Hiatus perturbation for a singular Schrödinger operator with an interaction supported by a curve in \mathbb{R}^3
aa r X i v : . [ m a t h - ph ] D ec Hiatus perturbation for a singular Schr¨odingeroperator with an interaction supported by acurve in R a,b and S. Kondej c a) Nuclear Physics Institute, Academy of Sciences, 25068 ˇReˇznear Prague, Czech Republicb) Doppler Institute, Czech Technical University, Bˇrehov´a 7,11519 Prague, Czech Republicc) Institute of Physics, University of Zielona G´ora, ul. Szafrana 4a,65246 Zielona G´ora, Poland [email protected] , [email protected] We consider Schr¨odinger operators in L ( R ) with a singular interac-tion supported by a finite curve Γ. We present a proper definition ofthe operators and study their properties, in particular, we show thatthe discrete spectrum can be empty if Γ is short enough. If it is notthe case, we investigate properties of the eigenvalues in the situationwhen the curve has a hiatus of length 2 ǫ . We derive an asymptoticexpansion with the leading term which a multiple of ǫ ln ǫ . Singular Schr¨odinger operators with interactions supported by manifolds ofa lower dimension are not a new topic; their properties were investigatedalready in the beginning of the nineties [BT] or even earlier in cases of a par-ticular symmetry, see, e.g., [AGS, Sha]. Recently, a new motivation appearedwhen people realized that such operators with attractive interaction provideus with a model of “leaky” quantum graphs which have the nice properties of1raph description of various nanostructures — see, e.g., the proceedings vol-umes [BCFK], [EKST], and references therein — but they are more realistictaking possible tunneling between the involved quantum wires into account.A series of papers devoted to this problem started by [EI] and we refer to[Ex] for a bibliography; among the questions addressed were geometricallyinduced spectral properties [EI], scattering [EK05], approximations by pointinteraction Hamiltonians [EN, BO] or strong-coupling asymptotic behavior[EY02]. Another result concerns perturbations of such Hamiltonians causedby alterations of the interaction support. In [EY03] the asymptotic behaviorfor the eigenvalue shift was derived in the situation when the support is amanifold of codimension one with a “hole” which shrinks to zero, in partic-ular, a curve in R with a hiatus; it was shown that in the leading orderthe perturbation acts as a repulsive δ interaction with the coupling strengthproportional to the hole measure (in particular, the hiatus length).The aim of this paper is to analyze the analogous question in the situationwhere the codimension of the manifold is two, specifically, for an interactionsupported by a curve in R . The extension is by far not trivial since thecodimension of the singular interaction support influences properties of suchSchr¨odinger operators substantially [AGHH]. In our particular case we knowthat to define such a Hamiltonian for a curve in R one cannot use, in contrastto the codimension one case, the “natural” quadratic form and has to resortto appropriate generalized boundary conditions [EK02].We are going to demonstrate that the asymptotic behavior of the eigen-values with respect to the hiatus length ǫ is of the following form, λ j ( ǫ ) = λ L − s j ( λ L ) ǫ ln ǫ + o ( ǫ ln ǫ ) , j = m, ..., n , where λ L is an unperturbed eigenvalue of the Hamiltonian corresponding tothe absence of the hiatus, the indices run through a basis in the correspond-ing eigenspace so that n − m + 1 is the multiplicity of λ L , and s j ( λ L ) arecoefficients specified in Theorem 6.6. This shows that the asymptotics isin the case of codimension two is substantially different — recall that forcodimension one the second term is linear in ǫ — due to the more singularinteraction involved. The dependence on the codimension is manifested alsoin other ways. For instance, while a nontrivial and attractive interactionsupported by any manifold of codimension one gives rise to bound states, inthe situation discussed here a minimum curve length is needed to producebinding as we will demonstrate in Section 4.2 Preliminaries
As mentioned in the introduction, we are interested in generalized Schr¨odingeroperators with singular potentials supported by sets of lower dimensions. Inour case the support of the singular potential will be a finite C smoothcurve in R of length L without self-intersections which may and may not bea loop; the corresponding Schr¨odinger operator can be formally written as − ∆ − ˜ αδ ( x − Γ) with ˜ α < . (2.1)We mark the parameter in this formal expression by a tilde to stress thatit is different from the true “coupling constant” which will introduce below.It is a natural requirement that the operator which gives a mathematicalmeaning to (2.1) should act as the Laplacian on the domain C ∞ ( R \ Γ), whichmotivates us to look for self-adjoint extensions of the symmetric operator − ˙∆ : C ∞ ( R \ Γ) L := L ( R ) such that ˙∆ f = ∆ f . The deficiency indicesof − ˙∆ are infinite, of course, and looking for operators giving a meaning to(2.1) we will restrict ourselves to a certain “local” one-parameter family ofextensions which will be specified in the next section.We have to say also something more about Γ and to introduce a familyof auxiliary “comparison” curves in its vicinity. Since Γ is C smooth byassumption it admits a parameterization by the arc length. This means thatΓ is a graph of a C function γ : [0 , L ] R such that | ˙ γ ( s ) | = 1, where ˙ γ stands for the derivative. Moreover, we assume that(a) there exist c > µ > | γ ( s ) − γ ( t ) | ≥ | s − t | (1 − c | s − t | µ ) for c | s − t | µ < . If Γ is not a closed curve then, of course, one of its endpoints is given by γ (0). However, if Γ is a loop then there is no such natural “starting point”and we assume that the above property is valid independently of the way theloop is parametrized.We will say that a family of curves { Γ d } is neighboring with Γ if theyare graphs of functions γ d : [0 , L ] R with the following properties for any s ∈ [0 , L ] and d small enough(b1) | γ ( s ) − γ d ( s ) | = d ,(b2) | ˙ γ ( s ) − ˙ γ d ( s ) | = O ( d ) as d → γ ( s ) − γ d ( s ) is perpendicular to t d ( s ) := ˙ γ d ( s ) ;the error term is assumed to be uniform on [0 , L ]. For instance, if the Fr´enetframe (t , n , b) is defined globally for Γ then any family of “shifted” curvedefined as the graphs of γ + η n + η b : [0 , L ] R , | η | = q η + η = d , is neighboring with Γ. In this section we shall construct an operator corresponding to the formalexpression (2.1). As mentioned above, it will be defined as a self-adjoint ex-tension of − ˙∆. To this aim we will follow the scheme proposed by Posilicano[Po01, Po04] which