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Boundary triples for Schrödinger operators with singular interactions on hypersurfaces
J. Behrndt1, M. Langer2 and V. Lotoreichik3
1Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria 2Department of Mathematics and Statistics, University of Strathclyde,
26 Richmond Street, Glasgow Gl 1XH, United Kingdom 3Department of Theoretical Physics, Nuclear Physics Institute CAS, 250 68 ReZ near Prague, Czech Republic behrndt@tugraz.at, m.langer@strath.ac.uk, lotoreichik@ujf.cas.cz
PACS 02.30.Tb, 03.65.Db DOI 10.17586/2220-8054-2016-7-2-290-302
The self-adjoint Schrödinger operator As,a with a ¿-interaction of constant strength a supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L2(Rn). The aim of this note is to construct a boundary triple for S* and a self-adjoint parameter Qs,a in the boundary space L2(C) such that As,a corresponds to the boundary condition induced by &s,a. As a consequence, the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of As,a in terms of the Weyl function and &s,a. Keywords: Boundary triple, Weyl function, Schrodinger operator, singular potential, ¿-interaction, hypersurface. Received: 22 January 2016
1. Introduction
Boundary triples and their Weyl functions are efficient and frequently used tools in the extension theory of symmetric operators and the spectral analysis of their self-adjoint extensions. Roughly speaking, a boundary triple consists of two boundary mappings that satisfy an abstract second Green's identity and a maximality condition. With the help of a boundary triple, all self-adjoint extensions of a symmetric operator can be parameterized via abstract boundary conditions that involve a self-adjoint parameter in a boundary space. In addition, the spectral properties of these self-adjoint extensions can be described with the help of the Weyl function and the corresponding boundary parameters. We refer the reader to [1-5] and Section 2 for more details on boundary triples and their Weyl functions.
The main objective of this note is to provide and discuss boundary triples and their Weyl functions for self-adjoint Schrodinger operators in L2(Rn) with ^-interactions of strength a E R supported on a compact smooth hypersurface C that separates Rn into a smooth bounded domain Q and an unbounded smooth exterior domain Qe. In an informal way, such an operator is often written in the form
As,a = -A - aöa, (1)
where Sc denotes the ^-distribution supported on C. A precise definition of the self-adjoint operator As,a in terms of boundary or interface conditions is given at the beginning of Section 3 below; see also [6,7] for an equivalent definition via quadratic forms. Schrodinger operators with S-interactions are frequently used in mathematical physics to model interactions of quantum particles; we refer to the monographs [8] and [9], to the review article [10] and to [6,11-25] for a small selection of related papers on spectral analysis of such operators.
Let Afree be the usual self-adjoint realization of —A in L2(Rn) and let As,a be the self-adjoint operator with 5-interaction on C in (1). We consider the densely defined, closed symmetric operator S = Afree n As,a in L2(Rn) and its adjoint S*, and we construct a boundary triple {L2(C), r0, ri} for S * and a self-adjoint parameter es,a in L2(C ) such that
Afree = S * f ker ro and As,« = S * f ker(ri — e^).
Although it is clear from the general theory that such a boundary triple and a self-adjoint parameter es,a exist, its construction is not trivial. Our idea here is based on a coupling of two boundary triples for elliptic PDEs which involve the Dirichlet-to-Neumann map as a regularization (see [26-28]), the restriction of this coupling to a suitable intermediate extension, and certain transforms of boundary triples and corresponding parameters. These efforts and technical considerations are worthwhile for various reasons. In particular, if y and M denote the Y-field and Weyl function corresponding to the boundary triple {L2(C), r0, r1} (see Section 2 for more details), then it follows immediately from the general theory in [3,4] that the resolvent difference of Afree and As,a admits the representation
(As,a — A)-1 — (Afree — A)-1 = Y (A) ^s,« — M (A))^ (A)*
for all A G p(As,a) and belongs to some operator ideal in L2(Rn) if and only if the resolvent of es,a belongs to the analogous operator ideal in L2(C); see Theorem 3.5. As a special case, the Schatten-von Neumann properties of the resolvent difference of Afree and As,a carry over to the resolvent of es,a, and vice versa. Moreover, the spectral properties of As,a can be described with the help of the perturbation term (es,a — M (A))-1. We mention that in the context of the more general notion of quasi boundary triples and their Weyl functions from [29, 30] a similar approach as in this note and closely related results can be found in [6,31]; we also refer to [27,28,32-37] for other methods in extension theory of elliptic differential operators.
