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Multi-dimensional versions of a determinant formula due to Jost and Pais
Vol. 59 (2007)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
MULTI.DIMENSIONAL VERSIONS OF A DETERMINANT FORMULA DUE TO JOST AND PAIS F. G E S Z T E S Y Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (e-mail: fritz @math.missouri.edu, http://www.math.missouri.edu/personnel/faculty/gesztesyf.html) M.
MITREA
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (e-mail: marius @math.missouri.edu, http://www.math.missouri.edu/personnel/faculty/mitream.html)
and M.
ZINCHENKO
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA (e-mail: [email protected]) (Received October 5, 2006)
We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schr6dinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schr6dinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set f2 C R n, n = 2, 3, where [2 has a compact, nonempty boundary 0f2 satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on 0f2 and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants perturbation associated with operators in L 2(~; dnx) to modified Fredholm determinants associated with operators in L2(0fl; dn-lcr), n = 2, 3. MSC2000: Primary: 47B10, 47G10, Secondary: 34B27, 34L40. Keywords: Fredholm determinants, non-self-adjoint operators, multi-dimensional Schr6dinger operators, Dirichlet-to-Neumann maps.
*Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS0405526 and DMS-0400639, FRG-0456306. [365]
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F. GESZTESY,M. MITREA and M. ZINCHENKO
1.
Introduction To illustrate the reason behind the title of this note, we briefly recall a celebrated result due to Jost and Pais [23] in 1951, who proved a spectacular reduction of a Fredholm determinant associated with the Birmann-Schwinger kernel of a onedimensional Schr6dinger operator on a half-line, to a simple Wronski determinant of appropriate solutions of the underlying Schr6dinger equation. This Wronsld determinant also equals the so-called Jost function of the corresponding half-line Schr6dinger operator. As the title of our paper suggests, we are seeking a certain multi-dimensional variant of this result. To describe the result due to Jost and Pais [23], we need a few preparations. Denoting by HOD,+and HOW,+the one-dimensional Dirichlet and Neumann Laplacians in L2((0, c~); dx), and assuming V C LI((0, ~ ) ; dx),
(1.1)
we introduce the perturbed Schr6dinger operators H ff and H N in L2((0, c~); dx) by = -f"
+ V f,
f ~ dom(HoD+) = {g E L2((0, oo); dx) lg, g' ~ AC([0, R]) for all R > 0,
(1.2)
g(0) = 0, ( - g " d- Vg) E L2((0, ~ ) ; dx)},
H ~ f = - f " + V f, f E dom(H~+) = {g a L2((0, c~); dx) lg, g' ~ AC([0, R]) for all R > 0,
(1.3)
g'(0) = O, (-g" + Vg) ~ L2((0, oo); dx)}. Thus, Hff and H+N are self-adjoint if and only if V is real-valued, but since the latter restriction plays no special role in our results, we will not assume real-valuedness of V throughout this paper. A fundamental system of solutions ~b+ D(z, .), 0+° (z, .), and the Jost solution f+ (z, .) of
- ~ ' ( z , x) + V~(z, x) : zap(z, x),
z ~ C\{0},
x ~ 0,
(1.4)
are then introduced via the standard Volterra integral equations cp+O(z,x) = z -1/2 sin(zl/2x) + / dx' z -1/2 sin(zl/2(x - x'))V(x')cp~(z, x'),
(1.5)
0 x
oD+(z, X)
cos(zl/2x) + [ dx' z -1/2 sin(zl/2(x - x'))V(x')O~(z, x'),
(1.6)
0 oO
f+(z, X)
e izl/2x -- f dx' Z-1/2 sin(zl/2(x - x'))V(x')f+(z, x'),
(1.7)
x
Z E C\{0},
~(Z 1/2) ~ 0,
X ~ 0.
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ON A DETERMINANT FORMULA BY JOST AND PAIS
In addition, we introduce
u = e x p ( i a r g ( V ) ) l V I 1/2,
v = IVI 1/2,
so that V = u v ,
(1.8)
and denote by I+ the identity operator in L2((0, oc); dx). Moreover, we denote by
W ( f , g)(x) = f ( x ) g ' ( x ) - f ' ( x ) g ( x ) ,
x > O,
(1.9)
the Wronskian of f and g, where f , g E cl([0, ec)). We also use the standard convention to abbreviate (with a slight abuse of notation) the operator of multiplication in L2((0, ec); dx) by an element f E L~oc((O, oc); dx) (and similarly in the higherdimensional context later) by the same symbol f (rather than MI, etc.). In the remainder of this note •p(7-{), p >_ 1, denote the usual trace ideals associated to the complex, separable Hilbert space 7-/, and det(.) and det2(.) denote the Fredholm and modified Fredholm determinant in connection with trace class (p = 1) and Hilbert-Schmidt operators (p = 2), respectively. Then, the following result holds. THEOREM 1.1 ([23], see also [211 and the references cited therein). Assume V 6 LI((0, ec); dx) and let z ~ C\[0, oc) with ,,~(Z1/2) > 0. Then,
u ( n ~ + - z I + ) - l v , u ( H g + - z I + ) - l p E/31 (L2((O, ~ ) ; d x ) )
(1.10)
and det (I+ + u(H~,+ - z I + ) - l v ) = 1 + z -1/2
f ,x sin(zl/2x)V(x)f+(z,
x)
0
= W(f+(z, .), dp~(z, .)) = f+(z, 0),
(1.11)
f
oo
det (I+ + u(HoU+ -- z I + ) - l v ) = 1 + iz -1/2
0
W(f+(z, .), O°+(z, .))
f ; ( z , O)
izl/2
izl/2
(1.12)
Eq. (1.11) is the modern formulation of the classical result due to Jost and Pais [23]. We emphasize that (1.11) and (1.12) exhibit the remarkable fact that the Fredholm determinant associated with trace class operators in the infinitedimensional space L2((0, ~ ) ; dx) is reduced to a simple Wronski determinant of C-valued distributional solutions of (1.4). This fact goes back to Jost and Pais [23] (see also [21, 28, 29, 30, Sect. 12.1.2, 36, 37, Proposition 5.7], and the extensive literature cited in these references). The principal aim of this note is to explore the extent to which this fact may generalize to higher dimensions n = 2, 3. While a direct generalization of (1.11), (1.12) appears to be difficult, we will next derive a formula for the ratio of such determinants which indeed permits a natural extension to higher dimensions.
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For this purpose we introduce the boundary trace operators YD (Dirichlet trace) and Yw (Neumann trace) which, in the current one-dimensional half-line situation, are just the functionals, YD: {C([0, oo)) C ,~---> g g(O),
YN: { CI([0' OO)) C '~h'''~ -h'(0).
(1.13)
In addition, we denote by m~,+, m0, m~,+, and m~ the Weyl-Titchmarsh m-functions corresponding to HOD+, H+°, /-/ON,+, and H~, respectively, that is, m~,+(z) = iz 1/2, m~(z) - f~(z, O) f+(z, 0)'
m~,+(z) -m:~(z) --
1
m~,+(z----~= iz-1/2' ~ mD+(z) --
f+(z, 0) f~_(z,O)"
(1.14) (1.15)
In the case where V is real-valued, we recall that m 0 is a Herglotz function (i.e. it maps the open complex upper half-plane C+ analytically into itself) and the measure dp~ in its Herglotz representation is then the spectral measure of the operator H+D and hence encodes all spectral information of H~. Similarly, m+° also encodes all spectral information of H~ since - 1 / n O = mN+ is also a Herglotz function and the measure dpN+ in its Herglotz representation represents the spectral measure of the operator H~v. Then we obtain the following result for the ratio of the perturbation determinants in (1.11) and (1.12). THEOREM 1.2 ([19]). Assume V ~ LI((O, o0); dx) and let z ~ C\cr(H~9) with ~(z U2) > 0. Then, det (I+ + u ( H ~ + - z I + ) - l v ) det (I+ + u(Ho°,+ - zI+)-lv ) = 1 - (ys(H+D - zI+)-lV[yo(H~,+ -- 71+)-1]*)1
(1.16)
W(f+(z),qbN(z)) f+(z,O) n°(z) m~,+(z) = izl/2W(f+(z), qb~(z)) -- izU2f+(z, O) - m~,+(z~) - mN(z)
(1.17)
The proper multi-dimensional generalizations to Schr6dinger operators in L2(f2; dnx), corresponding to an open set f2 C R n with compact, nonempty boundary 0f2, more precisely, the proper operator-valued generalization of the Weyl-Titchmarsh function mO(z ) is then given by the Dirichlet-to-Neumann map, denoted by MR(z ). This operator-valued map indeed plays a fundamental role in our extension of (1.17) to the higher-dimensional case. In Section 3, which contains our principal result, we present a direct multi-dimensional extension of Theorem 1.2.