generalizes the standard Krein’s theory. The self-adjointextensions are parametrized in it by Birman-Schwinger-type operators enter-ing into expression of their resolvents. As usual, such a resolvent consists of a‘free’ term and a ‘perturbative’ remainder. Since we are in three dimensions,the ‘free’ resolvent is given by R z = ( − ∆ − z ) − : L L , z ∈ ρ ( − ∆),which is an integral operator with the kernel G z ( x, y ) = 14 π e −√− z | x − y | | x − y | . (3.1)As there is no risk of confusion we will use the same notation G z ( · ) for thefunction of a scalar argument, i.e. G z ( ρ ) = e −√− z | ρ | (4 π | ρ | ) − , ρ ∈ R \ { } .To construct the second term of the resolvent we need an embeddingto the Hilbert space associated with the support of our singular potential.Such a space is naturally defined by L ( R , µ Γ ), where µ Γ denotes the Diracmeasure on Γ. It is convenient, however, to use a natural identification L ( R , µ Γ ) ≃ L ( I ), I ≡ (0 , L ) which we will do in the following. Is is wellknown that R z defines a unitary map between L and W , ( R ) ≡ W , .Moreover, with reference to the Sobolev theorem we claim that the traceoperator τ : W , → L ( I ) is continuous, and consequently, the followingoperators, R z := τ R z : L → L ( I ) , R ∗ z : L ( I ) → L , where R ∗ z is the adjoint to R z , are continuous as well.4 .1 Birman–Schwinger operator The mentioned Birman-Schwinger operators are defined as symmetric oper-ators Θ z in L ( I ) parameterized by z ∈ ρ ( − ∆) and satisfying the pseudo-resolvent equivalenceΘ w − Θ z = ( w − z )R w R ∗ z for w , z ∈ ρ ( − ∆) . (3.2)Furthermore, if the set Z := { z ∈ ρ ( − ∆) : Θ − z exists and is bounded } isnonempty then the following operator R z ; α = R z − R z (Θ z ) − R ∗ z for z ∈ Z (3.3)defines the resolvent of certain self-adjoint extension of − ˙∆, cf. [Po01]. Ouraim is now to find such an operator Θ z satisfying (3.2) and corresponding tothe singular potential defined by a certain coupling constant. The explicitform of such operator was discussed by [BT] but for our purpose it is usefulto derive the other, albeit equivalent form of Θ z (recall that in the mentionedpaper the potential was defined more generally, as a function on I ; in ourmodel it is just a “coupling” constant which we will denote as α .)The most natural way of determining Θ z would be to take the embeddingof R z to L ( I ), as it is done on the codimension one case [BEKˇS]. However,the explicit formula for G z , the kernel of R z , shows that the expression τ R ∗ z does not make sense because G z has a singularity; to make use of the ap-proach sketched above the singularity has to be removed by an appropriateregularization. To put it differently, the operator τ can not be canonically ex-tended onto L which is the range of R ∗ z . On the other hand, to preserve theequivalence (3.2) we have to consider a special type of regularization whichdoes not depend on z . Using standard facts from the Sobolev space theory[RS] we claim that R ∗ z f ∈ W , ( R \ Γ) ∩ C ∞ ( R \ Γ) for f ∈ L ( I ), thusthe embedding R ∗ z f ↾ Γ d is a C ∞ function on I ; recall that Γ d was introducedin Section 2 as a neighbooring curve with Γ. With these facts in mind weintroduce a logarithmic regularization defined through the pointwise limits˘ f ( s ) = lim d → h R ∗ z f ↾ Γ d ( s ) + 12 π f ( s ) ln d i for s ∈ I . (3.4)By virtue of the following lemma, the relation (3.4) defines a function be-longing to L ( I ) provided f ∈ W , ( I ).5 emma 3.1 The operator Q z defined by the relation ( Q z f )( s ) := 14 π (cid:18)Z I f ( t ) − f ( s ) | t − s | d t + f ( s ) ln 4 s ( L − s ) (cid:19) + Z I R z ( s, t ) f ( t )d t , (3.5) where R z ( s, t ) := G z ( γ ( s ) − γ ( t )) − (4 π | s − t | ) − , (3.6) maps W , ( I ) L ( I ) and Q z f = ˘ f . In the following we will also employ the decomposition of the kernel of thelast term in (3.5), namely R z ( s, t ) = A z ( | s − t | ) + D z ( s, t ), where A z ( ρ ) := G z ( ρ ) − (4 π | ρ | ) − , D z ( s, t ) := G z ( γ ( s ) − γ ( t )) − G z ( s − t ) . (3.7)Before starting the proof of the lemma let us make a couple of comments.The operator Q z defined on the space W , ( I ) with the topology inheritedfrom L ( I ) is essentially self-adjoint. Taking its closure we obtain its uniqueself-adjoint extension in L ( I ) for which we will use the same notation.Furthermore, Q z satisfies the relation (3.2). Indeed, let us note first thatthe first resolvent formula for R z can be extended by the continuity to theequivalence R ∗ w − R ∗ z = ( w − z ) R w R ∗ z . Since R w R ∗ z f is a continuous functionas an element of W , we can take the limitlim d → (R ∗ w − R ∗ z ) f ↾ Γ d = ( w − z ) R w R ∗ z f ↾ Γ d , f ∈ W , ( I ) , w, z ∈ ρ ( − ∆) , which consequently, in view of (3.4), gives( Q w − Q z ) f = ( w − z )R w R ∗ z f . (3.8) Proof of Lemma 3.1.
We break the argument into two parts:
Step 1:
Assume first that Γ is a straight line segment, i.e. we have D z = 0.Let us decompose R ∗ z f ↾ Γ d into the sum of two terms,R ∗ z f ↾ Γ d ( s ) = Z I G dz ( s − t ) f ( t ) d t = Z I S d ( s − t ) f ( t ) d t + Z I A dz ( s − t ) f ( t ) d t , (3.9)where G dz ( ρ ) := G z (( d + ρ ) / ) and S d ( ρ ) := (4 π ( d + ρ ) / ) − , A dz ( ρ ) := G z ( ρ ) − S d ( ρ ) . Z I S d ( s − t ) f ( t )d t = Z I ( f ( t ) − f ( s )) S d ( s − t )d t + f ( s ) Z I S d ( s − t )d t . (3.10)The integrated function in the first term at the r.h.s. of the last relation canbe bounded by | f ( t ) − f ( s ) | (4 π | t − s | ) − which belongs to L ( I ) in view of thefact that f ∈ W , ( I ). Hence employing Lebesque’s dominated convergencetheorem we can conclude that the first term at the r.h.s. of (3.10) tends to R I ( f ( t ) − f ( s ))(4 π | t − s | ) − d t for d →