2. Boundary triples and Weyl functions
In this preparatory section, we recall the notion of boundary triples, associated Y-fields and Weyl functions, and discuss some of their properties. For a more detailed exposition, we refer the reader to [1-5,38].
In the following, let H be a Hilbert space, let S be a densely defined, closed symmetric operator in H, and let S * be the adjoint operator.
Definition 2.1. A triple {G, r0, r1} is called a boundary triple for S * if G is a Hilbert space and r0, r1 : dom S * ^ G are linear mappings that satisfy the abstract second Green's identity
(S *f,g)« — (f,S * g)« = (rf, r0g)g — (rf, r1g)o
for all f, g G dom S *, and the mapping r := (r0, r1)T : dom S * ^ G x G is surjective.
Recall that a boundary triple {G, r0, r1} for S * exists if and only if the defect numbers of S coincide or, equivalently, S admits self-adjoint extensions in H. Moreover, a boundary triple is not unique (except in the trivial case S = S*). The following special observation will be used in Section 3: suppose that {G, r0, r1} is a boundary triple for S* and let G be a bounded self-adjoint operator in G ; then {G, r0, r1}, where
(I G\ (r
rj 1J W ' (2)
is also a boundary triple for S*. Recall also that dom S = ker r0 fl kerTi and that the mapping
8 ^ A© := S * [ {/ G dom S * : ff = (^j/, rif )T G 6} (3)
establishes a bijective correspondence between the closed linear subspaces (relations) in G x G and the closed linear extensions A© c S* of S. In the case when 6 is (the graph of) an operator, the closed extension A© in (3) is given by
A© = S* \ ker(r - 6r„). (4)
It is important to note that the identity (A©)* = A©» holds and hence A© in (3)-(4) is self-adjoint in H if and only if 6 is self-adjoint in G. It follows, in particular, that the extension
Ao = S* \ kerTo (5)
is self-adjoint. This extension often plays the role of a fixed extension within the family of self-adjoint extensions of S. We also mention that 6 in (3) is an unbounded operator if and only if the extensions Ao and A© are disjoint but not transversal, that is,
S = A© n Ao and A© + Ao C S*, (6)
where + denotes the sum of subspaces. Note that this appears only in the case when G is infinite-dimensional, that is, the defect numbers of S are both infinite.
The next theorem can be found in [39]. Very roughly speaking, it can be regarded as converse to the above considerations. Here, the idea is to start with boundary mappings defined on the domain of some operator T that satisfy the abstract second Green's identity and some additional conditions, and to conclude that T coincides with the adjoint of the restriction of T to the intersection of the kernels of the boundary mappings. Theorem 2.2 will be used in the proof of Lemma 3.1.
Theorem 2.2. Let T be a linear operator in H, let G be a Hilbert space and assume that r0, r : dom T m G are linear mappings that satisfy the following conditions:
(i) there exists a self-adjoint restriction A0 of T in H such that dom A0 c ker r0;
(ii) ran(r0, Ti)t = G x G;
(iii) for all f, g G dom T the abstract Green's identity
(Tf,g)H - (f,Tg)H = (rif, r>g)g - (rf, r^g
holds.
Then S := T \ (kerT0 n kerr1) is a densely defined, closed, symmetric operator in H such that S* = T and {G, r0, r1} is a boundary triple for S* with the property A0 = S* \ ker r0.
In the following, we assume that S is a densely defined, closed, symmetric operator in H and that {G, r0, r1} is a boundary triple for S*. Let A0 = S* \ kerr0 be as in (5) and observe that the following direct sum decomposition of dom S* is valid:
domS* = dom A0 + ker(S* - A) = kerT + ker(S* - A), A G p(A0).
