation with some confining potential, and therefore by a formalism which contains a small parameter ~, the ratio between the linear size of the support of the wave function in the direction(s) transversal to the graph and its size in the direction of the graph. An alternative description introduces boundary conditions (mostly Dirichet b.c.) at the boundary of a small tubular neighbourhood of the graph; in this case the small parameter is the aspect ratio, namely the relative size of the diameter of this neighbourhood versus the typical length of the edges. The presence of a small parameter suggests to analyze the problem by adiabatic (or multiscale) methods [1], with some care because the limit set is not a smooth manifold but rather a metric graph. [267]
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G. E DELL'ANTONIO
On the other hand, the theory of quantum graphs, i.e. of quantum mechanical theoretical structures associated to one-dimensional metric graphs, has been much developed in the past few years [2-4]. It is based on Krein's theory of self-adjoint extensions of second-order differential operators defined on open metric graphs, and it is now a fairly complete theory, including properties of the scattering matrix (for connected graphs that have one or more semi-infinite edges); in the latter context, a rather accurate analysis has been made of inverse scattering problems. If one could prove that the physical problem of interest is well approximated, for small values of the parameter E, by quantum mechanics on a metric graph, one could take over the mathematics developed for quantum graphs. Moreover, since the limit problem can be treated exactly, one could estimate the error as a function of E and perhaps even set up a recurrence scheme. In the physical literature of the past few years such possibility has been given for granted, with some degree of success. More specifically, it is commonly assumed that the limit dynamics for E ~ 0 exists and is characterized by giving at each vertex Kirchhoff's conditions, namely continuity of the wave function and vanishing of sum of the directional derivatives at each vertex along oriented links. As we shall see if one does not choose properly the structure of the confining potential in a neighbourhood of the vertices of the graph (or the shape of the boundary) it is not evident that there is a simple limit dynamics, i.e. a dynamics which has as generator a self-adjoint operator specified by some boundary conditions at the vertices. It is fair to say that the mathematical analysis of this problem is still at a primitive stage. One should notice that most of the mathematical research on this problem has been devoted in the past to the study of the existence of the limit of the spectrum. In fact, in some applications, mostly in chemistry, one is interested in energy estimates, and therefore in properties of the spectrum. This analysis makes use of compactness (to ensure existence of the limit) and mini-max techniques and is in general limited to the lowest lying eigenvalues. On the other hand, if one is interested e.g. in charge conduction or in scattering problems, one would have to prove that the dynamics has a limit and that this limit coincides with the dynamics defined on the quantum graph by a well-defined self-adjoint operator. Altematively one may set up a scattering theory for the system and prove that in some sense the scattering matrix can be reduced to a transition matrix on the graph. Here we are interested in the problem of existence of a limit dynamics. Some preliminary results about the possibility of defining the dynamics on a quantum graph as limit of the (regular) dynamics defined by the Schrrdinger equation with a constraining potential have been obtained in collaboration with L. Tenuta in [5]. We are aware of a few recent papers in which the problem of limit dynamics is posed. The related parabolic problem of diffusion processes on graphs is analyzed in [6]. For quantum mechanics, if the operator is the Laplacian and if the constraint is given in terms of Neumann boundary conditions at the border of a fattened tube Saito [7] establishes in some cases weak resolvent convergence, which is not enough
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269
to guarantee a limit dynamics. In the case of Dirichlet b.c. Post [8] proves norm resolvent convergence, and therefore existence of limit dynamics with Dirichlet b.c. at the vertices, in some cases of fattened graphs shrinking at the vertices. In [9] Molchanov and Veinberg study the same problem from the point of view of comparison between scattering matrices. Although the model these authors study is somewhat remote from the physical situation, the paper is very inspiring. A relevant analysis of the limit problem is given in [10] where a simplified problem is considered. We also mention a recent publication [11] where the problem of a limit dynamics is treated. Following [5] we prove that for any given initial condition (subject to natural condition, e.g. to be of finite energy) the projection (in a sense that will be specified) of the flow on the graph defines (not necessarily in a unique way) on the graph a function of time which away from the vertices satisfies the Schrrdinger equation. But each function in this collection depends on two times (the time in which the initial condition has been set and the running time) and therefore it may not define a flow. The existence of a limit dynamics depends in an essential (and so far largely unknown) way on the shape of the constraining potential at the vertices. We also comment briefly on the possibility of introducing wave operators W~ and scattering operators S, for small E ~ 0, and the corresponding reductions ff'E and SE on the graph (it will be essential for that purpose to have at least weak decoupling for channels corresponding to different eigenvalues of an operator which acts on the degrees of freedom transversal to the graph). If under suitable conditions the family S, converges along subsequences when E ~ 0 to an isometry So one could find conditions on the constraining potential at the vertices under which the limit flow can be described by a Schrrdinger's operator on the graph since the S matrix for the SchrSdinger evolution on the graph for a self-adjoint extension of the Laplacian under some boundary condition on the vertices are known [2, 3] (in particular their dependence on the energy is completely specified). 1.