0. To handle the second term wedecompose it integrating separately along I δ = I δ,s := { t ∈ I : | t − s | < δ } and I cδ := I \ I δ for δ small enough. As a result we arrive at Z I δ S d ( s − t ) d t = 12 π [ ln ω d ( δ ) − ln d ] , ω d ( a ) := a + ( a + d ) / , (3.11) Z I cδ S d ( s − t ) d t = 14 π ln ω d ( s ) ω d ( L − s ) − π ln ω d ( δ ) . (3.12)Combining the above results with (3.10) we getlim d → (cid:18)Z I S d ( s − t ) f ( t ) d t + 12 π f ( s ) ln d (cid:19) = 14 π (cid:18)Z I f ( t ) − f ( s ) | t − s | d t + f ( s ) ln 4 s ( L − s ) (cid:19) . (3.13)To obtain the result we have to handle the limit R I A dz ( s, t ) f ( t ) as d → d → Z I A dz ( s, t ) f ( t )d t = Z I A z ( s, t ) f ( t )d t , (3.14)which reproduces the remaining term at the r.h.s. of (3.5). Putting together(3.13) and (3.14) we arrive at the sought result,lim d → (cid:18) R ∗ z f ↾ Γ d ( s ) + 12 π f ( s ) ln d (cid:19) = 14 π (cid:18)Z I f ( t ) − f ( s ) | t − s | d t + f ( s ) ln 4 s ( L − s ) (cid:19) + Z I A z ( s, t ) f ( t )d t . tep 2: Consider next the general case when Γ is a finite curve which mayand may not be closed. Then we employ the decompositionlim d → (R ∗ z f ↾ Γ d ( s )) = Z I G z ( γ d ( s ) − γ ( t )) f ( t ) d t = Z I G dz ( s − t ) f ( t ) d t + Z I D dz ( s, t ) f ( t ) d t , where D dz ( s, t ) := G z ( γ d ( s ) − γ ( t )) − G dz ( s − t ) and γ d is the function whosegraph is the neighbooring curve Γ d with Γ. Using the result of the first stepand the limit lim d → Z I D dz ( s, t ) f ( t ) d t = Z I D z ( s, t ) f ( t ) d t discussed in Remark 8.2 below we get the claim, concluding thus the proofof the lemma.Now we can express the resolvent of the Hamiltonian . As was alreadymentioned one can parameterize self-adjoint extensions of certain symmetricoperators by means of operators satisfying the pseudo-resolvent formula. Inour model we are specifically interested in extensions of − ˙∆ : C ∞ ( R \ Γ) L := L ( R ). The operators Θ z = Q z − α are suitable candidates for the roleof Birman–Schwinger operators because they satisfy the relation (3.2). Theparameter α ∈ R appearing here will be referred to as the coupling constant.It is certainly different from the ˜ α appearing in (2.1); it is enough to noticethat the absence of the interaction is associated with the value α = ∞ .To complete the argument one has to make an a posteriori claim that theset Z is nonempty, which will be done in Section 4 below. Summing up thediscussion, the operator R z ; α = R z − R z ( Q z − α ) − R ∗ z for z ∈ Z (3.15)is in view of the mentioned result in [Po01] the resolvent of a self-adjointextension of − ˙∆. We will regard it as a rigorous counterpart of the formalHamiltonian (2.1) and denote it in the following as H α, Γ . Q z The need to introduce a renormalization makes the use of Birman–Schwingerapproach more complicated than in the codimension one case. In addition,8he way we have chosen above, with the limit taken over a family of compar-ison curves “parallel” to the entire Γ is not a particularly elegant one. It ispossible to think of other regularizations defining Q z by Z I G z (˜ γ d ( s ) − γ ( t )) f ( t ) d t + 12 π f ( s ) ln d , where ˜ γ d correspond to another curve family. One possibility is to considercurves which coincide with Γ everywhere except in the vicinity of the sin-gularity, the point with s = t , where they have a recess the size of which iscontrolled by the parameter d . To describe this and other possible regular-izations we will look at a more general class into which all of them fit.Given s we consider a family of C curves ˜Γ d,s which are graphs of ˜ γ d,s ≡ ˜ γ d : [0 , L ] R with d := | ˜ γ d ( s ) − γ d ( s ) | = k ˜ γ d − γ k ∞ . The assumptions(b) of Section 2 will be then replaced by the following modified ones; for any t ∈ [0 , L ] and d small enough,( e b1) | γ ( t ) − ˜ γ d ( t ) | = O ( d ) as d → e b2) | ˙ γ ( t ) − ˙˜ γ d ( t ) | = O ( d ) as d → e b3) γ ( s ) − ˜ γ d ( s ) is perpendicular to ˜t d ( s ) := ˙˜ γ d ( s ) ,where we suppose also that the error terms are uniform on [0 , L ]. Let us stressthat, in distinction to Γ d , the curve Γ d,s is in general not parameterized byits arc length. Then we have the following theorem the proof of which wepostpone to Section 8. Theorem 3.2
Under the stated assumptions, ( Q z f )( s ) = lim d → (cid:20)Z I G z ( γ ( s ) − ˜ γ d ( t )) j d ( t ) f ( t ) dt + 12 π f ( s ) ln d (cid:21) , (3.16) where j d ( s ) := (cid:0)P i =1 ( ˙˜ γ d,i ( s ) (cid:1) / . We have said that in distinction to the codimension one case an attractiveinteraction supported by a finite curve may not induce bound states. Theaim of this section is to make this claim precise and to find conditions under9hich the Hamiltonian H α, Γ has a nonempty discrete spectrum. Since thesingular potential in our model is supported by a compact set it is easy tocheck the stability of the essential spectrum, σ ess ( H α, Γ ) = σ ess ( − ∆) = [0 , ∞ ) , see [BT]. This means that the negative halfline can contain only the discretespectrum σ d ( H α, Γ ), and consequently the set Z of (3.3) is nonempty. Lookingfor negative eigenvalues we put z = λ with λ <
0. We employ the Birman–Schwinger philosophy, specifically the following result, λ ∈ σ d ( H α, Γ ) ⇔ ker( Q λ − α ) = { } , (4.1)where the multiplicity of λ is equal to dim ker( Q λ − α ) – cf. [Po04]. Inaddition, the eigenfunctions of H α, Γ corresponding to an eigenvalue λ aregiven by ψ λ = R ∗ λ φ λ , where φ λ ∈ ker( Q λ − α ) . (4.2)It is well-known that a point interaction in R always attractive, i.e. itgives rise for any α ∈ R to exactly one bound state with the eigenvalue ξ = ξ ( α ) = − − πα + ψ (1)) , (4.3)where ψ (1) = − , ... is Euler–Mascheroni constant, cf. [AGHH]. Askingabout existence of bound states in our model, one may naively expect thesame behavior as the perturbation is again of codimension two. It appears,however, that it is not so due to the presence of the third dimension whichmakes a finite curve in R “more singular” than a point in R . We will showthat if the length of curve is small enough then our system has no boundstates. We need the following result auxiliary result. Lemma 4.1
Suppose that σ d ( H α, Γ ) = ∅ . Then the ground-state eigenvalue λ = min { λ ∈ σ d ( H α, Γ ) } is simple and the corresponding eigenfunction ψ := ψ λ is a multiple of a positive function. Proof.
We will employ the form associated with − Q z + α , cf. [BT], ς z [ f ] = − Z I × I | f ( s ) − f ( t ) | G z ( γ ( s ) − γ ( t )) d t d s − Z I | f ( s ) | ( a z ( s ) + α ) d s , (4.4)10here, with the notation introduced above, a z ( s ) := − Z I δ G z ( γ ( s ) − γ ( t )) d t + Z I cδ (cid:18) π | s − t | − G z ( γ ( s ) − γ ( t )) (cid:19) d t − π log 2 δ . Let us note that the inequality || f ( s ) | − | f ( t ) || ≤ | f ( s ) − f ( t ) | implies ς z [ | f | ] ≤ ς z [ f ] . For a fixed z < inf σ ( H α, Γ ) the form ς z is strictly positive. Using the Beurling–Deny criterion together with the other results from [RS, vol. II, p. 204] wefind that ( − Q z + α ) − is positivity preserving. Moreover, the operators R z ,R z , and R ∗ z are positivity improving because they are defined by means ofthe kernel which is strictly positive. Hence referring to (3.15) we concludethat the resolvent R z ; α of H α, Γ is positivity improving, and using [RS] againwe get the sought positivity of ψ .We begin the discussion concerning the existence of bound state by ana-lyzing the simplest case, namely the situation when Γ is a line segment. Lemma 4.2
Suppose that Γ is a finite line segment of length L . If L < πα then there H α, Γ has no bound states. On the other hand, if L > π e πα − ψ (1) then there exists at least one bound state. Proof.