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ON A DETERMINANT FORMULA BY JOST AND PAIS
2.
Schr~dinger operators with Dirichlet and Neumann boundary conditions and some results on Dirichlet-to-Neumann maps
In this section we first recall some properties of Dirichlet, H~a, and Neumann, H~,a, Laplacians in Lz(g2;dnx) associated with open sets ~2 C N n, n = 2, 3, introduced in Hypothesis 2.1 below. Subsequently, we introduce the Dirichlet and Neumann Schr6dinger operators HaD and H N in LZ(f2; dnx), that is, perturbations of the Dirichlet and Neumann Laplacians H 0,~2 D and H 0,[2 N by a potential V satisfying Hypothesis 2.4. Finally, we cite some results on Dirichlet-to-Neumann maps to be used in Section 3. HYPOTHESIS 2.1. Let n = 2, 3 and assume that ~2 C ~n is an open set with a compact, nonempty boundary Of2. In addition, we assume that one of the following three conditions holds: (i) f2 is of class C l'r for some 1/2 < r < 1; (ii) £2 is convex; (iii) ~ is a Lipschitz domain satisfying a uniform exterior ball condition. We note that while 0f2 is assumed to be compact, ~2 may be unbounded in connection with conditions (i) or (iii). For more details in this context we refer to [19, Appendix A] and [20, Appendix A]. First, we introduce the boundary trace operator yo (Dirichlet trace) by go: C ( ~ ) ~ C(OQ),
you = ul0a.
(2.1)
Then there exists a bounded, linear operator YD,
YD: HS(~) --+ HS-(1/2)(O~) ~
L2(0~;
dn-lcr),
1/2 < s < 3/2,
(2.2)
whose action is compatible with that of yD °, that is, the two Dirichlet trace operators coincide on the intersection of their domains. Here d n - l a denotes the surface measure on 0S2 and we refer to [19, Appendix A] for our notation in connection with Sobolev spaces. In addition, we introduce the operator VN (Neumann trace) by YN :
V • ~/DV:
HS+l(f2) --+ L2(0f2; dn-l~r),
1/2 < s < 3/2,
(2.3)
where v denotes outward pointing normal unit vector to 0f2. It follows from (2.2) that YN is also a bounded operator. Given Hypothesis 2.1, we introduce the self-adjoint and nonnegative Dirichlet and Neumann Laplacians H0°,a and H0U,a associated with the domain f2 as follows: H DO,s2= - A ,
dom(HoDa) = {u • Hz(S2) I×DU = 0},
HNO,a= - A ,
dom(H~,a)
= {u •
H2(a) IyNU =
0}.
(2.4) (2.5)
A detailed discussion of H 0A2 D and H 0,f~ N is provided in [19, Appendix A]. LEMMA 2.2 ([19]). Assume Hypothesis 2.1. Then the operators H O,f~ ° and H 0,~2 N introduced in (2.4) and (2.5) are nonnegative and self-adjoint in L2(f2; dnx) and
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E GESZTESY,M. MITREAand M. ZINCHENKO
the following mapping properties hold for all q E [0, 1] and z E C\[0, o0), (HoD,~ - zlfa) -q, (H~,e - zls2) -q E ~(LZ(f2; dnx), Hzq(f2)).
(2.6)
Here Ia and Ioa denote the identity operators in LZ(ff2;dnx) and L2(0f2; dn-lcr), respectively. The fractional powers in (2.6) (and in subsequent analogous cases) are defined via the functional calculus implied by the spectral theorem for self-adjoint operators. As explained in [19, Appendix A], the key ingredients in proving Lemma 2.2 are the inclusions dom(HoD~2) C H2(f2),
dom(HNa) C H2(~)
(2.7)
and methods based on real interpolation spaces. For the remainder of this paper we agree to the simplified notation that the operator of multiplication by the measurable function f in L2(f2; dnx) is again denoted by the symbol f . LEMMA 2.3 ([19], [20]). Assume Hypothesis 2.1 and let 2 <_ p, (n/2p) < q <_ 1, f E LP(~; dnx), and z E C\[O, oo). Then, f(Hgza - zI~) -q, f ( H g f a - zI~2) -q E /3p(L2(~; dnx)).
(2.8)
Next, we turn to our assumptions on the potential V and the corresponding definition of Dirichlet and Neumann Schr6dinger operators Hg and H N in L2(f2; d~x). HYPOTHESIS 2.4. Suppose that f2 satisfies Hypothesis 2.1 and assume that V E LP(f2; dnx) for some p satisfying 4/3 < p < 2, in the case n = 2, and 3/2
Z ~ C\[O, co),
(2.9)
u(Ogf2 -- ZI~) -1/2, (Ogf2 - zla)-'/2u E ~2p(t2(a; dnx)),
Z E C\[0, oo). (2.10)
Moreover, (2.9) and (2.10) imply
u(Ogf2 - zlf2)-ll), u(Hgf2 - zlfa)-lv E ~p(L2(a; dnx)) C 1~2(L2(~-~; dnx)), z ~ C\[0, co),
(2.11)
and one can introduce densely defined, closed operators H~ and H~ (which are extensions of H0Oa+ V on dom(H0°a)Ndom(V) and HoN~+ V on dom(H~,a)Ndom(V), respectively) by the method of quadratic forms (cf. [19, Section 2] for precise details).
ON A DETERMINANT FORMULA BY JOST AND PAIS
371
Next, we turn to Dirichlet and Neumann boundary value problems associated with the Helmholtz differential expression - - A - z as well as the corresponding differential expression - A + V - z in the presence of a potential V, both in connection with the open set f2. In addition, we provide a detailed discussion of Dirichlet-to-Neumann M~,~, M D and Neumann-to-Dirichlet M~,~, M N, maps in L2(Og2;dn-l~). Denote by ~U : {U • Hl(f2) [ Au • (Hl(f2)) *} --+ H-'/2(Of2)
(2.12)
a weak Neumann trace operator defined by (~N u ,
4)) = f d"x Vu(x) . V ~ ( x ) + (Au, ~)
(2.13)
tt
fa
for all 4) • H1/2(O~) and • • Hl(f2) such that ~,Dqb = 4). We note that this definition is independent of the particular extension qb of ~b, and that ~U is an extension of the Neumann trace operator YN defined in (2.3). THEOREM 2.5 ([20]). Suppose f2 is an open Lipschitz domain with a compact nonempty boundary 0~2. Then for every f e Hl(Of2) and z • C\cr(HoD,~) the following Dirichlet boundary value problem, ( - A -- Z)U~ = 0 on f2, ~'ou~ = f on 0~2,
u~ • H3/2(f2),
(2.14)
has a unique solution ug satisfying ~NUg • L2(0~; dn-la). Similarly, for every g • L2(Oa; dn-l~r) and z • C\a(HoNa) the following Neumann boundary value problem,
( - A - z)u N = 0 on f2, = g on
u N • H3/2(a),
aa,
(2.15)
has a unique solution u N. Moreover, the solutions ug and u N are given by the formulae ug (z) = --(gu(Hff, a -- 2I~)-1) *f,
(2.16)
RoN(z) = (YD ( H'N o , ~ - z- I ~),-Ix*) g,
(2.17)
and hence (2.14)-(2.17) imply the following mapping properties, [?/u((Hffa -- zla)-l)*]* : H'(Of2)
--+
H3/2(~"~),
g • C\~(HoD,a),
[yo((H~,~ - zln)-l)*]* : LZ(Ofa; d n-la) --+ H3/2(f2),
z • C\~(H~,~).