Generalities
Consider a graph F with a finite number of vertices Ui, i = 1. . . . . V, each vertex has a finite number nvi of (open) edges ei attached to it; the graph is metric i.e. each edge is diffeomorphic to a segment (0, a) of the real line (a = + ~ is allowed). This defines the length of an edge and allows the introduction on each edge of differential operators, in particular second-order elliptic. The couple made of a metric graph and a differential (or pseudodifferential) operator defined on it is usually called quantum graph (a detailed review on this topic is given in [2]). By imposing suitable boundary conditions one associates to a quantum graph a self-adjoint operator, and therefore a dynamics. For a general procedure we refer to [3, 4]. This gives unitary evolutions on the graph and scattering matrices if at least one of the edges has infinite length.
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G. E DELL'ANTONIO
We consider the case in which the quantum graph is viewed as immersed in R d, d > 1, and we are interested in the possibility of recovering some of these evolutions as limit, in a suitable sense, of the solutions of the Schr6dinger equation in j~d in presence of a positive potential V which is zero on the graph and in the limit takes the value +c~ elsewhere. Whether this is possible depends crucially on the shape of the level sets of the potential (or the shape of the boundary of the fattened tube) near the vertices; this is not surprising, as the dynamics we expect to obtain in the limit depends on the boundary conditions for the differential operator on the graph. Call F ° the open set obtained from F by deleting the vertices. We shall prove under fairly general assumptions that the function obtained by projection from the solution of Schr6dinger's equation with the constraining potential via convergent subsequences satisfies Schrrdinger's equation on F°; the matching of this solution at the vertices to form a flow (a one-parameter group of unitaries) is much more difficult to achieve. The same result would be obtained in the case of Dirichlet boundary conditions. As in [5] we consider only the case in which n = 2, each edge is a (straight) open segment and the differential operator along each edge is - d 2 / d ~ 2 where ~ parameterizes the edge. We consider a constraining potential of the form V(e, x) - - D~r(x) - - ,
x 6 ~2,
(1.1)
E2
where E is a small parameter (we shall consider the limit E ~ 0) and the function D~ coincides with the distance function Dr from the graph at least in the complement of a neighbourhood of the vertices of linear size ~ where ot is a fixed number, 0 < ot < 1/2. Notice that the distance function is continuous but not differentiable at the vertices. Let O, be defined by (1.2)
OE -- {x c R 2 : D r ( x ) > ~} tO N~,
where N, is a small neighbourhood of the vertices of linear size E~, 0 < ot < 1/2. We introduce in O~ channels, corresponding for each point on q E F ° to the decomposition with respect to the eigenfunctions of a "transversal" operator Hq, q ~ r "°. For each edge ei denote by e~ the open set ei A O~ and denote by ,E~ = e~ ® ~ the cylinder in ]t~2 of base e~. We can choose the size of the open neighbourhoods in such a way that ~E tAi ,Ei is a covering of O~. In each E~ one can introduce orthogonal coordinates; we shall denote by xi the coordinate along ei and with Yi the coordinate along the direction perpendicular t o ei. We denote by Yi the subset of E~ defined by Yi =- {x : x 6 ~ ,
D(x, ei) < D ( x , ek)},
Yk#i.