It follows, in particular, that r0 \ ker(S* - A) is a bijective operator from ker(S* - A) onto G. The inverse is denoted by
7(A) = (r0 \ ker(S* - A))-1, A G p(A0);
when viewed as a function A m 7(A) on p(A0), we call 7 the 7-field corresponding to the boundary triple {G, r0, r1}. The Weyl function M associated with {G, r0, r1} is defined by
M(A) = riY(A) = ri(r0 \ ker(S* - A))-1, A G p(Ac).
It can be shown that the values M(A) of the Weyl function M are bounded, everywhere defined operators in G, that M is a holomorphic function on p(A0) with the properties M(A) = M(A)* and that Im M(A) is uniformly positive for A G C+, i.e. M is an operator-valued Nevanlinna or Riesz-Herglotz function that is uniformly strict; see [2].
3. Schrodinger operators with ¿-interactions on hypersurfaces
Let Qj c Rn, n > 2, be a bounded domain with C^-smooth boundary C = dQj and let Qe = Rn \ Qj be the corresponding exterior domain with the same C^-smooth boundary dQe = C. In the following, /i|C and /e|C denote the traces of functions in Qj and Qe, respectively; if /i|c = /e|c, we also set / |c := /j|c = /e|c. Moreover, <9^ /j|c and dve /e |c denote the traces of their normal derivatives; here we agree that the normal vectors vi and ve point outwards of the domains, so that, vi = —ve.
In the following, let a = 0 be a real constant and consider the Schrodinger operator with a 8-interaction of strength a supported on C defined by
Av/ = —A/,
dom= {/ = (f) G H2(Qj) x H2(Qe), f =,/e_!Cd f , ) . (7)
I \JeJ a/|C = /i|C + /e|cj
According to [6, Theorem 3.5 and Theorem 3.6] the operator A^,« is self-adjoint in L2(Rn) and corresponds to the densely defined, closed sesquilinear form
[/, g] = (V/, Vg)
(L2(Rn))n — a(/|C,g|C)L2(C),
dom o*,a = H i(Rn).
Observe that the normal derivatives of the functions in dom may have a jump at the interface C or, more precisely, that / G dom A^,« is contained in H2(Rn) if and only if /i|C = — dVe /e|C. We also recall that the essential spectrum of the operator is [0, ro) and that the negative spectrum consists of a finite number of eigenvalues of finite multiplicity; see [6,7]. In the following, we fix some point n such that
n G p(Ai>a) f (—ro, 0). (8)
In Proposition 3.3 below, we specify a boundary triple {L2(C), r0, ri} for the adjoint of the densely defined, closed, symmetric operator
S/ = —A/, domS = {/ G H2(Rn) : /|c = °}, (9)
such that the free or unperturbed Schrodinger operator
Afree/ = —A/, dom Afree = H2(Rn),
corresponds to the kernel of the first boundary mapping r0. Note that the operator in (7) is a self-adjoint extension of S and that the defect numbers dim(ran(S ^ i)x) are infinite. Hence, the abstract considerations in Section 2 ensure that there exists a self-adjoint operator or relation 8^,« such that
A*,a = S* [ ker(ri — 65,«ro). (10)
The parameter and further properties of the operator A5,a will be discussed in Lemma 3.4 and Theorem 3.5 below.
Some further notations and preparatory results are required before Proposition 3.3 can be stated and proved. Consider the densely defined, closed, symmetric operators
Sj/j = —A/j, dom Sj = H(2(Qj),
and
Se/e = —A/e, dom Se = Hj^Qe),
in L2(Qj) and L2(Qe), respectively. Their adjoints are given by the maximal operators
Sj*/j = —A/j, domS* = {/j G L2(Qj) : —A/j G L2(Qj)},
and
Sf = -Afe, domSe* = f G L2(^) : -Afe G L2(^)},
where the expressions -Af and -Afe are understood in the sense of distributions. It is important to note that H2(^i) and H2(He) are proper subsets of the maximal domains dom S* and dom S*, respectively, and that the symmetric operator S in (9) is an infinite-dimensional extension of the orthogonal sum Sj © Se, which is also a symmetric operator in L2 (Rn) = L2(Qi) © L2(Qe). Recall from [27,40] that the trace maps admit continuous extensions to the maximal domains (equipped with the graph norms):
domS* 9 fj M file G H-1/2(C), domS* 9 f M dVif»|c G H-3/2(C),
and
domS* 9 fe m fe|c G H-1/2(C), domS* 9 fe M dvefe|e G H-3/2(C).