(2.18) (2.19)
We temporarily strengthen our hypothesis on V and introduce the following assumption.
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F. GESZTESY,M. MITREA and M. ZINCHENKO
HYPOTHESIS 2.6. Suppose the set f2 satisfies Hypothesis 2.1 and assume that V E L2(f2; dnx) 7) LP(f2; dnx) for some p > 2. THEOREM 2.7 ([20]). Assume Hypothesis 2.6. Then for every f E Hi(Of2) and z E C\cr(H~) the following Dirichlet boundary value problem, ( - - A + V - - z ) u o = O on f2,
u ° E H3/2(f2),
(2.20)
you ° = f on Of2,
has a unique solution u ° satisfying ~Nu ° ~ L2(Of2; d n - l t 7 ) . Similarly, for every g ~ L2(0f2; dn-l~r) and z E C\cr(H N) the following Neumann boundary value problem, (--A .qt_ V -- Z)U N = 0 o n ~'2, ~NUN = g on 0f2,
u N E H3/2(~),
(2.21)
has a unique solution u N. Moreover, the solutions u ° and u N are given by the formulae u°(z) = - [ y N ( ( H ° -- zlo)-l)*]* f,
(2.22)
u N (Z) = [VD((H u - zI~)-l)*]*g.
(2.23)
Assuming Hypothesis 2.6, we now introduce the Dirichlet-to-Neumann maps, M~,o(z) and M~(z), associated with ( - A - - z ) and (--A + V - z ) on f2, as follows,
M~.~(z):, {Hl(Of2)f --+ _~NUg '
Z ~ C\a[H D ., o,~1"
(2.24)
where u~ is the unique solution of ( - A - z)u D = 0 on f2,
u~ E H3/Z(f2),
YDUD = f
on 0f2,
(2.25)
and M°(z): { Hl(0~)f ~--+LZ(0~;_~Nuo,d'-l~r)' where u ° is the unique solution of ( - - m + W - Z)U D -~- 0 on f2,
U D E H3/2(~),
Z 6 C\~r(H°),
~/oUD = f
on 0~.
(2.26)
(2.27)
In addition, still assuming Hypothesis 2.6, we introduce the Neumann-to-Dirichlet maps, MoN~(z) and MN(z), associated with ( - A - z ) and ( - A + V - z) on f2, as follows, MN (Z): [ L2(0~; '
[
HI(0~), g w-~ you~,
dn-l~y) ~
z ~ C\cr(H0U,~),
(2.28)
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ON A DETERMINANT FORMULA BY JOST AND PAIS
where u N is the unique solution of ( - A - z)u N = 0 on ~2,
u~v
• H3/2(~),
~NUu = g on 0f2,
(2.29)
Z • C\a(HN),
(2.30)
~NUy = g
(2.31)
and
yDuN,HI(Of2)'
MN(z): { L2(0~; dn-la)g e-->---> where u N is the unique solution of (-A+V-z)u
N=O on g2,
ug•H3/2(g2),
on 0f2.
It follows from Theorems 2.5 and 2.7, that under the assumption of Hypothesis 2.6, the operators M~,,n(z), MD(z), M~,,f2(z), and MN(z) are well-defined and satisfy the following equalities,
M~,~(z) = -M~,r~(z) -1, MN(z)=--MD(z)-I,
z • C\(a(HoD,r~) U a(H0Ur~)), Z • C \ ( a ( H D ) Ua(HN)),
(2.32) (2.33)
and
Mga(z) = Mg(z) =
VN[YN((Hga -- Z/a)-')*]*, ;N[~((Hg
m~,,f2(z ) = yD[~D((Oga
Mg(z) =
vo[×o((Hg
- z/a)-')*]*, --
zla)-l)*] *,
- zI.)-~)*] *,
z • C\a(H0°a), z • C\a(HD), Z • C\a(H~Va), z • C\a(HN).
(2.34) (2.35) (2.36) (2.37)
Next, we use (2.35) and (2.37) to extend the above definition of the operators
MD(z) and Mg(z) to the more general situation governed by Hypothesis 2.4: LEMMA 2.8 ([20]). Assume Hypothesis 2.4. Then the operators MD(z) and MN (z) defined by equalities (2.35) and (2.37) have the following mapping properties, Mg(Z)" Hl(O~'-2) ~ L2(0~"2; dn-l~), Z • C\cr(HD), (2.38) MN(z)
"
L2(0~; dn-lcr)
--+
HI(0~),
Z • C\a(HN).
(2.39)
Weyl-Titchmarsh operators, in a spirit close to ours, have recently been discussed by Amrein and Pearson [2] in connection with the interior and exterior of a ball in ]R3 and potentials V • L°°(]R3; d3x). For additional literature on Weyl-Titchmarsh operators, relevant in the context of boundary value spaces (boundary triples, etc.), we refer, for instance, to [1, 3-7, 14, 15, 18, Chapter 3, 22, 26, 33, 34]. For applications of the Dirichlet-to-Neumann map to Borg-Levinson-type inverse spectral problems we refer to [12, 27, 31, 35, 38, 39] (see also [25] for an alternative approach based on the boundary control method). Finally, we mention the following auxiliary result, which plays a crucial role in the proof of Theorem 3.2.
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F. GESZTESY, M. MITREA and M, ZINCHENKO
LEMMA 2.9 ([20]). Assume Hypothesis 2.4. Then the following identities hold, Mg~2(Z ) - MD(z) = ~N(H o -- zl~2)-lV[yN((Hgs2 - zlf2)-l)*] *,
z 6 C\(cr(H0°~)U or(H°)), Mg(z)M~,.(z)-l
= I~. - ~N(Hg -- zl~)-'V[×o((HoNa - zlf2)-')*]*, z 6C\(a(Hg,)Ua(H°)Ucr(H~,a)).
3.
(2.40) (2.41)
A m u l t i - d i m e n s i o n a l v a r i a n t o f a f o r m u l a d u e to Jost a n d Pais In this section w e present our multi-dimensional variants o f the Jost and Pais as discussed in the introduction. We start with an extension o f a result in
formula [19].
THEOREM 3.1 ([20]). Assume Hypothesis 2.4 and let z ~ C\(,~(ug)u,,(ng~)U,~(Hg~)).
Then,
×u(H~,~ - zl~)-' V ( H g - zI~)-l v[yo(H~,~
-~1f2)-l]
* E
~1 (L2(0~'-2;d"-lo')), (3.1)
yN(Ha° -- zle)-IV[yD(HoN, e -- 21~1-1] * e/32(LZ(0f2; dn-lo')),
(3.2) and
det 2 (lf2 + u ( H o N , ~ - - z l ~ ) - 1 v ) detz(l~+u(H~,~-zl~)-lv) =
det2(l~fz - y N ( H ~ -- zl~)-IV[yo(H~),.N - -1 z . )~-11. 1
)etr(T2(z)),
(3.3)
where , r 2 ( z ) = ~,N (Ho?~
J~)-lv(Hg
N - z - I ~),,-11. - z l ~ ) - ' V [ y , ( H o,~ E ~1 ( L 2 ( 0 ~ ; d n - l ° ' ) ) •
(3.4)
Combining Lemma 2.9 and Theorem 3.1 implies our principal new result (cf. [20]), the multi-dimensional analog of Theorem 1.2. THEOREM 3.2. Assume Hypothesis 2.4 and let z~C\(~(Hg)
N U ~ ( H g ~ ) U ~ ( H o,@.