(1.3)
Notice that the closure of tAiY,, covers ~,' and that by construction Yi N Yk = 13 if
i#k.
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In Y/ the hamiltonian takes the explicit form 02 ax 2 + I-Ii',
H --
d2 y{ . 2. +. E2. H iE =-- dy
1
I
(1.4)
(H/E the transverse operator ). A term - 7l I has been added so that the spectrum of H is bounded below by 0. We shall denote by ~n,i(Yi) the restriction to ~i~Eof the n-th eigenvector of H/~. We take initial data of the form OPO = ~ ( ~ i ( X i ) i
® ~,i(Yi),
~i ~ Cm(ei),
(1.5)
where the functions q~i have support in e~. We shall use the notation dPt -~- eitHop O. Using rearrangement inequalities to control the L 2 norm in terms of the energy, it is easy to prove that there is a positive constant Cl such that the L 2 norm of restriction of dot to OE is bounded by Cl E~ (notice that this may not be true for the H 1 norm). We c a l l (gen,i(xi, t) the projection of OPt on the edge ei along the n-th channel and shall denote by ~bn~(t) the collection of the ~b~,i. With these notation, we have the following result. THEOREM 1.1. Given any real T, for each t E [0, T] and for each value of the indices i the family (parameterized by E) q)~(xi, t) strongly converges along subsquences to an element ~i(t) C L2(ei) with ~ i ]tPi(t)[2 = E i Iq~i(0)l2. Moreover qb(t) E Hi(l-'0) uniformly in t and therefore ~b(x; t) -- {~bi(t)} is continuous up to the boundary in each edge. For every t there is a unitary map Utl,t 2 in L2(F)U i L2(ei) which is such that 4~(t2) = Ut 1,t2(p (tl).
(1.6)
In addition, the estimates we give are sufficient to prove that the limit gPi of the projection on the open edge ei satisfies in a weak sense the equation i~7(gi(Xi, t) --
02 O x i ~ i ( x i ' t).
REMARK 1. For more general initial data we have only a weaker result [5]. Indeed, if the initial data are of the form *o = ~
dPn,i(xi) ® ~n,i(Yi), i
~ [~bn,i[H1 < OO, i,n
given any real T, for each t e [0, T] and for each value of the indices i and n the family (parameterized by E) ~b/'/,l E .(xi, t) weakly converges along subsquences to an element qbn,i(t) E L2(ei). Moreover ~ ( t ) = {~bn~i} ~ Hi(F0) uniformly in 0 < t < T and therefore ~b(x; t) is continuous up to the boundary in each edge. Also in this
272
G. E DELL'ANTONIO
case the estimates we give are sufficient to prove thet for each n and ei the limit ~bi,n of the projection on the open edge ei satisfies in a weak sense the equation 0 02 i - ~ n , i (Xi, t) = -- Ox2 ~n,i (Xi, t ). We sketch briefly the proof of Theorem 1.1. One can easily prove that on overlapping patches the transformation between different sets of coordinates along different edges is a diffeomorphism. On each edge define the "projection operator" I-I~,n from C ~ ( R 2) to L2(ei) by
(Fli,n~)(x) -- dpn,i =
dyi~n.i(yi)dp(xi, yi)dyi.