Furthermore, consider the self-adjoint extensions Af and Af of Sj and Se, respectively, corresponding to Dirichlet boundary conditions on C:
Af fj = -Afj, dom Af = HK^j) n H2(a),
and
Af fe = -Afe, dom Af = Hl(^e) n H2(^).
Since Af and Af are both non-negative, it is clear that n in (8) belongs to p(Af) n p(Af), and hence, we have the direct sum decompositions
dom S* = dom Af + ker(S* - n) = (H1(ftj) n H2(a)) + ker(S* - n) (11)
and
dom Se* = dom Af + ker(Se* - n) = (H^) n H2(^e)) + ker(Se* - n). (12) We agree to decompose functions fj G dom S* and fe G dom S* in the form
fj = ff + fj7 and fe = ff + fe, (13)
where ff G dom Af, fj7 G ker(S* - n), and fef G dom Af, fe7 G ker(Se* - n). In the following, we often make use of the operators
I ! I
i = (-Ac + I) 4 and i-1 = (-Ac + I)-4,
where Ae denotes the Laplace-Beltrami operator on C. Both mappings i and i-1 are regarded as isomorphisms
i : Hs(C) M Hs-1 (C) and i-1 : H*(C) M Ht+2(C) for s, t G R, and also as operators that establish the duality
^)L2(C) = 1/2(e)
for ^ G H 1/2(C) and ^ G H-1/2(C), when the spaces H 1/2(C) and H-1/2(C) are equipped with the corresponding norms. Note also that i-1 can be viewed as a bounded self-adjoint operator in L2(C) with ran i-1 = H 1/2(C) and that i with domain dom i = H 1/2(C) is an unbounded self-adjoint operator in L2(C) with 0 G p(i).
Now we have finally collected all necessary notation to state the first lemma.
Lemma 3.1. Let S be the densely defined, closed, symmetric operator in (9). Then the adjoint S * of S is given by
S */ = -A/,
domS* = {/ = / G domS* x domS*, /j|c = /e|^ (14)
Further, let
To/ = i-1/|c and Ti/ = —/f |c + dve /eD |C) (15)
for / = (/j,/e)T G dom S* and with /jD, /f as in (13). Then {L2(C), Y0, Yi} is a boundary triple for S* with the property Af © Af = S* \ ker Y0.
Proof. The assertions in Lemma 3.1 will be proved with the help of Theorem 2.2. To this end, we set
T/ = -A/,
domT = j/ = / G domS* x domS*, /j|c = /e|cj ,
and consider the boundary mappings Y0, Yi : domT ^ L2(C) in (15). First of all, we note that item (i) in Theorem 2.2 is satisfied with the self-adjoint operator A0 = Af © Af since for any function / = (/j,/e)T G dom(Af © Af) c H2(Qj) x H2(Qe) one has /j G domS*, /e G dom S*, and /j|C = /e|C. In order to see that the mapping
(Y) : domT ^ L2(C) x L2(C) (16)
is suq'ective, let ^ G L2(C). Since the Neumann trace map is surjective from H2(Qj)nH0i(Qj) onto Hi/2(C) and from H2(Qe) f H0(Qe) onto Hi/2(C), there exist /f G dom Af and /f G dom Af such that dVi/f |c = dve/f |c = — 2G Hi/2(C). Next, we choose /j7 G ker(S* — n) and /en G ker(Se* — n) such that /C|c = /|c = ¿^ G H-i/2(C), which is possible by the surjectivity of the trace map from the maximal domain onto H-i/2(C); cf. [27,40,41]. Now, it is easy to see that / = (/f + /j7, /f + /en)T G domT satisfies
Y0 / = i-1/|c = ^ and Yi/ = — i(d,i /f |c + |c ) = and hence the map (16) is onto. Next, we verify that the abstract second Green's identity
(T/,g)L2(R") — (/,Tg)L2(Rn) = (Y1 ^ Y0g)L2(C) — (Y0 ^ Yig)L2(C), /'g G dom T' (17)
holds. For this, it is useful to recall that Green's identity for /j = /f + /j7 and gj = gf + g? yields
(S*/j,gj)L2(Hi) (/j, S*gj)L2(Qi) (18)
= (/j |c ,dvi gf |c) H-!/2(C)xH!/2(C) — ^ |c ,gj |c) H !/2(C)xH-!/2 (C) ,
and for /e = /f + / and ge = gf + gn in the analogous form
e e e e e e
(Se*/e,ge)L2(Qe) — ^ Se*ge)L2(He)
(/e|c,dvege |c)H—l/2(C)xHl/2(C) Ze |c,ge|c)hi/2(C)xH-1/2(C).