ON A DETERMINANT FORMULA BY JOST AND PAIS
375
Then, Mg(s)M~(Z)
-1 -- I ~
:
--yg(n~
I ZI~)IlW[yo(n~
-- Z ] ~ ) I 1] *
t 2(c2(of ;
(3.5)
and det 2 (I~
+u(H~-zI~)-lv)
det 2(I~
+u(H~,~ - z I ~ ) - l v )
= det2(Ia~ -
yN(Hff -- ZI~)-IV[yD(HoNf2 -- ~i~)-1]* )etr(T2(z))
[A.cD~'_~ ~nD i"_', l~_tr(T2(z)) = det2uvlf2~,~)lvl0,f2~) le
with
T2(z)
given by
(3.6) (3.7)
(3.4).
Rather than entering into details of the proof of Theorems 3.1 and 3.2, we offer an elementary comment on determinants which, however, lies at the heart of the matter of our multi-dimensional variant of the one-dimensional Jost and Pais result. Suppose A 6 ~('~t-/1,']-/2) , B E ~('~2,']--/1) with AB ~ ~1(']--/2) and BA c ~1(~--/1). Then, det(IT-t2 - AB) ----d e t ( l ~ - BA). (3.8) Eq. (3.8) follows from the fact that all nonzero eigenvalues of AB and BA coincide including their algebraic multiplicities. The latter fact, in turn, can be derived from the formula
A(BA-zI~)-IB=I~2+z(AB-zI~2)
-1,
z~C\(cr(aB) Ucr(Ba))
(3.9)
(and its companion with A and B interchanged), as discussed in detail by Deift [13]. In particular, ~1 and 7-/2 may have different dimensions. Especially, one of them may be infinite and the other finite, in which case one of the two determinants in (3.8) reduces to a finite determinant. This case indeed occurs in the original one-dimensional case studied by Jost and Pais [23] as described in detail in [21] and the references therein. In the proof of Theorem 3.1 the role of ~1 and ~2 is played by L2(f2; d"x) and L2(0f2; dn-l~r), respectively. In the context of KdV flows and reflectionless (i.e. generalizations of soliton-type) potentials represented as Fredholm determinants, a reduction of such determinants (in some cases to finite determinants) has also been studied by Kotani [24], relying on certain connections to stochastic analysis. The principal reduction in Theorem 3.2 reduces (a ratio of) modified Fredholm determinants associated with operators in L2(f2; dnx) o n the left-hand side of (3.6) to modified Fredholm determinants associated with operators in L2(0f2; dn-lcr) on the right-hand side of (3.6) and, especially, in (3.7). This is the analog of the reduction described in the one-dimensional context of Theorem 1.2, where f2 corresponds to the half-line (0, cx~) and its boundary 0f2 thus corresponds to the one-point set {0}.
376
E GESZTESY, M. MITREA and M. ZINCHENKO
In the context of elliptic operators on smooth k-dimensional manifolds, the idea of reducing a ratio of zeta-function regularized determinants to a calculation over the ( k - 1)-dimensional boundary has been studied by Forman [16] in 1987. He also pointed out that if the manifold consists of an interval, the special case of a pair of boundary points then permits one to reduce the zeta-function regularized determinant to the determinant of a finite-dimensional matrix. The latter case is of course an analog of the one-dimensional Jost and Pais formula mentioned in the introduction (cf. Theorems 1.1 and 1.2). Since then, this topic has been further developed in various directions and we refer, for instance, to Burghelea, Friedlander, and Kappeler [8]-[11], Friedlander [17], Park and Wojciechowski [32], and the references therein. Acknowledgements We are indebted to Yuri Latushkin and Konstantin A. Makarov for numerous discussions on this topic. Fritz Gesztesy would like to thank all organizers of the 21st Max Born Symposium, and especially, Witold Karwowski and Robert Olkiewicz, for their kind invitation, the stimulating atmosphere during the meeting, and the extraordinary hospitality extended to him during his stay in Wroclaw in June of 2006. He also gratefully acknowledges a research leave for the academic year 2005/06 granted by the Research Council and the Office of Research of the University of Missouri-Columbia. REFERENCES [1] S. Albeverio, J. E Brasche, M. M. Malamud and H. Neidhardt: Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions, J. Funct. Anal. 228 (2005), 144-188. [2] W. O. Amrein and D. B. Pearson: M operators: a generalisation of Weyl-Titchmarsh theory, J. Comp. Appl. Math. 171 (2004), 1-26. [3] J. Behrndt and M. Langer: Boundary value problems for partial differential operators on bounded domains, preprint, 2006. [4] J. Behrndt, M. M. Malamud and H. Neidhardt: Scattering matrices and Weyl functions, preprint, 2006. [5] J. E Brasche, M. M. Malamud and H. Neidhardt: Weyl functions and singular continuous spectra of self-adjoint extensions, in Stochastic Processes, Physics and Geometry: New Interplays. 11. A Volume in Honor of Sergio Albeverio, E Gesztesy, H. Holden, J. Jost, S. Paycha, M. R6ckner and S. Scarlatti (eds.), Canadian Mathematical Society Conference Proceedings, Vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 75-84. [6] J. E Brasche, M. M. Malamud and H. Neidhardt: Weyl function and spectral properties of self-adjoint extensions, Integral Eqs. Operator Theory 43 (2002), 264-289. [7] B. M. Brown and M. Marietta: Spectral inclusion and spectral excactness for PDE's on exterior domains, IMA J. Numer. Anal. 24 (2004), 21-43. [8] D. Burghelea, L. Friedlander and T. Kappeler: On the determinant of elliptic differential and finite difference operators in vector bundles over S 1, Commun. Math. Phys. 138 (1991), 1-18. Erratum: Commun. Math. Phys. 150 (1992), 431. [9] D. Burghelea, L. Friedlander and T. Kappeler: Meyer-Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992), 34-65. [10] D. Burghelea, L. Friedlander and T. Kappeler: Regularized determinants for pseudodifferential operators in vector bundles over S 1, Integral Eqs. Operator Theory 16 (1993), 496-513. [11] D. Burghelea, L. Friedlander and T. Kappeler: On the determinant of elliptic boundary value problems on a line segment, Proc. Amer. Math. Soc. 123 (1995), 3027-3038.
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[12] S. Chanillo: A problem in electrical prospection and a n-dimensional Borg-Levinson theorem, Proc. Amer. Math. Soc. 108 (1990), 761-767. [13] P. A. Deift: Applications of a commutation formula, Duke Math. J. 45 (1978), 267-310. [14] V.A. Derkach and M. M. Malamud: Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), 1-95. [15] V. A. Derkach and M. M. Malamud: The extension theory of Hermitian operators and the moment problem, J. Math. Sci. 73 (1995), 141-242. [16] R. Forman: Functional determinants and geometry, Invent. Math. 88 (1987), 447--493. [17] L. Friedlander: Determinant of the Schriidinger operator on a metric graph, preprint, 2005. [18] F. Gesztesy, N. J. Kalton, K. A. Makarov, and E. Tsekanovskii: Some applications of operator-valued Herglotz functions, in Operator Theory, System Theory and Related Topics. The Moshe Livgic Anniversary Volume, D. Alpay and V. Vinnikov (eds.), Operator Theory: Advances and Applications, Vol. 123, Birkh~user, Basel, 2001, p. 271-321. [19] E Gesztesy, Y. Latushkin, M. Mitrea and M. Zinchenko: Nonself-adjoint operators, infinite determinants and some applications, Russ. J. Math. Phys. 12 (2005), 443-471. [20] E Gesztesy, M. Mitrea and M. Zinchenko, Variations on a theme of Jost and Pais, in preparation. [21] E Gesztesy and K. A. Makarov: (Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited, Integral Eqs. Operator Theory 47 (2003), 457-497. (See also Erratum 48 (2004), 425-426 and the corrected electronic only version in 48 (2004), 561--602.) [22] V. I. Gorbachuk and M. L. Gorbachuk: Boundary Value Problems for Operator Differential Equations, Kluwer, Dordrecht 1991. [23] R. Jost and A. Pals: On the scattering of a particle by a static potential, Phys. Rev. 82 (1951), 840-851. [24] S. Kotani: KdV-flow and Floquet exponent, preprint, Osaka University, 2004. [25] Y. Kurylev, M. Lassas and R. Weder: Multidimensional Borg-Levinson theorem, Inverse Probl. 21 (2005), 1685-1696. [26] M. Marietta: Eigenvalue problems on exterior domains and Dirichlet to Neumann maps, J. Comp. Appl. Math. 171 (2004), 367-391. [27] A. Nachman, John Sylvester and G. Uhlmann: An n-dimensional Borg-Levinson theorem, Commun. Math. Phys. 115 (1988), 595-605. [28] R. G. Newton: Relation between the three-dimensional Fredholm determinant and the Jost function, J. Math. Phys. 13 (1972), 880-883. [29] R. G. Newton: Inverse scattering. I. One dimension, J. Math. Phys. 21 (1980), 493-505. [30] R. G. Newton: Scattering Theory of Waves and Particles, 2nd ed., Dover, New York 2002. [31] L. P~iv~irinta and V. Serov: An n-dimensional Borg-Levinson theorem for singular potentials, Adv. AppL Math. 29 (2002), 509-520. [32] J. Park and K. P. Wojciechowski: Adiabatic decomposition of the (-determinant and Dirichlet to Neumann operator, J. Geom. Phys. 55 (2005), 241-266. [33] B. Pavlov: The theory of extensions and explicitly-solvable models, Russ. Math. Surv. 42:6 (1987), 127-168. [34] B. Pavlov: S-matrix and Dirichlet-to-Neumann operators, Chapter 6.1.6 in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, Vol. 2, R. Pike and P. Sabatier (eds.), Academic Press, San Diego 2002, pp. 1678-1688. [35] M. Salo: Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field, preprint, 2005. [36] B. Simon: Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178 (2000), 396--420. [37] B. Simon: Trace Ideals and Their Applications, Mathematical Surveys and Monographs, Vol. 120, 2nd ed., Amer. Math. Soc., Providence, RI 2005. [38] J. Sylvester and G. Uhlmann: A uniqueness theorem for an inverse boundary value problem in electrical prospection, Commun. Pure Appl. Math. 39 (1986), 91-112. [39] J. Sylvester and G. Uhlmann: A global uniqueness theorem for an inverse boundary value problem, Ann. Math. 125 (1987), 153-169.