(1.7)
ei Explicit computations give the estimate, for each value of the index k, f dxllOxl-I.,iep(x)l[LZ(ei) k 2
2 k 2 <_ ][qbillL2(ei) IlOx~)l[L2(~2)
and therefore 1-I~n,i extends to a unique operator (of norm _< 1) from Hk(o~) to Hk(1R). From this it follows that sup ~bn ~ i(Xi)(t)llHlei [0,T]
<
(1.8)
We now make use of the following two properties: in Of the square of the distance from the graph is a polynomial of second order in the variables xi, Yi and ~,~ is a polynomial in yi/.C/-E times exp(-y/2/2e). Taking into account the structure of H~ and using the explicit expression of Fl,~,i, by lengthy estimates [5] one proves that there is a constant C such that, for sufficiently small value of e, one has on each edge ej 1
~
1
1
C
- - d2 rq~t~ _< -'E ~llOxj~bt I12 + ~llOyj4~ll 2 + +2E
(1.9)
Let 6 be a positive number, and let f8 be the characteristic function of the set of points which are at a distance greater that D ' from I'. From (1.9) one has, for a positive constant C1, 1 sup Ilfa4'~llL2(~2) -< C ~ = . (1.10) [0,T] On the other hand, if ~Pk 6 C2 has support in ek one has, exploiting (1.10), =
1 z
+
1
(clrz -- y2)¢k k,n + E"¢kE
k,n,
(1.11)
where En is the difference between the energy of the n-th excited state of the harmonic oscillator and its ground state (which corresponds to n = 1). From this,
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273
it is rather easy to deduce [5] exploiting (1.10) that for a dense set of functions ~t limc(q~, lp ~ n ) = 0,
E 0
n > 2,
(1.12)
i.e. if the initial data is in the first "channel" in all edges, the scalar product of the solution with functions supported on any edge ei and orthogonal to Xln, .E n >__2, vanishes with e. Since the number of links is assumed to be finite, this proves that in a weak sense there is asymptotic decoupling between the solution with initial data of the form (1.5) and each of the channels that correspond to n > 2. Notice that by itself this result does not imply strong convergence of t~t since the weak convergence is not uniform in j and m > 2 and in the set of functions in the scalar product in (1.12). In order to prove strong convergence of the projected flow in the (open) links ei we now exploit the fact that ~i,1 correspond to ground states. Consider the energy form QE associated to the operator HE. It is left invariant by the flow of HE, and is is bounded uniformly in • on the vectors we chose as initial states. We have seen that, for each t and with an error in L 2 n o r m of the order at most ~ we can write the eitHEqbO =--(a t in the form
(at = ZqSn,i(t, Xi) ®~,i(Yi), i
~
II~i,nll 2 < CX),
i,n
where q)n,i(t, Xi) are supported in the (open) sets ei. One has O(c~t) = O(q)o) ~ C. But 0 E En Q(~b~) = xi~k,t( t, xi)[I 2 + --I[qb~n(Xi)ll 2] + O(~/~). (1.13) • k.n
~[11
Since all terms on the right-hand side are positive, it follows for each value of i,
Z
ll(9~,n(t'xi)ll2 < C3~/-~.
n>l
On the other hand, the evolution is unitary. It follows Ill~bt - t~°ll2 ~ Y ~ II~,l (t' xi)H2 - Z i i
Ilqbi,l(O' xi)ll2l < C4~r-E"
For each k the sequence q~" k,t is equibounded in H 1 and therefore for each value of t one can extract from ~bk,t E a subsequence that converges in LZ(eg) to a function ~bk(t) which for each value of t belongs to H 1, and is therefore continuous up to the boundary of each link e~. If one defines q~(t) ~ {~bk(t)},
II~b(t)ll2 = ~
IIq~kll2, k
one has II~(t)ll2 = 11~(O)ll2.
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G. E DELL'ANTONIO
In a separable Hilbert space, weak convergence and convergence of the norm implies strong convergence; we have therefore proven strong convergence along subsequences in L2(F) of ~b~(t) to ~b(t) and that Ut : ~b(0) --+ ~b(t) is for each t a unitary map in L2(F). Notice that since we have only proven convergence along subsequences there is no reason why this map should possess the group property U~ .Us = Ut+s. To prove that the limit ~b(t) satisfies Schrtdinger's equation in each (open) link, remark that from our previous estimates it follows that q~" 1,k E C 1 ([0, T]), L2(R) is equicontinuous from [0, T[ to LZ(~). Taking the weak limit as • --> 0 of both sides of the equation for • > 0 one has on each edge e~ iOtq~l,k
:- --102~l k 2
in D'([0, T]) x ek.