Since T is a restriction of the orthogonal sum S* © S* and fj|e = fe|C, gj|e = ge|C for f, g G dom T, we conclude from (18) and (19) that
(Tf,g)L2(Rn) - (f,Tg)L2(R")
= (S*fj,gj)L2(ni) - ^S^L2^) + (Se*fe,ge)L2(Qe) - C^ ^e^2(Qe) = (fj|e gf |e) h-i/2(C)xHi/2(C) - fjC'|e , gj|e) H i/2(C)xH-i/2 (C)
+ (fe|e ,dve gf |e) H-1/2(C)xHl/2(C) - ff |e , ge | C ) H1/2 (C) X H-1/2 (C) = (f|egf|e + dvegfH-1/2(e)xH 1/2(e) - ff|e + dveff|e,g|e)h 1/2(e)xH-1/2(e)
= (i-1f |e,i(dvigf |e + dvegf |e))L2(e) - (i(dV<ff |e + ff |e),i-1g|e^
= (-i(dviff |e + dveff |e),i-1g|e)L2(e) - (i-1f |e, -¿(^gf |e + gf |e))L2(e)
= (Y1f, Y0g)L2(e) - (Tof, Y1g)L2(e)
holds. Thus, (17) is shown and item (iii) in Theorem 2.2 is satisfied. Hence, Theorem 2.2 implies that the symmetric operator
S? := T \ (kerXo n kerY^ (20)
is densely defined, closed and its adjoint coincides with T. We show that S coincides with the symmetric operator S in (9). Note first that Theorem 2.2 also implies that
Af © Af = T \ kerXo. (21)
Both operators, S and are restrictions of the operator in (21). We now let f = (fj, fe)T G dom(Af © Af) = ker Yo. For such f, we have
f g ker Y1 ^ dVifj|c + dvefe|e = 0 ^ f G H2(Rn) ^ f G domS.
Thus, S = S. Now, the remaining statements in Lemma 3.1 follow immediately from Theorem 2.2. □
Next, we specify the Weyl function N and the 7-field Z corresponding to the boundary triple {L2(C), Yo, Y1} in Lemma 3.1. It is clear from the definition of Yo that the 7-field acts as follows:
Z(A) : L2(C) M L2(Rn), p m fA, A G p(Af) n p(Af) = C \ [0, «>), where fA = (fj,A,fe,A)T G H2(Qj) x H2(ne) satisfies -Afj,A = Afj,A, -Afe,A = Afe,A and
fj,A|e = fe,A |e = ¿p.