REPORTS ON MATHEMATICAL PHYSICS
No. 3
MULTI.DIMENSIONAL VERSIONS OF A DETERMINANT FORMULA DUE TO JOST AND PAIS F. G E S Z T E S Y Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (e-mail: fritz @math.missouri.edu, http://www.math.missouri.edu/personnel/faculty/gesztesyf.html) M.
MITREA
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA (e-mail: marius @math.missouri.edu, http://www.math.missouri.edu/personnel/faculty/mitream.html)
and M.
ZINCHENKO
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA (e-mail: [email protected]) (Received October 5, 2006)
We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schr6dinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schr6dinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set f2 C R n, n = 2, 3, where [2 has a compact, nonempty boundary 0f2 satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on 0f2 and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants perturbation associated with operators in L 2(~; dnx) to modified Fredholm determinants associated with operators in L2(0fl; dn-lcr), n = 2, 3. MSC2000: Primary: 47B10, 47G10, Secondary: 34B27, 34L40. Keywords: Fredholm determinants, non-self-adjoint operators, multi-dimensional Schr6dinger operators, Dirichlet-to-Neumann maps.
*Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS0405526 and DMS-0400639, FRG-0456306. [365]
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F. GESZTESY,M. MITREA and M. ZINCHENKO
1.
Introduction To illustrate the reason behind the title of this note, we briefly recall a celebrated result due to Jost and Pais [23] in 1951, who proved a spectacular reduction of a Fredholm determinant associated with the Birmann-Schwinger kernel of a onedimensional Schr6dinger operator on a half-line, to a simple Wronski determinant of appropriate solutions of the underlying Schr6dinger equation. This Wronsld determinant also equals the so-called Jost function of the corresponding half-line Schr6dinger operator. As the title of our paper suggests, we are seeking a certain multi-dimensional variant of this result. To describe the result due to Jost and Pais [23], we need a few preparations. Denoting by HOD,+and HOW,+the one-dimensional Dirichlet and Neumann Laplacians in L2((0, c~); dx), and assuming V C LI((0, ~ ) ; dx),
(1.1)
we introduce the perturbed Schr6dinger operators H ff and H N in L2((0, c~); dx) by = -f"
+ V f,
f ~ dom(HoD+) = {g E L2((0, oo); dx) lg, g' ~ AC([0, R]) for all R > 0,
(1.2)
g(0) = 0, ( - g " d- Vg) E L2((0, ~ ) ; dx)},
H ~ f = - f " + V f, f E dom(H~+) = {g a L2((0, c~); dx) lg, g' ~ AC([0, R]) for all R > 0,
(1.3)
g'(0) = O, (-g" + Vg) ~ L2((0, oo); dx)}. Thus, Hff and H+N are self-adjoint if and only if V is real-valued, but since the latter restriction plays no special role in our results, we will not assume real-valuedness of V throughout this paper. A fundamental system of solutions ~b+ D(z, .), 0+° (z, .), and the Jost solution f+ (z, .) of
- ~ ' ( z , x) + V~(z, x) : zap(z, x),
z ~ C\{0},
x ~ 0,
(1.4)
are then introduced via the standard Volterra integral equations cp+O(z,x) = z -1/2 sin(zl/2x) + / dx' z -1/2 sin(zl/2(x - x'))V(x')cp~(z, x'),
(1.5)
0 x
oD+(z, X)
cos(zl/2x) + [ dx' z -1/2 sin(zl/2(x - x'))V(x')O~(z, x'),
(1.6)
0 oO
f+(z, X)
e izl/2x -- f dx' Z-1/2 sin(zl/2(x - x'))V(x')f+(z, x'),
(1.7)
x
Z E C\{0},
~(Z 1/2) ~ 0,
X ~ 0.
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ON A DETERMINANT FORMULA BY JOST AND PAIS
In addition, we introduce
u = e x p ( i a r g ( V ) ) l V I 1/2,
v = IVI 1/2,
so that V = u v ,
(1.8)
and denote by I+ the identity operator in L2((0, oc); dx). Moreover, we denote by
W ( f , g)(x) = f ( x ) g ' ( x ) - f ' ( x ) g ( x ) ,
x > O,
(1.9)
the Wronskian of f and g, where f , g E cl([0, ec)). We also use the standard convention to abbreviate (with a slight abuse of notation) the operator of multiplication in L2((0, ec); dx) by an element f E L~oc((O, oc); dx) (and similarly in the higherdimensional context later) by the same symbol f (rather than MI, etc.). In the remainder of this note •p(7-{), p >_ 1, denote the usual trace ideals associated to the complex, separable Hilbert space 7-/, and det(.) and det2(.) denote the Fredholm and modified Fredholm determinant in connection with trace class (p = 1) and Hilbert-Schmidt operators (p = 2), respectively. Then, the following result holds. THEOREM 1.1 ([23], see also [211 and the references cited therein). Assume V 6 LI((0, ec); dx) and let z ~ C\[0, oc) with ,,~(Z1/2) > 0. Then,
u ( n ~ + - z I + ) - l v , u ( H g + - z I + ) - l p E/31 (L2((O, ~ ) ; d x ) )
(1.10)
and det (I+ + u(H~,+ - z I + ) - l v ) = 1 + z -1/2
f ,x sin(zl/2x)V(x)f+(z,
x)
0
= W(f+(z, .), dp~(z, .)) = f+(z, 0),
(1.11)
f
oo
det (I+ + u(HoU+ -- z I + ) - l v ) = 1 + iz -1/2
0
W(f+(z, .), O°+(z, .))
f ; ( z , O)
izl/2
izl/2
(1.12)
Eq. (1.11) is the modern formulation of the classical result due to Jost and Pais [23]. We emphasize that (1.11) and (1.12) exhibit the remarkable fact that the Fredholm determinant associated with trace class operators in the infinitedimensional space L2((0, ~ ) ; dx) is reduced to a simple Wronski determinant of C-valued distributional solutions of (1.4). This fact goes back to Jost and Pais [23] (see also [21, 28, 29, 30, Sect. 12.1.2, 36, 37, Proposition 5.7], and the extensive literature cited in these references). The principal aim of this note is to explore the extent to which this fact may generalize to higher dimensions n = 2, 3. While a direct generalization of (1.11), (1.12) appears to be difficult, we will next derive a formula for the ratio of such determinants which indeed permits a natural extension to higher dimensions.