In order to prove the absence of bound states under the condition
L < πα it suffices in view of (4.1) to show thatsup σ ( Q λ ) < π ln L . (4.5)It is clear that the value sup σ ( Q λ ) is achieved by ( Q λ φ , φ ), where φ ∈ ker( Q λ − α ) is the normalized function corresponding to the ground state ψ by the relation ψ = R ∗ λ φ , cf. (4.2). Using the expression of Q λ given byLemma 3.1 we get the following asymptotics, ψ ↾ Γ d ( s ) = R ∗ λ φ ↾ Γ d ( s ) ≈ − π φ ( s ) ln d − ( Q λ φ )( s ) as d → , s ∈ I . (4.6)11ince ψ can be chosen positive by Lemma 4.1 and φ ∈ W , ( I ), as wewill demonstrate in Section 5 below, we come to the conclusion that φ ispositive as well because the leading term of (4.6) is determined by φ . Thusto estimate sup σ ( Q λ ) it is sufficient to consider the expression ( Q λ f, f ) forpositive functions f only. Using the relation (3.5) again we find( Q λ f, f ) = Z I × I ξ f ( s, t ) d s d t + Z I × I A λ ( s − t ) f ( s ) f ( t ) d s d t + (4 π ) − Z I f ( s ) ln 4 s d s , (4.7)where ξ f ( s, t ) := ( f ( t ) − f ( s )) f ( s )4 π | s − t | . A straightforward calculation yields the estimate ξ f ( s, t ) − ξ f ( t, s ) = − ( f ( t ) − f ( s )) π | s − t | ≤ , which in turn leads to the following inequality Z I × I ξ f ( s, t ) d s d t = Z I Z s The method we have used in the proof is not particularly pre-cise which explains the gap of π e − ψ (1) ≈ . 56 in the ratio of the lengths L for which the existence and nonexistence of the discrete spectrum wereestablished above.Let us discuss next the general situation and consider a nontrivial curvewhich again may or may not be closed. To be concrete we consider a familyof curves which are connected subsets of a fixed Γ corresponding to differentsubintervals of the arc length parameter. The deviation of each such curvefrom the corresponding straight segment is measured by the quantity D λ = 0given by (3.7). Since | γ ( s ) − γ ( t ) | ≤ | s − t | in view of the used parameterizationand the function ρ e − ρ /ρ is decreasing we find that D λ > D λ ( s, t ) ≤ π (cid:18) | γ ( s ) − γ ( t ) | − | s − t | (cid:19) . (4.9)Using the assumption (a) and mimicking the argument of [EK02] one canshow that the operator with kernel defined by the r.h.s. of (4.9) is bounded(or even Hilbert-Schmidt) and denote its norm as D , (see also Remark 8.3).Proceeding as in the proof of Lemma 4.2 we arrive at the conclusion thatthe operator H α, Γ has no bound states if L < πα − D . On the other hand,using arguments borrowed from [EK02] we can claim that in the case L > π e πα − ψ (1) the bound states do not disappear when a segment is replaced bya curve of the same length, since the bending acts as an effective attractiveinteraction. Summarizing this discussion we have the following result. Theorem 4.4 For a fixed α ∈ R in the described situation, there exists L α > such that the operator H α, Γ has no discrete spectrum for L < L α . Onthe other hand, if L > π e πα − ψ (1) then there is at least one bound state. Regularity of eigenfunction Before we proceed to our main result we need as a preliminary to investigatethe regularity of φ ∈ ker( Q λ L − α ), where λ L is an eigenvalue of H α, Γ ; specif-ically we will demonstrate that this function belongs to W , . The proof ofthis claim is involved and we divide it into several steps. To simplify the pre-sentation we will show first the regularity of the corresponding eigenfunctionin the case when Γ is a loop, and then we will comment on an extension ofthe result. The idea is to compare the loop with a circle of the same length.Suppose Γ is a closed curve satisfying the assumptions of Section 2 and Γ c is a circle of the length L ; up to Euclidean transformations, Γ c is thus thegraph of the function γ c ( · ) = L π (cos πL ( · ) , sin πL ( · ) , 0) : [0 , L ] R . Theoperator Q z can be defined in analogy with (3.4), i.e. Q z = T cz + D cz , (5.1)where T cz f = lim d → h R ∗ z f ↾ Γ cd + 12 π f ln d i for s ∈ (0 , L ) (5.2)and D cz is given by the kernel D cz ( s, t ) := G z ( γ ( s ) − γ ( t )) − G z ( γ c ( s ) − γ c ( t ));in the above expression Γ cd stands for a neighbooring curve with Γ c and theproperties described in Section 2. Lemma 5.1 Assume that the assumption (a) is satisfied; then for any func-tion f ∈ L ( I ) we have D cz f ∈ W , ( I ) . The proof is quite technical and we postpone it to the appendix. Lemma 5.2 For φ ∈ ker( Q λ L − α ) we have ( T cz − α ) φ ∈ W , ( I ) . Proof. Using the pseudo-resolvent formula (3.8) for w = λ L and the fact thatR w R ∗ z φ ∈ W , ( I ) we get ( Q z − α ) φ ∈ W , ( I ). Applying then the result ofthe previous lemma and the decomposition (5.1) we get the claim.This allows us finally to formulated the indicated result. Proposition 5.3 Any eigenfunction φ ∈ ker( Q λ L − α ) belongs to W , ( I ) . Proof. Using the radial symmetry valid for Γ c one finds T cz f = X k ∈ Z b k ( z ) f k e i πk ( · ) /L , f k are Fourier coefficients of f and b k ( z ) ∈ C . Hence T cz commuteswith the derivative operator D , which implies for z ∈ C + k Dφ k ≤ C k ( T cz − α ) Dφ k = C k D ( T cz − α ) φ k < ∞ , (5.3)where C is a positive constant; we have used the fact that T cz − α is invertiblewith a bounded inverse in combination with Lemma 5.2. The sought claimfollows directly from (5.3). Remark 5.4 In a similar way one can deal with the situation when thecurve Γ is not closed; the idea is to compare it to a circular segment. Tobe precise we introduce Γ c which is, as before, a circle defined as the graphof γ c : [0 , L + d ] R , d > c,r being the graph of γ c,r : [0 , L ] R such that γ c,r ( s ) = γ c ( s ) for any s ∈ [0 , L ]. In analogy with(5.1) we can decompose the operator Q z corresponding to Γ as Q z = T c,rz + D c,rz , where T c,rz and D c,rz are defined as in (5.1) but by means of γ c,r , with thevariable appropriately restricted. The proofs of Lemmata 5.1, 5.2 can bemimicked directly for the operators T c,rz and D c,rz . On the other hand, theproof of Proposition 5.3 requires some comments. Given δ > I : L (0 , L + δ ) L (0 , L ) and ˘ I ∗ : L (0 , L ) L (0 , L + δ ). Using the explicit form of Q z given by (3.5) we can easily checkthat T c,rz = ˘ I T cz ˘ I ∗ . Now can repeat the reasoning