In order to specify the Weyl function N, we recall the definition of the Dirichlet-to-Neumann maps Dj(A) and De(A) associated with the Laplacians on Qj and Qe, respectively. Note first that for p,^ G H-1/2(C) and A G p(Af) and p G p(Af) the boundary value problems
-Afj = f fj|e = p and - Afe = f fe |C = ^
admit unique solutions fj)A G dom S* and fe>M G dom S*. Hence, the operators
Dj>-1/2(A)fj>A|e = dvi fj,A |e, dom ^-^(A) = H-1/2(C), (22)
and
De,-1/2Gu)feJc = dve fe,A|c, dom D*,-1/2(p) = H-1/2(C), (23)
are well defined, and map H-1/2(C) into H-3/2(C). We have used the index -1/2 in the definition of the Dirichlet-to-Neumann maps in (22) and (23) to indicate that their domain is H-1/2(C). For the following, it is important that the restrictions
d(A)/a|c = dvi /a|c , dom D(A) = H 1(C),
and
DeM/eJc = dve /ejc, dom De (p) = H 1(C), of Di)-1/2(A) and De,-:L/2(p) to H 1(C) are densely defined, closed, unbounded operators in L2(C) that satisfy
D(A)* = Di(A) and De(p)* = De(/z)
for all A G p(Af) and for all p G p(Af), respectively. For A G p(Af) n p(Af) = C \ [0, w), it is convenient to introduce the operators
E-1/2(A) := Di,-1/2(A) + De,-1/2(A) and E(A) := D(A) + De(A). (24)
Furthermore, the restrictions of Di,-1/2(A) and De -1/2(p) to H3/2(C) will be used. These restrictions are denoted by Di 3/2(A) and De,3/2(p), respectively; they map H3/2(C) into H 1/2(C), and as above the index 3/2 is used to indicate that their domain is H3/2 (C).
Lemma 3.2. Let S be the symmetric operator in (9), let {L2(C), Y0, Y1} be the boundary triple for S * in Lemma 3.1 and fix n as in (8). For A G p(Af) n p(Af) = C \ [0, w) the operators E-1/2(A) in (24) have the property
ran(E-1/2(A) - E-1/2(n)) C H 1/2(C) (25)
and the Weyl function corresponding to the boundary triple {L2(C), Y0, Y1} is given by N (A) = -i(e-1/2(A) - E-1/2(n))i, A G C \ [0, w).
Proof Let A G p(Af) n p(Af) and let /A = (/a, /e>A) G ker(S* - A). Then /i>A|c = /e>A|c and according to (11)-(13) we have
/i,A = /Da + /Ja and /e,A = /Da + /¡A,
where /A G dom Af, /^ G ker(S* - n), /efA G dom Af and /enA G ker(Se* - n). Hence, it follows with the help of /¿,A|c = /i?A|c and /e,A|C = /e'?A|c, and the definition of the Dirichlet-to-Neumann maps that
(E-1/2(A) - E-1/2(n))lYo/A
= (Di,-1/2(A) - Di,-1/2(n) + De,-1/2(A) - De,-1/2(n))/A|c
= Di>-1/2(A)/i>A|c - Di,-1/2(n)/jA|c + De,-1/2(A)/e,A|c - De,-^)/^
= /i,A|C - /¿^C + dVe/e,A|C - dve = dvi (/i,A - /Ja)|c + dve (/e,A - /e,A)|c = /iDA|C + /fA |C
(26)
and hence
-1 (E-1/2(A) - f-x/2(n» ITq/a = /Ale + dve/dJc) = Y/ Also, the inclusion (25) follows from (26) since /A G H2^) and /eDA G H2(Qe), and hence, dvi/daIc + dve/êdaIc G H 1/2(C) in (26), and for any' '
^ G domE-i/2(A) = domE-i/2(n) = H-1/2(C),
there exists /a = (/¿,a, /e,A)T e ker(S* - A) and / = /, /e,n) e ker(S* - n) such that
/i,A|C = /e,A|C = P = An |C = fe,n |C.
□
In the following proposition, we provide a boundary triple for S * such that the operator Afree corresponds to the first boundary mapping. For this, we modify the boundary triple in Lemma 3.1 in a suitable manner.
Proposition 3.3. Let S be the densely defined, closed, symmetric operator in (9) with adjoint S * in (14), and let E(n) = iE (n)i. Then, E(n)-1 is a bounded self-adjoint operator in L2(C ) and {L2(C), r0, r1}, where
r„/ = i-1/|c + E(n)-1i(dvi /D |c + dve /D |c ) and ri/ = -¿(^ /D |c + dve /D |c ), (27)
is a boundary triple for S * with the property Afree = S * ï kerr0. For A e C \ [0, œ) the Weyl function corresponding to {L2(C), r0, r1} is given by
M (A) = -¿(E-1/2 (A) - E-1/2 (n))i(l + E(n)-1i(E-1/2(A) - E-V2(n))i).