'
(1.14)
It is not difficult to see that Oxqb~,k converges strongly to 0xtPl,k in the open edge ek. This concludes the brief sketch of the proof of Theorem 1.1. [] REMARK 2. Eq. (1.6) is not sufficient to guarantee the existence of a limit (Hamiltonian) flow. Indeed, without strong hypothesis on the shape of the confining potential in a neighhourhood of the vertices, the group property need not hold, Ui
tl,t2
= Ui
tl,r
Ui
r,t2"
(1.15)
Our methods, which rely on embedding and compactness, are not suited to prove uniqueness. It may be possible to prove it if one makes strong assumptions on the shape of the level sets of the potential V, (or on the shape of the boundaries of a fattened tube at the 1 scale). Uniqueness implies that the group property holds; by Theorem 1 the one-parameter group U ( t ) is weakly continuous and therefore has a generator, which by (1.14) must be a self-adjoint extension of the free Laplacian defined on I'0. In general, it is difficult to control the properties (and even the existence) of the limit of the derivative of the solution along different directions at a vertex. We remark that some of the statements one finds in the literature should be taken with care, due to the singular nature of the solution in ~2 in the limit when • --+ 0. Indeed, the projection on the graph of the solution tends to a function that is at least continuous up to the vertices, but the t r a n s v e r s a l part of the solution becomes very singular. This makes very difficult a correct use e.g. of Gauss' law at the vertices.
2.
Particular shapes and some remarks
Consider first the graph composed of a single edge e of length 1, and consider the problem of deriving, in the limit E ~ O, the dynamics induced on e by
DYNAMICS ON QUANTUM GRAPHS AS CONSTRAINED SYSTEMS
275
Schr6dinger's equation in S---[0, 1] x IR (with coordinates x, y) with Hamiltonian HE =-- Ha + KEx '
d2
KEx =
dy 2
+ y2er2(x )
-
-1I ,
(2.1)
Ixl < 8(E),
(2.2)
E
where erE(x) -- 1,
1>
Ixl > ~(~),
Ixl
erE(x) -- 2 - 8(~---)'
and Hd = - 0 x 2 with Neumann boundary conditions at 0 and 1. The function 8(E) will be chosen later, with limE_+0S(E)= 0. For each value of x the operator /~x has discrete spectrum and the lowest eigenvalue is erE(x). We denote by ~x",E(y) the n-th eigenfunction of the operator KE(x), depending parametrically on x. If the harmonic potential did not depend on x, the problem would be solvable by separation of variables, and the free flow with Neumann boundary conditions along the x-line would be recovered trivially. Since the potential depends on x only in a neighbourhood of the origin which vanishes when E--+ 0, we expect that in our case the limit motion be on (0, 1] (with Neumann b.c. at 1) and that at the origin different boundary conditions may appear. We study only the projection onto the first eigenfunction of KE(x) and choose an initial datum of the form lP0(x, y) = 4~(X)~xI'E (y),
(2.2)
where the support of 4,(x) is contained in Ix[ > 8(e). Let QE be the quadratic form associated to the operator HE,
if
QE0P) = (Vx~P, Vx~P) + -
IqSxl2(erE(lxl)- 1)dx.
(2.3)
E
[xl_<6E
We choose 4~(x) with support in IxL > 8E and such that Iq~lff = C. Then Q0P0) = C and, since the energy form is conserved one will have Q ( ~ t ) = C, Vt. Let ~tE(t, x) =--eitHElpO be the solution of Schr6dinger's equation with initial datum ~P0. For each value of x we decompose ¢tE(t,x) in the eigenfunctions of HE,x, ~t(x, y, t) = Z
~n(X' t ) ) ~ ( y ) .
(2.4)
tl
We want to prove the lemma. LEMMA 1. For each value of n > 2 one has, in the strong sense, lim ~b~(x, t) = 0
E--+0
(2.5)
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G. E D E L L ' A N T O N I O
Moreover the function ~l(X, t) - limE~0~b~(x, t) defined on (0, 1) has L 2 norm equal to one, is continuous at the origin, is twice differentiable away from the origin, and it solves d 1 d2 i--;-u(X,at t) -- 2 dx 2u(x' t), u(0, t) = 0, 3u(1, t) = 0. (2.6) Proof: From (2.3) one has C = (Vx~t, -Vx~t) + (Vy~Pt, VyTh) + f f dxdy(Ttt, ~[y2cre(x)2 - 1 ] ~ t ) .