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E GESZTESY, M. MITREA and M. ZINCHENKO
For this purpose we introduce the boundary trace operators YD (Dirichlet trace) and Yw (Neumann trace) which, in the current one-dimensional half-line situation, are just the functionals, YD: {C([0, oo)) C ,~---> g g(O),
YN: { CI([0' OO)) C '~h'''~ -h'(0).
(1.13)
In addition, we denote by m~,+, m0, m~,+, and m~ the Weyl-Titchmarsh m-functions corresponding to HOD+, H+°, /-/ON,+, and H~, respectively, that is, m~,+(z) = iz 1/2, m~(z) - f~(z, O) f+(z, 0)'
m~,+(z) -m:~(z) --
1
m~,+(z----~= iz-1/2' ~ mD+(z) --
f+(z, 0) f~_(z,O)"
(1.14) (1.15)
In the case where V is real-valued, we recall that m 0 is a Herglotz function (i.e. it maps the open complex upper half-plane C+ analytically into itself) and the measure dp~ in its Herglotz representation is then the spectral measure of the operator H+D and hence encodes all spectral information of H~. Similarly, m+° also encodes all spectral information of H~ since - 1 / n O = mN+ is also a Herglotz function and the measure dpN+ in its Herglotz representation represents the spectral measure of the operator H~v. Then we obtain the following result for the ratio of the perturbation determinants in (1.11) and (1.12). THEOREM 1.2 ([19]). Assume V ~ LI((O, o0); dx) and let z ~ C\cr(H~9) with ~(z U2) > 0. Then, det (I+ + u ( H ~ + - z I + ) - l v ) det (I+ + u(Ho°,+ - zI+)-lv ) = 1 - (ys(H+D - zI+)-lV[yo(H~,+ -- 71+)-1]*)1
(1.16)
W(f+(z),qbN(z)) f+(z,O) n°(z) m~,+(z) = izl/2W(f+(z), qb~(z)) -- izU2f+(z, O) - m~,+(z~) - mN(z)
(1.17)
The proper multi-dimensional generalizations to Schr6dinger operators in L2(f2; dnx), corresponding to an open set f2 C R n with compact, nonempty boundary 0f2, more precisely, the proper operator-valued generalization of the Weyl-Titchmarsh function mO(z ) is then given by the Dirichlet-to-Neumann map, denoted by MR(z ). This operator-valued map indeed plays a fundamental role in our extension of (1.17) to the higher-dimensional case. In Section 3, which contains our principal result, we present a direct multi-dimensional extension of Theorem 1.2.
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ON A DETERMINANT FORMULA BY JOST AND PAIS
2.
Schr~dinger operators with Dirichlet and Neumann boundary conditions and some results on Dirichlet-to-Neumann maps
In this section we first recall some properties of Dirichlet, H~a, and Neumann, H~,a, Laplacians in Lz(g2;dnx) associated with open sets ~2 C N n, n = 2, 3, introduced in Hypothesis 2.1 below. Subsequently, we introduce the Dirichlet and Neumann Schr6dinger operators HaD and H N in LZ(f2; dnx), that is, perturbations of the Dirichlet and Neumann Laplacians H 0,~2 D and H 0,[2 N by a potential V satisfying Hypothesis 2.4. Finally, we cite some results on Dirichlet-to-Neumann maps to be used in Section 3. HYPOTHESIS 2.1. Let n = 2, 3 and assume that ~2 C ~n is an open set with a compact, nonempty boundary Of2. In addition, we assume that one of the following three conditions holds: (i) f2 is of class C l'r for some 1/2 < r < 1; (ii) £2 is convex; (iii) ~ is a Lipschitz domain satisfying a uniform exterior ball condition. We note that while 0f2 is assumed to be compact, ~2 may be unbounded in connection with conditions (i) or (iii). For more details in this context we refer to [19, Appendix A] and [20, Appendix A]. First, we introduce the boundary trace operator yo (Dirichlet trace) by go: C ( ~ ) ~ C(OQ),
you = ul0a.
(2.1)
Then there exists a bounded, linear operator YD,
YD: HS(~) --+ HS-(1/2)(O~) ~
L2(0~;
dn-lcr),
1/2 < s < 3/2,
(2.2)
whose action is compatible with that of yD °, that is, the two Dirichlet trace operators coincide on the intersection of their domains. Here d n - l a denotes the surface measure on 0S2 and we refer to [19, Appendix A] for our notation in connection with Sobolev spaces. In addition, we introduce the operator VN (Neumann trace) by YN :
V • ~/DV:
HS+l(f2) --+ L2(0f2; dn-l~r),
1/2 < s < 3/2,
(2.3)
where v denotes outward pointing normal unit vector to 0f2. It follows from (2.2) that YN is also a bounded operator. Given Hypothesis 2.1, we introduce the self-adjoint and nonnegative Dirichlet and Neumann Laplacians H0°,a and H0U,a associated with the domain f2 as follows: H DO,s2= - A ,
dom(HoDa) = {u • Hz(S2) I×DU = 0},
HNO,a= - A ,
dom(H~,a)
= {u •
H2(a) IyNU =
0}.
(2.4) (2.5)
A detailed discussion of H 0A2 D and H 0,f~ N is provided in [19, Appendix A]. LEMMA 2.2 ([19]). Assume Hypothesis 2.1. Then the operators H O,f~ ° and H 0,~2 N introduced in (2.4) and (2.5) are nonnegative and self-adjoint in L2(f2; dnx) and
370
E GESZTESY,M. MITREAand M. ZINCHENKO
the following mapping properties hold for all q E [0, 1] and z E C\[0, o0), (HoD,~ - zlfa) -q, (H~,e - zls2) -q E ~(LZ(f2; dnx), Hzq(f2)).
(2.6)
Here Ia and Ioa denote the identity operators in LZ(ff2;dnx) and L2(0f2; dn-lcr), respectively. The fractional powers in (2.6) (and in subsequent analogous cases) are defined via the functional calculus implied by the spectral theorem for self-adjoint operators. As explained in [19, Appendix A], the key ingredients in proving Lemma 2.2 are the inclusions dom(HoD~2) C H2(f2),
dom(HNa) C H2(~)
(2.7)
and methods based on real interpolation spaces. For the remainder of this paper we agree to the simplified notation that the operator of multiplication by the measurable function f in L2(f2; dnx) is again denoted by the symbol f . LEMMA 2.3 ([19], [20]). Assume Hypothesis 2.1 and let 2 <_ p, (n/2p) < q <_ 1, f E LP(~; dnx), and z E C\[O, oo). Then, f(Hgza - zI~) -q, f ( H g f a - zI~2) -q E /3p(L2(~; dnx)).
(2.8)
Next, we turn to our assumptions on the potential V and the corresponding definition of Dirichlet and Neumann Schr6dinger operators Hg and H N in L2(f2; d~x). HYPOTHESIS 2.4. Suppose that f2 satisfies Hypothesis 2.1 and assume that V E LP(f2; dnx) for some p satisfying 4/3 < p < 2, in the case n = 2, and 3/2
Z ~ C\[O, co),
(2.9)
u(Ogf2 -- ZI~) -1/2, (Ogf2 - zla)-'/2u E ~2p(t2(a; dnx)),
Z E C\[0, oo). (2.10)
Moreover, (2.9) and (2.10) imply
u(Ogf2 - zlf2)-ll), u(Hgf2 - zlfa)-lv E ~p(L2(a; dnx)) C 1~2(L2(~-~; dnx)), z ~ C\[0, co),
(2.11)
and one can introduce densely defined, closed operators H~ and H~ (which are extensions of H0Oa+ V on dom(H0°a)Ndom(V) and HoN~+ V on dom(H~,a)Ndom(V), respectively) by the method of quadratic forms (cf. [19, Section 2] for precise details).