Proof. First, we show that E(n)-1 = i-1E(n)-1i-1 is a bounded self-adjoint operator in L2(C). Observe that E(n) = Dj(n) + De(n) is injective. In fact, we assume that E(n)p = 0 for some p e H1 (C), p = 0. Then, there exists /n = /,/en)T e ker(S* - n) such that /n|C = p and hence
0 = E (n)p = E (n)/n |c = Di(n)/n |c + De(n)/en |c = dVt /n |c + /en |c. (28)
Together with /n|C = /e|C, this implies that /n e dom Afree and hence ker(Afree - n) = {0}. This is impossible, as n < 0. Thus E(n) is injective. It follows from [6, Proposition 3.2 (iii)] that E(n) is surjective. Hence, E(n)-1 is a bounded self-adjoint operator in L2(C). Since ¿-1 is also a bounded self-adjoint operator in L2(C), it is clear that E(n)-1 = i-1E(n)-1i-1 is a bounded self-adjoint operator in L2(C).
Now, let {L2(C), T0, Y1} be the boundary triple in Lemma 3.1. Note that the boundary mappings r„ and r1 in (27) satisfy:
T„\ = /I -E(n)-1^ (Yo
rj 1 ) \,T1
Hence, it follows that {L2(C), r0, r1} is a boundary triple for S*; see (2) in Section 2. Again, we let N denote the Weyl function corresponding to the boundary triple {L2(C), Y0, T1}. It is not difficult to see that the Weyl function corresponding to {L2(C), r0, r1} is given by
M (A) = N (A) (I - E(n)-1N (A))-1, A e C \ [0, œ).
Hence, the form of the Weyl function M follows from Lemma 3.2.
It remains to be shown that Afree = S * ï ker r0 holds. Assume that for some / e dom S * we have ( )
1-1/|c + E(n)-1i(dvi /D |c + dve /D |c ) = 0. (29)
As ker E-1/2(n) = {0} (this can be seen as in (28)), this is equivalent to
E-1/2(n)/|c + (dvi /D |c + dve /D |c ) =0.
Furthermore, since
E-1/2(n)/|c = Di,-1/2(n)/i7 |c + De,_1/2(n)/en |c = ^ tf |C + dve / |C holds for / decomposed as in (13), we conclude that (29) is equivalent to
dvi/n |c + dve/? |c + dvi /D |c + dve/D |c = 0,
which in turn is equivalent to
dvi fi|e + dve fe|e = 0.
Therefore, f G kerr0 if and only if f G dom Afree. □
Our next goal is to identify the self-adjoint parameter ©s>Q, such that (10) holds with the boundary triple in Proposition 3.3.
Lemma 3.4. Let S be the densely defined, closed, symmetric operator in (9) with adjoint S* in (14), and let {L2(C), r0, r} be the boundary triple in Proposition 3.3. Then
©s,a = i(Di,3/2(n) + De,3/2(n) - «)i(l - E(n)-1i(Di,3/2(n) + D^n) - a)i) '
is an unbounded self-adjoint operator in L2(C) such that the Schrodinger operator As>Q, in (7) corresponds to ©s,a, that is,
As,« = S* \ ker(ri - ©5;«ro). (30)
Proof. We make use of the fact that the boundary triple {L2(C), Yo, Y1} in Lemma 3.1 and the boundary triple {L2(C), r0, r1} in Proposition 3.3 are related via
To = Yo - E(n)-1Yi and ri = Yi, (31)
and we also make use of the operator
As,« = i(Di,3/2(n) + De,3/2(n) - a)i, dcmAi)ft = H2(C). (32)
Our first task is to show that
As,« = S* \ ker(Y1 - A^Yo) (33)
holds. In fact, f G ker(Y1 - As,«Y0) if and only if f G dom S* and
i(Di,3/2(n) + De,3/2(n) - a)f |e = ff |e + ff |e),
where f |e G domDi)3/2(n) = domDe,3/2(n) = H3/2(C), together with elliptic regularity, also implies that f = (f ,fe)T with f G H2(^) and fe G H2(Qe). With f decomposed as in (13) we have
(Di,3/2(n) + De,3/2(n) - a)f |e = /?|e + d,efe"|e - af |e.