which leads to (5.3) butinstead of the norm k · k in L (0 , L ) we consider the norm k · k δ in L ( δ, L − δ ),where δ > k Dφ k δ ≤ C k ( T c,rz − α ) Dφ k δ = C k ˘ I ( T cz − α ) ˘ I ∗ Dφ k δ = C k D ˘ O ( T cz − α ) ˘ O ∗ φ k δ < ∞ . This means that Proposition 5.3 extends to the case when Γ is not a loop,by which the eigenfunction regularity is finally established generally. Now we finally come to our main topic. In this section we consider the eigen-value problem for a curve with a short hiatus. Suppose that we have thesystem with the singular interaction supported by a curve Γ of length L and15atisfying the assumptions of Section 2. Naturally we have to exclude thetrivial case assuming that H α, Γ has bound states; we know from Theorem 4.4a sufficient condition for that is L > π e πα − ψ (1) . For simplicity we will sup-pose first that there is exactly one bound state with corresponding eigenvalue λ L ; the generalization will be provided at the end of this section.Consider now a family of curves Γ ǫ which coincides with Γ everywhereapart a short hiatus placed symmetrically w.r.t x = Γ( s ), in other words,Γ ǫ is a graph of function γ ǫ : [0 , s − ǫ ) ∪ ( s + ǫ, L ] R and γ ǫ ( s ) = γ ( s )for s ∈ [0 , s − ǫ ) ∪ ( s + ǫ, L ]. In the following we will use the notations I cǫ ≡ (0 , s − ǫ ) ∪ ( s + ǫ, L ) and I ǫ for ( s − ǫ, s + ǫ ). Our aim is to deriveasymptotics of eigenvalue λ ( ǫ ) of H α, Γ ǫ for ǫ small. Of course, we may expectthat λ ( ǫ ) → λ L for ǫ → 0. Since, as discussed above, the eigenvalue problemcan be reduced in view of 4.1 to analysis of the Birman–Schwinger operator,we will seek the function λ ( ǫ ) such that ker( Q ǫλ ( ǫ ) − α ) is nontrivial where Q ǫλ denotes the Birman–Schwinger operator corresponding to Γ ǫ . The firststep towards that is to relate Q λ and Q ǫλ . It is convenient to introduce thenatural embedding maps acting between L ( I ) and L ( I cǫ ). Let I ǫ stand forthe canonical embedding from L ( I ) to L ( I cǫ ) and I cǫ for its adjoint actingfrom L ( I cǫ ) to L ( I ). We will also use the abbreviation Q ǫcλ := I cǫ Q ǫλ I ǫ . Lemma 6.1 The asymptotic expansion ( Q ǫλ I ǫ f, I ǫ f ) = ( Q λ f, f ) + 2 π | f ( s ) | ǫ ln ǫ + o ( ǫ ln ǫ ) (6.1) holds for ǫ → ∞ and any f ∈ D ( Q λ ) ∩ W , ( I ) . Proof. Let us first note that for any f ∈ L ( I ) such that I ǫ f ∈ D ( Q ǫλ ) wehave ( Q ǫλ I ǫ f, I ǫ f ) = ( Q ǫcλ f, f ) and Q ǫcλ f can be decomposed as, Q ǫcλ f = lim d → (cid:20)Z I cǫ G λ ( γ d ( · ) − γ ( t )) f ( t )d t + 12 π ln d f (cid:21) χ cǫ = Q λ f − J f − J ′ f − T f , (6.2)where J f := (cid:20) lim d → Z I ǫ G λ ( γ d ( · ) − γ d ( t )) f ( t )d t (cid:21) χ cǫ , (6.3) J ′ f := (cid:20) lim d → Z I cǫ G λ ( γ d ( · ) − γ ( t )) f ( t )d t (cid:21) χ ǫ T f = lim d → (cid:20)Z I ǫ G λ ( γ d ( · ) − γ ( t )) f ( t ) d t + 12 π ln d f (cid:21) χ ǫ . The symbols χ ǫ , χ cǫ stand for the characteristic functions of I ǫ and I cǫ , respec-tively. Let us show how the last term of (6.2) emerges. In analogy with theproof of Lemma 3.1, see eq. (3.5), one shows that( T f )( s ) = 14 π f ( s ) ln 4( s − s + ǫ )( s − s + ǫ ) χ ǫ ( s )+ (cid:18)Z I ǫ f ( t ) − f ( s )4 π | s − t | d t + Z I ǫ R λ ( s, t ) f ( t ) d t (cid:19) χ ǫ ( s ) ; (6.4)recall that R λ ( s, t ) = lim d → R λd ( s, t ) = lim d → ( G λ ( γ d ( s ) − γ ( t )) − S d ( s − t ))and S d ( s − t ) = (4 π ( d + ( s − t ) ) / ) − . Using the identity Z I ǫ ln 4( s − s + ǫ )( s − s + ǫ )d s = 8 ǫ ln 2 ǫ together with the expansion f ( s ) = f ( s ) + o (1) for s ∼ s , which can beperformed in view of the fact that f ∈ W , ( I ) we obtain( T f, f ) = 2 π | f ( s ) | ǫ ln ǫ + O ( ǫ ) ; (6.5)note that the second and the third term of (6.4) can be uniformly boundedw.r.t. s , cf. Remark 8.3 below, and consequently, they contribute in (6.5) tothe error term only. The latter depends on λ , however, it is important for usthat it can be uniformly bounded together with its derivative being O ( ǫ ).Let us now consider the term J f appearing in (6.2). Applying to (6.3)the decomposition G λ ( γ d ( s ) − γ ( t )) = S d ( s − t ) + R λd ( s, t ) we get by astraightforward computation( J f )( s ) = (cid:18) ( f ( s ) + o ǫ (1)) j ǫ ( s ) + Z I ǫ R λ ( s, t ) f ( t ) d t (cid:19) χ cǫ ( s ) , (6.6)where the error term o ǫ (1) means the asymptotics for ǫ → 0, and j ǫ ( s ) := lim d → Z I ǫ S d ( s − t ) f ( s ) d s = 14 π ln | s − s | + ǫ | s − s | − ǫ for | s − s | > ǫ . J f, f ) = ( f ( s ) + o ǫ (1)) Z I cǫ j ǫ ( s ) f ( s ) d s + Z I cǫ Z I ǫ R λ ( s, t ) f ( t ) f ( s ) d t d s . (6.7)By an analogous argument as in the first step of proof we can check that thelast term of (6.7) contributes to O ( ǫ ). To handle the first term at the r.h.s.of (6.7) we integrate by parts Z I cǫ j ǫ ( s ) f ( s )d s = ˆ j ǫ ( s ) f ( s ) | I cǫ − Z I cǫ ˆ j ǫ ( s ) f ′ ( s )d s , (6.8)ˆ j ǫ ( s ) := 14 π X k = {− , } k ( | s − s | − kǫ ) h ln( | s − s | − kǫ ) − i | s − s | s − s . Consequently, the first term of (6.8) takes the following formˆ j ǫ ( s ) f ( s ) | I cǫ = − π ǫ ln ǫf ( s ) + o ( ǫ ln ǫ ) for s ∈ I cǫ . Furthermore, the second term can be estimated as (cid:12)(cid:12)(cid:12) Z I cǫ ˆ j ǫ ( s ) f ′ ( s ) d s (cid:12)(cid:12)(cid:12) ≤ k ˆ j ǫ k L ( I cǫ ) k f k W , ( I ) . One can check directly that k ˆ j ǫ k L ( I cǫ ) = o ( ǫ ln ǫ ). Summarizing, we get Z I cǫ j ǫ ( s ) f ( s ) d s = − π ǫ ln ǫf ( s ) + o ( ǫ ln ǫ ) , and consequently, ( J f, f ) = − π | f ( s ) | ǫ ln ǫ + o ( ǫ ln ǫ ). Using the fact that( J f, f ) = ( J ′ f, f ) in combination with (6.5) we get the claim.With the above lemma we are ready to demonstrate the following result. Lemma 6.2 The eigenvalues of Q ǫλ tend to the eigenvalues of Q λ for ǫ → .Moreover, if ǫ and λ − λ L are small enough the operator Q ǫλ has an eigenvalue η ( λ, ǫ ) which tends to α as ǫ → and λ → λ L . roof. Since Q ǫcλ is the natural embedding of Q ǫλ to space L ( I ) it suffices toshow the claim for Q ǫcλ . Let us make the following decomposition( Q ǫcλ f, f ) = (( Q ǫcλ f, f ) − ( Q λ f, f )) + (( Q λ f, f ) − ( Q λ L f, f )) + ( Q λ L f, f ) . (6.9)The convergence of the first term at the r.h.s. of (6.9) is proved in the pre-vious lemma, precisely we have 0 < ( Q λ f, f ) − ( Q ǫcλ f, f ) → ǫ → Q