ON A DETERMINANT FORMULA BY JOST AND PAIS
371
Next, we turn to Dirichlet and Neumann boundary value problems associated with the Helmholtz differential expression - - A - z as well as the corresponding differential expression - A + V - z in the presence of a potential V, both in connection with the open set f2. In addition, we provide a detailed discussion of Dirichlet-to-Neumann M~,~, M D and Neumann-to-Dirichlet M~,~, M N, maps in L2(Og2;dn-l~). Denote by ~U : {U • Hl(f2) [ Au • (Hl(f2)) *} --+ H-'/2(Of2)
(2.12)
a weak Neumann trace operator defined by (~N u ,
4)) = f d"x Vu(x) . V ~ ( x ) + (Au, ~)
(2.13)
tt
fa
for all 4) • H1/2(O~) and • • Hl(f2) such that ~,Dqb = 4). We note that this definition is independent of the particular extension qb of ~b, and that ~U is an extension of the Neumann trace operator YN defined in (2.3). THEOREM 2.5 ([20]). Suppose f2 is an open Lipschitz domain with a compact nonempty boundary 0~2. Then for every f e Hl(Of2) and z • C\cr(HoD,~) the following Dirichlet boundary value problem, ( - A -- Z)U~ = 0 on f2, ~'ou~ = f on 0~2,
u~ • H3/2(f2),
(2.14)
has a unique solution ug satisfying ~NUg • L2(0~; dn-la). Similarly, for every g • L2(Oa; dn-l~r) and z • C\a(HoNa) the following Neumann boundary value problem,
( - A - z)u N = 0 on f2, = g on
u N • H3/2(a),
aa,
(2.15)
has a unique solution u N. Moreover, the solutions ug and u N are given by the formulae ug (z) = --(gu(Hff, a -- 2I~)-1) *f,
(2.16)
RoN(z) = (YD ( H'N o , ~ - z- I ~),-Ix*) g,
(2.17)
and hence (2.14)-(2.17) imply the following mapping properties, [?/u((Hffa -- zla)-l)*]* : H'(Of2)
--+
H3/2(~"~),
g • C\~(HoD,a),
[yo((H~,~ - zln)-l)*]* : LZ(Ofa; d n-la) --+ H3/2(f2),
z • C\~(H~,~).
(2.18) (2.19)
We temporarily strengthen our hypothesis on V and introduce the following assumption.
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F. GESZTESY,M. MITREA and M. ZINCHENKO
HYPOTHESIS 2.6. Suppose the set f2 satisfies Hypothesis 2.1 and assume that V E L2(f2; dnx) 7) LP(f2; dnx) for some p > 2. THEOREM 2.7 ([20]). Assume Hypothesis 2.6. Then for every f E Hi(Of2) and z E C\cr(H~) the following Dirichlet boundary value problem, ( - - A + V - - z ) u o = O on f2,
u ° E H3/2(f2),
(2.20)
you ° = f on Of2,
has a unique solution u ° satisfying ~Nu ° ~ L2(Of2; d n - l t 7 ) . Similarly, for every g ~ L2(0f2; dn-l~r) and z E C\cr(H N) the following Neumann boundary value problem, (--A .qt_ V -- Z)U N = 0 o n ~'2, ~NUN = g on 0f2,
u N E H3/2(~),
(2.21)
has a unique solution u N. Moreover, the solutions u ° and u N are given by the formulae u°(z) = - [ y N ( ( H ° -- zlo)-l)*]* f,
(2.22)
u N (Z) = [VD((H u - zI~)-l)*]*g.
(2.23)
Assuming Hypothesis 2.6, we now introduce the Dirichlet-to-Neumann maps, M~,o(z) and M~(z), associated with ( - A - - z ) and (--A + V - z ) on f2, as follows,
M~.~(z):, {Hl(Of2)f --+ _~NUg '
Z ~ C\a[H D ., o,~1"
(2.24)
where u~ is the unique solution of ( - A - z)u D = 0 on f2,
u~ E H3/Z(f2),
YDUD = f
on 0f2,
(2.25)
and M°(z): { Hl(0~)f ~--+LZ(0~;_~Nuo,d'-l~r)' where u ° is the unique solution of ( - - m + W - Z)U D -~- 0 on f2,
U D E H3/2(~),
Z 6 C\~r(H°),
~/oUD = f
on 0~.
(2.26)
(2.27)
In addition, still assuming Hypothesis 2.6, we introduce the Neumann-to-Dirichlet maps, MoN~(z) and MN(z), associated with ( - A - z ) and ( - A + V - z) on f2, as follows, MN (Z): [ L2(0~; '
[
HI(0~), g w-~ you~,
dn-l~y) ~
z ~ C\cr(H0U,~),
(2.28)
373
ON A DETERMINANT FORMULA BY JOST AND PAIS
where u N is the unique solution of ( - A - z)u N = 0 on ~2,
u~v
• H3/2(~),
~NUu = g on 0f2,
(2.29)
Z • C\a(HN),
(2.30)
~NUy = g
(2.31)
and
yDuN,HI(Of2)'
MN(z): { L2(0~; dn-la)g e-->---> where u N is the unique solution of (-A+V-z)u
N=O on g2,
ug•H3/2(g2),
on 0f2.
It follows from Theorems 2.5 and 2.7, that under the assumption of Hypothesis 2.6, the operators M~,,n(z), MD(z), M~,,f2(z), and MN(z) are well-defined and satisfy the following equalities,
M~,~(z) = -M~,r~(z) -1, MN(z)=--MD(z)-I,
z • C\(a(HoD,r~) U a(H0Ur~)), Z • C \ ( a ( H D ) Ua(HN)),
(2.32) (2.33)
and
Mga(z) = Mg(z) =
VN[YN((Hga -- Z/a)-')*]*, ;N[~((Hg
m~,,f2(z ) = yD[~D((Oga
Mg(z) =
vo[×o((Hg
- z/a)-')*]*, --
zla)-l)*] *,
- zI.)-~)*] *,
z • C\a(H0°a), z • C\a(HD), Z • C\a(H~Va), z • C\a(HN).
(2.34) (2.35) (2.36) (2.37)
Next, we use (2.35) and (2.37) to extend the above definition of the operators
MD(z) and Mg(z) to the more general situation governed by Hypothesis 2.4: LEMMA 2.8 ([20]). Assume Hypothesis 2.4. Then the operators MD(z) and MN (z) defined by equalities (2.35) and (2.37) have the following mapping properties, Mg(Z)" Hl(O~'-2) ~ L2(0~"2; dn-l~), Z • C\cr(HD), (2.38) MN(z)
"
L2(0~; dn-lcr)
--+
HI(0~),
Z • C\a(HN).
(2.39)
Weyl-Titchmarsh operators, in a spirit close to ours, have recently been discussed by Amrein and Pearson [2] in connection with the interior and exterior of a ball in ]R3 and potentials V • L°°(]R3; d3x). For additional literature on Weyl-Titchmarsh operators, relevant in the context of boundary value spaces (boundary triples, etc.), we refer, for instance, to [1, 3-7, 14, 15, 18, Chapter 3, 22, 26, 33, 34]. For applications of the Dirichlet-to-Neumann map to Borg-Levinson-type inverse spectral problems we refer to [12, 27, 31, 35, 38, 39] (see also [25] for an alternative approach based on the boundary control method). Finally, we mention the following auxiliary result, which plays a crucial role in the proof of Theorem 3.2.
374
F. GESZTESY, M. MITREA and M, ZINCHENKO
LEMMA 2.9 ([20]). Assume Hypothesis 2.4. Then the following identities hold, Mg~2(Z ) - MD(z) = ~N(H o -- zl~2)-lV[yN((Hgs2 - zlf2)-l)*] *,
z 6 C\(cr(H0°~)U or(H°)), Mg(z)M~,.(z)-l
= I~. - ~N(Hg -- zl~)-'V[×o((HoNa - zlf2)-')*]*, z 6C\(a(Hg,)Ua(H°)Ucr(H~,a)).