(2.7)
According to (2.4) we expand the second and third terms in (2.7). We remark that all terms are positive and that the sum is convergent since the flow is unitary. One has 1
C = IVx~tl 2 + y ~ 3(dPt", (na,(x)- 1)q~tX). n
(2.8)
K_
With our choice of ~r~(x) one has
n~r,(x) > 1,
Vx,
Vn >_2.
From (2.8) Z n k b t ' ~ 1 2 < caE.
(2.9)
n>l
We conclude that, in the limit E --+ 0, if one starts with a wave function that is in the lowest transversal channel, then there is strong decoupling of the other channels. Moreover, (2.8) provides an explicit estimate of the coupling when E is small. Remark also that (2.8) implies II~b°(.,t)ll2 = 1, Vt. We prove now that q~0(., t) c H 1. From (2.9) we have C > rlVx~Pt]l2. Making use of the explicit form of the eigenvectors of the harmonic oscillator one has from (2.9) C _> IV~b'(x, t)lz+Z(Vx logcr~(x))+ ~[Vx 3 logcrE(x)[qbE(x)[Zdx > l lVq~E(x, t)122.
(2.10)
This proves that ~b~(., t) is in H 1 uniformly in e and in t and therefore its limit (which we have proved to exist) belongs to H 1. The fact that it is a weak solution of the free Schr6dinger equation outside any neighbourhood of the origin follows from weak* convergence; further estimates show that the function is actually C 2 outside the origin. We now analyze the behaviour of the solution in a neighbourhood of the origin. By (2.8)
1(9~(x)12(crE(x)- 1)dx < --.
(2.11)
E
Let 8 be chosen so that l i r n ~ 0 ( e / 6 ( e ) ) - - + 0 . From (2.8) we conclude
f lVx(O~(x,t))ldx <_C.
(2.12)
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277
Consider now 4~1(x2) - 4~l(xl) for 0 < xl < x2 < BE. In view of (2.12) sup
I~b(x2)-4~(xl)1 < C1/21xl -x211/2 < C1/2 2 v / ~ E ) .
~(e)>_lx21>lxll>_0
Suppose now that lim. SUpx~0 lim. super0 ~b~ = K ~= 0 Then for some C1 > 0 one has lim. sup ~p~(x) > C1, 0 ~ x ~ BE, for some 3, limE~0S(e)= 0. This contradicts (2.11). We conclude that the projection of the solution converges pointwise to a function which is zero at the origin, and it has a subsequence that converges to a function in H 1. Since we have already proven that away from the origin the sequence converges to a solution of the free Schr6dinger equation, we can conclude that the sequence converges to a solution of the Schr6dinger equation in (0, 1) with Dirichlet b.c. at the origin and Neumann b.c. at 1. This ends the proof of Lemma 1. [] From Lemma 1 we can derive another lemma. LEMMA 2. Let F be any graph with a (locally)finite number of vertices, and consider the problem of finding a dynamics on I" as limit of the dynamics on ~ 2 given by the Hamiltonian HE, HE = --A + Ve,
V(~, x) -- D~(x) ~2 '
X E ]t~2,
where E is a small parameter (we shall consider the limit ~ -+ 0), D~ coincides with the distance function from the graph Dr at least in the complement of a neighbourhood of the vertices of linear size E~ where ot is a fixed number 0 < o r < 1/2. Assume moreover on one of the edges, e, ending in a vertex v, the constraining potential has the form given in Lemma 1, i.e. 1
2 2y2cr2(x)~re(x)
~
1,
1 > Ixl > 8(e),
Ixl ~rE(x) ~- 2 - 8(e~'
Ixl < 8(e),
(2.2)
where we have denoted by x the coordinate along the edge e, with origin at the vertex v, and by y the coordinate in the transversal direction. The function cr is as in Lemma 1 (i.e. the strength of the potential is increasing towards 0). The limit solution on the edge e is zero (i.e. it satisfies Dirichlet b.c.) at the vertex v. The proof is obtained from Lemma 2 by an easy localization argument making use of the estimates one has obtained on the fast decay of the solution away from the graph I'. [] We now briefly comment on the possibility of making use of scattering theory to compare the limit process with the Hamiltonian flow generated by some self-adjoint extension at the vertices.