Therefore, f G ker(Y1 - As«Yo) if and only if f = (f, fe)T G dom S* with f G H2and
fe G H2(^e) and
dvifi7|e + dvefe|e - af |e = -($,<ff |e + dVe ff |e), and the latter can be rewritten in the form
fi|e + fe |e = af |e.
We have shown (33), and as As,« is a self-adjoint operator in L2(Rn), it follows that As,« in (32) is an unbounded self-adjoint operator in L2(C). Next, we consider the operator
©s,a = As,«(/ - E(n)-1A5,„)-1 (34)
on its natural domain; note that ker(1 - £(n)-1As,«) = {0} as otherwise E(n)p = As,«p for some non-trivial p G H2(C), which is a contradiction to a = 0. Now, we assume that
/ G ker(r - 6^0). Then, (31) and (34) yield
Y/ - A^Yo/ = r/ - A5,«(ro + E(n)-1Y1)/ = r1/ - A5,«(ro + Efa)-1^)/ = r1/ - Av(ro + E(n)-165,«ro)/ = r1/ - Av(/ + E(n)-1A5,«(/ - e(n)-%,«)-1)ro/
= r1/ - a5,«(/ - e(n)-%,«)-1ro/ = r1/ - 6vro/
= 0,
and hence / G ker(Y1 - A5,aYo). The converse inclusion is shown in the same way and therefore
ker(r - 6«s,aro) = ker(Y1 - A^Yo)
and thus the extensions
S* [ ker(r1 - 6^) and S* f ker(Y1 - A^Yo),
coincide. Therefore, (33) implies (30). Since A5,a is self-adjoint in L2(Rn), it also follows from (30) that 6,5,0, is self-adjoint in L2 (C). Moreover, as S in (9) coincides with the intersection of Afree and A5,a, that is, Afree and A5,a are disjoint, and since Afree and A5,a are not transversal, one concludes that 65,a is an unbounded operator in L2 (C); cf. (6). □
We are now able to obtain some immediate and important consequences from the previous considerations, well-known results for boundary triples and Weyl functions [3,4] and the resolvent estimates in [6,42].
Theorem 3.5. Let S be the densely defined, closed, symmetric operator in (9) with adjoint S* in (14), let {L2(C), ro, r1} be the boundary triple in Proposition 3.3 with
Afree = S* \ kerTo,
and let y and M be the y-field and Weyl function corresponding to {L2(C), ro, r1}. Furthermore, let 65,a be as in Lemma 3.4 so that
A,,« = S* \ ker(r - 6,,aro).
Then, the following assertions hold for all AG [0, w):
(i) A G ap(A,,a) if and only if 0 G ap(6,,a - M(A));
(ii) A G p(A5,a) if and only if 0 G p(65,a - M(A));
(iii) for all A G p(A,,a) the resolvent formula
(A,,« - A)-1 - (Afree - A)-1 = y(A) (6,,a - M(A))-1y(A)*
is valid, and the resolvent difference of A5,a and Afree belongs to the Schatten-von Neumann ideal &p(L2(Rn)) for all p > ;
(iv) for all £ G p(6,,a) the operator (6,,a - £)-1 belongs to the Schatten-von Neumann ideal 6P(L2(C)) for all p > ^. '
Acknowledgements
J. Behrndt and V. Lotoreichik gratefully acknowledge financial support by the Austrian Science Fund (FWF): Project P 25162-N26. J. Behrndt also wishes to thank Professor Igor
Popov for the pleasant and fruitful research stay at the ITMO University in St. Petersburg in September 2015. V. Lotoreichik was also supported by the Czech Science Foundation (GACR) under the project 14-06818S.
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