3.
(2.40) (2.41)
A m u l t i - d i m e n s i o n a l v a r i a n t o f a f o r m u l a d u e to Jost a n d Pais In this section w e present our multi-dimensional variants o f the Jost and Pais as discussed in the introduction. We start with an extension o f a result in
formula [19].
THEOREM 3.1 ([20]). Assume Hypothesis 2.4 and let z ~ C\(,~(ug)u,,(ng~)U,~(Hg~)).
Then,
×u(H~,~ - zl~)-' V ( H g - zI~)-l v[yo(H~,~
-~1f2)-l]
* E
~1 (L2(0~'-2;d"-lo')), (3.1)
yN(Ha° -- zle)-IV[yD(HoN, e -- 21~1-1] * e/32(LZ(0f2; dn-lo')),
(3.2) and
det 2 (lf2 + u ( H o N , ~ - - z l ~ ) - 1 v ) detz(l~+u(H~,~-zl~)-lv) =
det2(l~fz - y N ( H ~ -- zl~)-IV[yo(H~),.N - -1 z . )~-11. 1
)etr(T2(z)),
(3.3)
where , r 2 ( z ) = ~,N (Ho?~
J~)-lv(Hg
N - z - I ~),,-11. - z l ~ ) - ' V [ y , ( H o,~ E ~1 ( L 2 ( 0 ~ ; d n - l ° ' ) ) •
(3.4)
Combining Lemma 2.9 and Theorem 3.1 implies our principal new result (cf. [20]), the multi-dimensional analog of Theorem 1.2. THEOREM 3.2. Assume Hypothesis 2.4 and let z~C\(~(Hg)
N U ~ ( H g ~ ) U ~ ( H o,@.
ON A DETERMINANT FORMULA BY JOST AND PAIS
375
Then, Mg(s)M~(Z)
-1 -- I ~
:
--yg(n~
I ZI~)IlW[yo(n~
-- Z ] ~ ) I 1] *
t 2(c2(of ;
(3.5)
and det 2 (I~
+u(H~-zI~)-lv)
det 2(I~
+u(H~,~ - z I ~ ) - l v )
= det2(Ia~ -
yN(Hff -- ZI~)-IV[yD(HoNf2 -- ~i~)-1]* )etr(T2(z))
[A.cD~'_~ ~nD i"_', l~_tr(T2(z)) = det2uvlf2~,~)lvl0,f2~) le
with
T2(z)
given by
(3.6) (3.7)
(3.4).
Rather than entering into details of the proof of Theorems 3.1 and 3.2, we offer an elementary comment on determinants which, however, lies at the heart of the matter of our multi-dimensional variant of the one-dimensional Jost and Pais result. Suppose A 6 ~('~t-/1,']-/2) , B E ~('~2,']--/1) with AB ~ ~1(']--/2) and BA c ~1(~--/1). Then, det(IT-t2 - AB) ----d e t ( l ~ - BA). (3.8) Eq. (3.8) follows from the fact that all nonzero eigenvalues of AB and BA coincide including their algebraic multiplicities. The latter fact, in turn, can be derived from the formula
A(BA-zI~)-IB=I~2+z(AB-zI~2)
-1,
z~C\(cr(aB) Ucr(Ba))
(3.9)
(and its companion with A and B interchanged), as discussed in detail by Deift [13]. In particular, ~1 and 7-/2 may have different dimensions. Especially, one of them may be infinite and the other finite, in which case one of the two determinants in (3.8) reduces to a finite determinant. This case indeed occurs in the original one-dimensional case studied by Jost and Pais [23] as described in detail in [21] and the references therein. In the proof of Theorem 3.1 the role of ~1 and ~2 is played by L2(f2; d"x) and L2(0f2; dn-l~r), respectively. In the context of KdV flows and reflectionless (i.e. generalizations of soliton-type) potentials represented as Fredholm determinants, a reduction of such determinants (in some cases to finite determinants) has also been studied by Kotani [24], relying on certain connections to stochastic analysis. The principal reduction in Theorem 3.2 reduces (a ratio of) modified Fredholm determinants associated with operators in L2(f2; dnx) o n the left-hand side of (3.6) to modified Fredholm determinants associated with operators in L2(0f2; dn-lcr) on the right-hand side of (3.6) and, especially, in (3.7). This is the analog of the reduction described in the one-dimensional context of Theorem 1.2, where f2 corresponds to the half-line (0, cx~) and its boundary 0f2 thus corresponds to the one-point set {0}.
376
E GESZTESY, M. MITREA and M. ZINCHENKO
In the context of elliptic operators on smooth k-dimensional manifolds, the idea of reducing a ratio of zeta-function regularized determinants to a calculation over the ( k - 1)-dimensional boundary has been studied by Forman [16] in 1987. He also pointed out that if the manifold consists of an interval, the special case of a pair of boundary points then permits one to reduce the zeta-function regularized determinant to the determinant of a finite-dimensional matrix. The latter case is of course an analog of the one-dimensional Jost and Pais formula mentioned in the introduction (cf. Theorems 1.1 and 1.2). Since then, this topic has been further developed in various directions and we refer, for instance, to Burghelea, Friedlander, and Kappeler [8]-[11], Friedlander [17], Park and Wojciechowski [32], and the references therein. Acknowledgements We are indebted to Yuri Latushkin and Konstantin A. Makarov for numerous discussions on this topic. Fritz Gesztesy would like to thank all organizers of the 21st Max Born Symposium, and especially, Witold Karwowski and Robert Olkiewicz, for their kind invitation, the stimulating atmosphere during the meeting, and the extraordinary hospitality extended to him during his stay in Wroclaw in June of 2006. He also gratefully acknowledges a research leave for the academic year 2005/06 granted by the Research Council and the Office of Research of the University of Missouri-Columbia. REFERENCES [1] S. Albeverio, J. E Brasche, M. M. Malamud and H. Neidhardt: Inverse spectral theory for symmetric operators with several gaps: scalar-type Weyl functions, J. Funct. Anal. 228 (2005), 144-188. [2] W. O. Amrein and D. B. Pearson: M operators: a generalisation of Weyl-Titchmarsh theory, J. Comp. Appl. Math. 171 (2004), 1-26. [3] J. Behrndt and M. Langer: Boundary value problems for partial differential operators on bounded domains, preprint, 2006. [4] J. Behrndt, M. M. Malamud and H. Neidhardt: Scattering matrices and Weyl functions, preprint, 2006. [5] J. E Brasche, M. M. Malamud and H. Neidhardt: Weyl functions and singular continuous spectra of self-adjoint extensions, in Stochastic Processes, Physics and Geometry: New Interplays. 11. A Volume in Honor of Sergio Albeverio, E Gesztesy, H. Holden, J. Jost, S. Paycha, M. R6ckner and S. Scarlatti (eds.), Canadian Mathematical Society Conference Proceedings, Vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 75-84. [6] J. E Brasche, M. M. Malamud and H. Neidhardt: Weyl function and spectral properties of self-adjoint extensions, Integral Eqs. Operator Theory 43 (2002), 264-289. [7] B. M. Brown and M. Marietta: Spectral inclusion and spectral excactness for PDE's on exterior domains, IMA J. Numer. Anal. 24 (2004), 21-43. [8] D. Burghelea, L. Friedlander and T. Kappeler: On the determinant of elliptic differential and finite difference operators in vector bundles over S 1, Commun. Math. Phys. 138 (1991), 1-18. Erratum: Commun. Math. Phys. 150 (1992), 431. [9] D. Burghelea, L. Friedlander and T. Kappeler: Meyer-Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992), 34-65. [10] D. Burghelea, L. Friedlander and T. Kappeler: Regularized determinants for pseudodifferential operators in vector bundles over S 1, Integral Eqs. Operator Theory 16 (1993), 496-513. [11] D. Burghelea, L. Friedlander and T. Kappeler: On the determinant of elliptic boundary value problems on a line segment, Proc. Amer. Math. Soc. 123 (1995), 3027-3038.
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