λ − Q λ L → λ → λ L and the convergence is understood in thenorm sense. Since α is an eigenvalue of Q λ L we get the final claim.Relying on the last lemma and 4.1 we state that the eigenvalue of H α, Γ ǫ approaches the eigenvalue of H α, Γ . Furthermore, for ǫ and λ − λ L smallenough we can introduce the eigenprojector P ǫλ onto the spaces spanned bythe eigenvectors of Q ǫλ corresponding to η ( λ, ǫ ). In the following we will usethe representation of P ǫλ by means of the resolvent of Q ǫλ , i.e. P ǫλ = 12 πi I C R ǫλ ( z ) d z with R ǫλ ( z ) := ( Q ǫλ − z ) − (6.10)and C := { α + r e iϕ : ϕ ∈ [0 , π ) , < r < | α |} . Furthermore, R ǫλ ( z ) satisfiesa first-resolvent-type identity of the following form R ǫλ ( z ) = I ǫ R λ ( z ) I cǫ + R ǫλ ( z )( I ǫ Q λ − Q ǫλ I ǫ ) R λ ( z ) I cǫ . (6.11)According to the previous discussion the eigenvalue λ ( ǫ ) is a zero of thefunction η ( λ, ǫ ) − α by (4.1), i.e. we have η ( λ ( ǫ ) , ǫ ) − α = 0. Thus to derivethe asymptotics of λ ( ǫ ) the most natural way is employ the implicit functiontheorem which requires to know the asymptotics of η ( λ, ǫ ). Let us note that η ( λ, ǫ ) = ( Q ǫλ P ǫλ I ǫ φ, P ǫλ I ǫ φ ) k P ǫλ I ǫ φ k − , (6.12)where φ ∈ ker( Q λ L − α ). To recover the asymptotics of η ( λ, ǫ ) we write it as η ( λ, ǫ ) = A ( λ, ǫ ) + B ( λ, ǫ ) + C ( λ, ǫ ) − α , (6.13)where A ( λ, ǫ ) := η ( λ, ǫ ) − ( Q ǫcλ φ, φ ), B ( λ, ǫ ) := ( Q ǫcλ φ, φ ) − ( Q λ φ, φ ), and C ( λ, ǫ ) := ( Q λ φ, φ ) − ( Q λ L φ, φ ). The asymptotics of B ( λ, ǫ ) was alreadyderived in Lemma 6.1, now we want to find the asymptotics of A ( λ, ǫ ). Tothis aim we first prove the following lemma.19 emma 6.3 As ǫ → and λ − λ L → , we have the relation k ( P ǫλ − I ) I ǫ φ k = O ( ǫ ln ǫ ) + O ( λ − λ L ) . (6.14) Proof. Applying (6.10), (6.11) and using the fact that I cǫ I ǫ φ = χ cǫ φ we getby a straightforward calculation k ( P ǫλ − I ) I ǫ φ k (6.15) ≤ kI ǫ ( P λ χ cǫ − I ) φ k + 12 π I C k R ǫλ ( z )( I ǫ Q λ − Q ǫλ I ǫ ) R λ ( z ) χ cǫ φ k| d z | . To handle the first r.h.s. term in (6.15) we employ the triangle inequality, kI ǫ ( P λ χ cǫ − I ) φ k ≤ kI ǫ ( P λ − P λ L ) χ cǫ φ k + kI ǫ ( P λ L χ cǫ − I ) φ k . (6.16)Using the pseudo-resolvent formula (3.2) and the representation of the pro-jector by means of the resolvent we get kI ǫ ( P λ − P λ L ) χ cǫ φ k = O ( λ − λ L ).Moreover, since P λ L is the eigenprojector onto the space spanned by φ wehave kI ǫ ( P λ L χ cǫ − I ) φ k = O ( ǫ ). To estimate the second term of (6.15) weconsider k ( I ǫ Q λ − Q ǫλ I ǫ ) f k where f ∈ D ( Q λ ) ∩ W , ( I ). Using (6.2), (6.3)and the results of Lemma 6.1 we obtain k ( I ǫ Q λ − Q ǫλ I ǫ ) f k = k J f k = | f ( s ) |O ( ǫ ln ǫ ) . (6.17)Moreover, let us note that the function g = R λ ( z ) χ cǫ φ belongs to W , ( I ).Indeed, to see this consider ( Q λ − z ) g which is a function from W , ( I )because χ cǫ φ ∈ W , ( I ) by Lemma 5.3. Now we can repeat the argumentsfrom Lemmata 5.1 and 5.3, i.e. we have D cλ g ∈ W , ( I ), and therefore( T cλ − z ) g ∈ W , ( I ), so finally k Dg k ≤ C k D ( T cλ − z ) g k < ∞ ;see (5.3). Since g ∈ W , ( I ) it makes sense to consider g ( s ) and to employ(6.17). Consequently, the second term in (6.15) can be estimated as k R ǫλ ( z )( I ǫ Q λ − Q ǫλ I ǫ ) g k ≤ r k ( I ǫ Q λ − Q ǫλ I ǫ ) g k = O ( ǫ ln ǫ ) , (6.18)where r = | z − α | . Combining these estimates we get the sought claim.The asymptotics for A ( λ, ǫ ) is given in the following lemma.20 emma 6.4 In the limits ǫ → and λ − λ L → we have | A ( λ, ǫ ) | = | η ( λ, ǫ ) − ( Q ǫλ I ǫ φ, I ǫ φ ) | = o ( ǫ ln ǫ )+ O (( λ − λ L ) )+ O ( ǫ ln ǫ ) O ( λ − λ L ) . Proof. Let us note that using the properties of the eigenprojector and theasymptotics k P ǫλ I ǫ φ k = 1 + O ( ǫ ln ǫ ) + O ( λ − λ L ) which is a consequence ofthe previous lemma we can estimate | A ( λ, ǫ ) | = | ( Q ǫλ P ǫλ I ǫ φ, P ǫλ I ǫ φ ) k P ǫλ I ǫ φ k − − ( Q ǫλ I ǫ φ, I ǫ φ ) |≤ k Q ǫλ ( P ǫλ − I ) I ǫ φ kk ( P ǫλ − I ) I ǫ φ k (1 + O ( ǫ ln ǫ ) + O ( λ − λ L )) . (6.19)The asymptotics for k ( P ǫλ − I ) I ǫ φ k was explicitly derived in Lemma 6.3.Furthermore, proceeding in analogy with (6.15) we find k Q ǫλ ( P ǫλ − I ) I ǫ φ k ≤ k Q ǫλ I ǫ ( P λ χ cǫ − I ) φ k + 12 π I C k Q ǫλ R ǫλ ( z )( I ǫ Q λ − Q ǫλ I ǫ ) R λ ( z ) χ cǫ φ k| d z | . (6.20)Mimicking now the argument of (6.16) we estimate the first term on the r.h.s.of (6.20) obtaining k Q ǫλ I ǫ ( P λ χ cǫ − I ) φ k ≤ k Q ǫλ I ǫ ( P λ − P λ L ) χ cǫ φ k + k Q ǫλ I ǫ ( P λ L χ cǫ − I ) φ k . (6.21)Furthermore k Q ǫλ I ǫ ( P λ − P λ L ) χ cǫ φ k ≤ k ( Q ǫλ I ǫ − I ǫ Q λ )( P λ − P λ L ) χ cǫ φ k + kI ǫ Q λ ( P λ − P λ L ) χ cǫ φ k , (6.22)where k ( Q ǫλ I ǫ − I ǫ Q λ )( P λ − P λ L ) χ cǫ φ k = O ( ǫ ln ǫ ) O ( λ − λ L ) and kI ǫ Q λ ( P λ − P λ L ) χ cǫ φ k = O ( λ − λ L ) + O ( ǫ ). Proceeding analogously as with the secondterm of (6.21) we get k Q ǫλ I ǫ ( P λ L χ cǫ − I ) φ k = O ( ǫ ln ǫ ), and therefore k Q ǫλ I ǫ ( P λ χ cǫ − I ) φ k = O ( ǫ ln ǫ ) + O ( λ − λ L ) + O ( ǫ ln ǫ ) O ( λ − λ L ) . (6.23)To handle the second term in (6.20) let us note that k Q ǫλ R ǫλ ( z ) k ≤ | z | r ≤ | α | r , hence using (6.18) we obtain k Q ǫλ R ǫλ ( z )( I ǫ Q λ − Q ǫλ I ǫ ) R λ ( z ) χ cǫ φ k = O ( ǫ ln ǫ ) , k Q ǫλ ( P ǫλ − O ǫ φ k = O ( ǫ ln ǫ ) + O ( λ − λ L ) + O ( ǫ ln ǫ ) O ( λ − λ L ) . (6.24)Putting the above results together and applying Lemma 6.3 to (6.19) we getthe claim of the lemma.Putting the results of Lemmata 6.1, 6.3, 6.4 together and applying (6.12)we get η ( λ, ǫ ) = 2 π | φ ( s ) | ǫ ln ǫ + ( Q λ φ, φ ) + o ( ǫ ln ǫ ) + O (( λ − λ L ) ) + O ( ǫ ln ǫ ) O ( λ − λ L ) , (6.25)as the hiatus half-length ǫ and the eigenvalue difference λ − λ L tend to zero.Let us keep the notation λ L for the eigenvalue of H α, Γ which means thatker ( Q λ L − α ) is nontrivial and suppose as before that φ ∈ ker ( Q λ L − α ) is thenormalized function in L ( I ). Our goal is to find an asymptotic expressionfor the eigenvalue of H Γ ǫ ,α by means of λ L and φ . Theorem 6.5 The eigenvalue of H α, Γ admits the following asymptotic ex-pansion as ǫ → , λ ( ǫ ) = λ L − ω ( κ L ) | φ ( s ) | ǫ ln ǫ + o ( ǫ ln ǫ ) , (6.26) where ω ( λ L ) = 16 κ L (cid:18)Z I × I e − κ L | γ ( s ) − γ ( t ) | φ ( s ) φ ( t ) d s d t (cid:19) − , κ L := p − λ L . Proof. Due to (4.1) the eigenvalue λ ( ǫ ) is determined by the conditionker ( Q ǫλ ( ǫ ) − α ) = { } . It is convenient to putˆ η ( λ, δ ) ≡ η ( λ, ǫ ) − α : U × C C where δ := ǫ ln ǫ and U is a neighborhood of zero. Our aim is to find where the function ˆ η vanishes. Using the fact that ˆ η ( λ L , 0) = 0 and ˆ η ∈ C × C ∞ and relying onthe implicit function theorem we can evaluate λ ( ǫ ) = λ L − ( ∂ δ ˆ η ) | θ L ( ∂ λ ˆ η ) − | θ L δ + o ( δ ) , θ L ≡ ( λ L , . 