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G. E DELL'ANTONIO
We shall consider only a graph with a single vertex, which we take to be the origin of a system of coordinates in the plane, and n > 2 semi-infinite edges ei. Suppose that uniqueness of the limit flow holds. It is reasonable to consider the scattering limit, which of course will depend on the shape of the potential inside the ball of radius one, and to expect that for E very small the corresponding scattering operator be close to the scattering matrix for one of the self-adjoint extensions (in 13] these scattering matrices have been analyzed in detail). The equation satisfied by the limit flow on F may be derived by the inverse scattering method. To define the wave operator (and therefore the scattering operator) in our setting one must resort to multichannel scattering theory and make use of a suitable reference operator H~. Consider a graph with a single vertex and N semi-infinite edges attached to it. A reasonable choice of asymptotic space is the direct sum 7"/as = @~=1(L2(~+ x R)). One can take as a reference Hamiltonian 1
H~ = .
-~ -
N
A x -
2
Oy --~ E 2
12 I,
where AN is the one-dimensional Laplacian on (e, +cx~) with Neumann boundary conditions at the origin. In order to construct a suitable map Y' between 7"/as and L2(~ 2) (or rather with the subspace in L2(]~3) of absolute continuity for H) one has to take into account the fact that the support of the region O, becomes vanishingly small when --+ 0 and that the support of the ground state of the transversal Hamiltonian is exponentially small with E. This map Y' can be chosen in such a way that lim,__,0 IIY'q~ll - IIq~ll = 0 for a dense set of functions 4~. In view of the strong asymptotic decoupling of the higher transversal channels of the transversal hamiltonian from the one corresponding to the ground state, it is natural to consider the "wave operator" W =- lim
lim eitHI"IE~,Y'e -itHO,
E---~0t---~
where FIE is the projection operator on the first channel and ~, is the characteristic function of the F,, (notice that it converges to 1 as E --+ 0). If the limit flow is unique, this sequence of operators should converge, for E --+ 0 to the one of wave operators described in [3]. Acknowledgements
Part of this work has been done at Gakushuin University, Tokio; I am grateful to prof. K. Yajima for the very warm hospitality. REFERENCES [1] G. E Dell'Antonio and L. Tenuta: Dynamics on Quantum Graphs as constrained systems, J. Phys. A37 (2004), 5605-5624. [2] P. Kuchement: Quantum Graphs I: some basic structures, Waves in Random Media 12 (2002), 1-24. [3] V. Kostrigin and R. Schrader: Kirchhoff's rule for quantum wires, J Phys. A32 (1999), 595-630.
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[4] S. Novikov: Schr6dinger operators on graphs and symplectic structures, Fields Inst. Commun. (Providence) 24 (1999), 397413. [5] G. E Dell'Antonio and L. Tenuta: Quantum graphs as limits of constrained systems, J. Math. Phys. 47 (2006), 072102-072129. [6] H. Wentzell and M. Freidlin: Diffusion processes on graphs and the averaging principle, Ann. Probab. 21 (1993), 2215-2245. [7] Y. Saito: Convergence of the Neumann Laplacian on shrinking domains, Electronic J. Diff. Equations 2000 (2000), 1-25. [8] O. Post: Branched quantum wires with Dirichlet b.c., J. Phys. A38 (2005), 4917-4932. [9] S. Molchanov and B. Vainberg: Transition from a network of thin fibers to a metric graph, to appear in Contemporary Mathematics 2006. [10] U. Smilansky and M. Solomyak: The quantum graph as limit of a network of physical wires, to appear in Contemporary Mathematics 2006 [11] V. Belov, S. Dobrokhotov and T. Tudorowskii: Asymptotic solution of nonrelativistic equations of quantum mechanics in curved nanotubes, Theor. Math. Physics 141 (2004), 1562-1592.