22o find ∂ δ ˆ η | θ L we use the asymptotics (6.25)1 δ (cid:16) ˆ η ( λ L , δ ) − ˆ η ( λ L , (cid:17) → π | φ ( s ) | as δ → . To find the other derivative we use (6.25) to state( ∂ λ ˆ η ) | θ L = ( ∂ λ ( Q λ φ, φ )) | θ L . On the other hand the derivative of Q λ w.r.t. λ coincides with the derivativeof G λ because the regularization we made was independent of the spectralparameter λ ; therefore we have( ∂ λ Q λ ( s, t )) | θ L = 18 πκ L e − κ L | γ ( s ) − γ ( s ) | . (6.27)Putting together (6.27), (6.27) we get the sought result.As the final step of is this section we return to the general questionand extend the above theorem to the case when H α, Γ have more than oneeigenvalue; recall that since Γ is finite by assumption we have ♯σ d ( H α, Γ ) < ∞ .Suppose that λ L < λ L ≤ ... ≤ λ NL , N ∈ N are the eigenvalues of H α, Γ and { φ i } Ni =1 is the corresponding eigenfunction system which is assumed to benormalized. Given λ L ∈ σ d ( H α, Γ ) define m ( λ L ) := min { j = 1 , ..., N : λ j = λ L } , n ( λ L ) := max { j = 1 , ..., N : λ j = λ L } and the matrix C ( λ L ) given by[ C ( λ L )] ij := φ i ( s ) φ j ( s ) ω ij , i, j = m ( λ L ) , ..., n ( λ L ) , where ω ij ( λ L ) := (cid:18)Z I × I e − κ L | γ ( s ) − γ ( t ) | φ i ( s ) φ j ( t ) d s d t (cid:19) − . Using this notation we can state our main result: Theorem 6.6 Let λ L ∈ σ d ( H α, Γ ) . Then the corresponding eigenvalues of H α, Γ ǫ have the following asymptotic expansion, λ j ( ǫ ) = λ L − s j ( λ L ) ǫ ln ǫ + o ( ǫ ln ǫ ) , m ( λ L ) ≤ j ≤ n ( λ L ) , as ǫ → , where s j ( λ L ) are the eigenvalues of matrix C ( λ L ) . The proof of essentially repeats the reasoning used above; the only newelement is that different eigenfunctions corresponding to the same eigenvalue λ L correspond to the appropriate scalar products. This consequently leadsto the appearance of the matrix C ( λ L ) which reduces to | φ ( s ) | ω ( λ L ) if λ L is a simple eigenvalue. 23 Concluding remarks First we note that the hiatus perturbation of a curve in R can be regardedas an effective repulsive interaction. The presence of a hiatus pushes theeigenvalues up which can be easily seen from (6.26) since ω ( λ L ) > ǫ ln ǫ < ǫ . This might be expected, of course, becausethe interaction supported by a curve in R is attractive as manifested by thefact that it produces bound states, at least if the curve is sufficiently long,cf. Theorem 4.4.Comparing the eigenvalue asymptotics (6.26) with the analogous resultfor a curve in R derived in [EY03] we can see the difference in the firstasymptotic term, which in the codimension one case behaves as O ( ǫ ) in con-trast to O ( ǫ ln ǫ ) obtained here. The former result is a natural consequenceof the additive character of the singular potential manifested by the sum-type quadratic form representation of the corresponding Hamiltonian. Sucha representation does not exists if the potential is supported by a set of codi-mension two. To find a self-adjoint realization of the δ interaction in thiscase we have to perform, for instance, a logarithmic regularization of theappropriate quantities, and consequently, the eigenvalue asymptotics w.r.t.the length of the hiatus, as well as its derivation, are more involved. To prove Theorem 3.2 we need the following lemma. Lemma 8.1 Given s ∈ [0 , L ] corresponding to ˜ γ d,s = ˜ γ d and d > , andmaking | s − t | small we have | γ ( s ) − ˜ γ d ( t ) | = ( s − t ) (1 + O ( d )) + d + O (( s − t ) ) . Proof. An elementary cosine formula gives | γ ( s ) − ˜ γ d ( t ) | = | ˜ γ d ( s ) − ˜ γ d ( t ) | + d − ι ( s, t ) , where ι ( s, t ) := ( γ ( s ) − ˜ γ d ( s ) , ˜ γ d ( s ) − ˜ γ d ( t )). Note that ι ( s, t ) | t = s = 0 and ∂ t ι ( s, t ) | t = s = 0 holds in view of the assumption ( e b3). Furthermore, theTaylor expansion in the corresponding “shifted” points of the coordinate24rojections of the curve ˜ γ d yields | ˜ γ d ( s ) − ˜ γ d ( t ) | = X i =1 ˙˜ γ d,i ( θ i ) ( s − t ) + O (( s − t ) . Using the Taylor expansion again and combining it with the asymptoticsgiven by ( e b1), ( e b2) and the fact that P i =1 ˙ γ i ( s ) = 1 we get the claim. Proof of Theorem 3.2 . The first step is show that in the following limitslim d → Z I (cid:20) G z ( γ ( s ) − ˜ γ d ( t )) − π | γ ( s ) − ˜ γ d ( t ) | (cid:21) f ( t ) d t (8.1)and lim d → Z I (cid:20) | γ ( s ) − ˜ γ d ( t ) | − s − t ) + d ) / (cid:21) f ( t ) d t (8.2)we can interchange the limit with the integration. Using the inequality | (e − κx − x − | ≤ κ for κ and x positive we can use the dominated con-vergence to prove claim concerning (8.1). To handle the second limit we canuse Lemma 8.1 and show that1 | γ ( s ) − ˜ γ d ( t ) | − s − t ) + d ) / = (cid:16) ( s − t ) O ( d ) + O (( s − t ) ) (cid:17)(cid:16) ( s − t ) + d (cid:17) − ≤ const . (8.3)Therefore using the dominated convergence again we can perform the inter-change in (8.2). The resulting limit of the sum of both the expressions (8.1)and (8.2) is given bylim d → Z I (cid:20) G z ( γ ( s ) − ˜ γ d ( t )) − π (( s − t ) + d ) / (cid:21) f ( t ) d t = Z I A z ( s − t ) f ( t )d t + Z I D z ( s, t ) f ( t ) d t , (8.4)where A z and D z are defined in Lemma 3.5. Repeating the argument fromthe proof of this lemma, see (3.11) and (3.12), we getlim d → (cid:20)Z I δ π (( s − t ) + d ) / f ( t ) d t + 12 π f ( s ) ln d (cid:21) = 14 π (cid:16) Z I f ( t ) − f ( s ) | t − s | d t + ln 4 s ( L − s ) f ( s ) (cid:17) . (8.5)25he final step is to note that Z I G z ( γ ( s ) − ˜ γ d ( t ))(1 − j d ( t )) f ( t ) d t = o (1)as d → 0, because j d = 1 + O ( d ) and R I G z ( γ ( s ) − ˜ γ d ( t )) f ( t ) d t has a singu-larity of the type f ( s ) ln d . Combing this with (8.4) and (8.5) we concludethe proof of Theorem 3.2 Remark 8.2 Using the same arguments as in (8.3) we can estimate1 | γ d ( s ) − γ ( t ) | − s − t ) + d ) / = (cid:16) ( s − t ) O ( d ) + O (( s − t ) ) (cid:17)(cid:16) ( s − t ) + d (cid:17) − ≤ const , (8.6)which directly implies |D dz ( s, t ) | = | G z ( γ d ( s ) − γ ( t )) − G dz ( s − t ) | ≤ const , for any s ∈ [0 , L ]. Proof of Lemma 5.1. Recall that our goal is to show that Z I D cz ( s, t ) f ( t ) d t ∈ W , for f ∈ L ( I ) , (8.7)where D cz ( s, t ) := G z ( γ ( s ) − γ ( t )) − G z ( γ c ( s ) − γ c ( t ))and G z ( ρ ) = e −√− z | ρ | (4 π | ρ | ) − . We proceed in three steps: Step 1: We show that the following inequality holds || γ ( s ) − γ ( t ) | − | γ c ( s ) − γ c ( t ) || ≤ c | s − t | µ +1 (8.8)for c | s − t | µ < 1. By a straightforward calculation one can find that | γ c ( s ) − γ c ( t ) | = L π (cid:18) − cos 2 π ( s − t ) L (cid:19) . Consequently, there exists a positive constant ˜ c such that | γ c ( s ) − γ c ( t ) | ≥ | s − t | (1 − ˜ c | s − t | ) (8.9)26or ˜ c | s − t | < 1. Using the above inequality we have | γ ( s ) − γ ( t ) | ≤ | s − t | ≤ | γ c ( s ) − γ c ( t ) | + ˜ c | s − t | . (8.10)On the other hand, using the assumption (a) we obtain | γ ( s ) − γ ( t ) | ≥ | s − t | − c | s − t | µ +1 ≥ | γ c ( s ) − γ c ( t ) | − c | s − t | µ +1 . (8.11)Combining (8.10) and (8.11) we arrive at (8.8). Step 2: The aim of this partof the proof is to show the following asymptotics, D cz ( s, t ) = O ( | s − t | µ − ) . (8.12)Using (8.8), (8.9) and the assumption (a) we get | T ( s, t ) | := (cid:12)(cid:12)(cid:12)(cid:12) | γ ( s ) − γ ( t ) | − | γ c ( s ) − γ c ( t ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | s − t | µ +1 | s − t | (1 − c | s − t | µ )(1 − ˜ c | s − t | ) ≤ c | s − t | µ − , (8.13)where c is a positive constant. Furthermore, using the the exponentialfunction expansion and (8.8) we find D cz ( s, t ) = T ( s, t ) + O ( | s − t | µ +1 ) , which in view of (8.13) implies (8.12). Step 3: Let us note that for f ∈ L ( I )we have (cid:12)(cid:12)(cid:12) Z I D cz ( s, t ) f ( t ) d t (cid:12)(cid:12)(cid:12) ≤ a ( s ) k f k L ( I ) , where a ( s ) := Z I |D cz ( s, t ) | d t . Using (8.12) we claim that a ′ ( s ) is an integrable function, and therefore wecan use the dominated convergence to show that Z I (cid:12)(cid:12)(cid:12) D s Z I D cz ( s, t ) f ( t ) d t (cid:12)(cid:12)(cid:12) d s = Z I (cid:12)(cid:12)(cid:12) Z I D s D cz ( s, t ) f ( t ) d t (cid:12)(cid:12)(cid:12) d s ≤ Z I Z I | D s D cz ( s, t ) | d t d s k f k L ( I ) < ∞ , (8.14)where D s stands for the derivative; in the above estimates we have again used(8.12) to check that the last term in the cahin (8.14) is finite. This finallyproves (8.7), and by that Lemma 5.1.27 emark 8.3 Let us note that in analogy with (8.13) we can estimate (cid:12)(cid:12)(cid:12) | γ ( s ) − γ ( t ) | − | s − t | (cid:12)(cid:12)(cid:12) ≤ c | s − t | µ +1 | s − t | (1 − c | s − t | µ ) ≤ c | s − t | µ − , (8.15)where we have again used the assumption (a). This implies |R λ ( s, t ) | = | G λ ( γ ( s ) − γ ( t )) − (4 π | s − t | ) − | ≤ c | s − t | µ − . Acknowledgments The research was partially supported by Ministry of Education, Youth andSports of the Czech Republic under the project LC06002. S.K. thanks to herson Antek that he had allowed her to work on this paper in the first monthsof his life. References [AGHH] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden: SolvableModels in Quantum Mechanics , 2nd printing, AMS, Providence, R.I.,2004.[AGS] J.-P. Antoine, F. Gesztesy, J. Shabani: Exactly solvable models ofsphere interactions in quantum mechanics, J. Phys. A20 (1987), 3687-3712.[BCFK] G. Berkolaiko, R. Carlson, S. Fulling, P. Kuchment, eds.: QuantumGraphs and Their Applications , Contemporary Math., vol. 415, AMS,Providence, R.I., 2006.[BEKˇS] J.F. Brasche, P. Exner, Yu.A. Kuperin, P. ˇSeba: Schr¨odinger op-erators with singular interactions, J. Math. Anal. Appl. (1994),112-139.[BO] J.F. Brasche, K. Oˇzanov´a: Convergence of Schr¨odinger operators, SIAM J. Math. Anal. (2007), 281-297.[BT] J.F. Brasche, A. Teta: Spectral analysis and scattering theory forSchr¨odinger operators with an interaction supported by a regular curve,in Ideas and Methods in Quantum and Statistical Physics , ed. by S. Al-beverio, J.E. Fenstadt, H. Holden, T. Lindstrøm, Cambridge Univ. Press1992, pp. 197-211.[Ex] P. Exner: Leaky quantum graphs: a review, a contribution to [EKST]; arXiv: 0710.5903 [math-ph] R , Ann. H. Poincar´e (2007), 241-263.[EI] P. Exner, T. Ichinose: Geometrically induced spectrum in curved leakywires, J. Phys. A34 (2001), 1439-1450.[EKST] P. Exner, J. Keating, P. Kuchment, T. Sunada, A. Teplyaev, eds.: Analysis on Graphs and Applications , Proceedings of an Isaac NewtonInstitute programme, AMS, a volume in preparation[EK02] P. Exner, S. Kondej: Curvature-induced bound states for a δ inter-action supported by a curve in R , Ann. H. Poincar´e (2002), 967-981.[EK05] P. Exner, S. Kondej: Scattering by local deformations of a straightleaky wire, J. Phys. A38 (2005), 4865-4874.[EN] P. Exner, K. Nˇemcov´a: Leaky quantum graphs: approximations bypoint interaction Hamiltonians, J. Phys. A36 (2003), 10173-10193.[EY02] P.Exner, K.Yoshitomi: Asymptotics of eigenvalues of the Schr¨odingeroperator with a strong δ -interaction on a loop, J. Geom. Phys. (2002),344–358.[EY03] P. Exner, K. Yoshitomi: Eigenvalue asymptotics for the Schr¨odingeroperator with a δ -interaction on a punctured surface, Lett. Math. Phys. (2003), 19-26; erratum (2004), 81-82.[Ka] T. Kato: Perturbation Theory for Linear Operators , 2nd edition,Springer, Berlin 1976.[Po04] A. Posilicano: Boundary triples and Weyl Functions for singular per-turbations of self-adjoint operator, Methods of Functional Analysis andTopology (2004), 57-63.[Po01] A. Posilicano: A Krein-like formula for singular perturbations of self-adjoint operators and applications, J. Funct. Anal. (2001), 109-147.[RS] M. Reed and B. Simon: Methods of Modern Mathematical Physics,II. Fourier Analysis, Self-Adjointness, IV. Analysis of Operators , Aca-demic Press, New York 1975, 1978.[Sha] J. Shabani: Finitely many delta interactions with supports on concen-tric spheres, J. Math. Phys.29