Approximation of point interactions by geometric perturbations in two-dimensional domains
aa r X i v : . [ m a t h - ph ] A ug Approximation of point interactions by geometricperturbations in two-dimensional domains
D.I. Borisov ∗ , P. Exner Institute of Mathematics, Ufa Federal Research Center, Russian Academy of Sciences, Ufa, Russia,Bashkir State Pedagogical University named after M. Akhmulla, Ufa, Russia,University of Hradec Kr´alov´e, Hradec Kr´alov´e, Czech Republic [email protected] Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University in Prague,Bˇrehov´a 7, 11519 PragueNuclear Physics Institute, Czech Academy of Sciences, 25068 ˇReˇz near Prague, Czech Republic [email protected]
Abstract
We present a new type of approximation of a second-order elliptic operator in a planardomain with a point interaction. It is of a geometric nature, the approximating familyconsists of operators with the same symbol and regular coefficients on the domain with asmall hole. At the boundary of it Robin condition is imposed with the coefficient whichdepends on the linear size of a hole. We show that as the hole shrinks to a point and theparameter in the boundary condition is scaled in a suitable way, nonlinear and singular,the indicated family converges in the norm-resolvent sense to the operator with the pointinteraction. This resolvent convergence is established with respect to several operator normsand order-sharp estimates of the convergence rates are provided.
Operators with singular, point-like perturbations attracted attention in the early days of quan-tum mechanics as idealized models for Hamiltonians of systems in which the interaction isconcentrated in a small area [10]. The advantage of such an idealized description is that one cansimplify considerably spectral analysis of such operators. From the mathematical point of view,point interactions are easy to deal with in the case of ordinary differential operators where theyare described by appropriate boundary conditions. In the practically important cases of dimen-sions two and three the question is more difficult, however, and it took time before Berezin andFaddeev [3] showed how to describe these operators in terms of self-adjoint extensions. In therecent decades point interactions were a subject of an intense interest; we refer to the monograph[1] for the presentation of the theory and an extensive bibliography.The key thing in application of the point interaction models is to understand how they canbe approximated by operators with regular coefficients. This is again easy in case of the ordinary ∗ Corresponding author describing systems in one spatial dimension, where such an interaction isthe limit of naturally scaled potentials. In dimensions two and three the procedure is much moresubtle because scaling of the coefficients leads generically to a trivial result. One has to use aparticular way of nonlinear scaling starting from the situation when the initial operator has aspectral singularity at the threshold of the continuous spectrum; a physicist would speak abouta particular way of ‘coupling constant renormalization’.With the importance of the point interaction models in mind, it would be certainly useful tohave approximations other than the standard one mentioned above and described in [1, Sec.I.1and I.5]. The aim of the present paper is to present an alternative approximation to two-dimensional point interactions, which is of a geometric nature. It employs families of operatorswith the same differential expression as the unperturbed one but restricted to the exterior ofa small hole containing the support of the point interaction; at the boundary of the hole weimpose Robin boundary condition with a coefficient depending in a singular way on a parametercharacterizing the linear size of the hole. Shrinking the hole and scaling properly the parameterin the boundary condition, we obtain an operator family that converges, in the norm-resolventsense, to an operator with a point interaction in the domain without the hole. The convergenceis established in terms of several operators norms and for each of them, we obtain order-sharpestimates for the convergence rate. As a consequence, we also obtain the convergence of theoperator spectra and the associated spectral projectors.It should be noted that elliptic boundary value problems with small holes represent a clas-sical example in the singular perturbation theory. Situations when the boundary of the hole issubject to one of the classical boundary conditions were investigated, for instance, in [12, 13],where typically a weak or strong resolvent convergence was established. Asymptotic expansionsfor solutions to such problems, in the first place, for the corresponding eigenvalues and eigenfunc-tions, were found under appropriate smoothness assumptions. Recent results on norm-resolventconvergence in the boundary homogenization theory [4, 5, 6] inspired results on the same conver-gence for operators in domains with small holes [7, 8], however, in these papers a fixed classicalboundary conditions was always imposed at the boundary of the hole, in particular, the Robincondition was used with the coefficient independent of the hole size.As we have mentioned, we work in the two-dimensional setting. The way we present our resultis particular and general at the same time. The particularity reflects the fact that we deal withapproximation of a single point interaction, and moreover, that our result also has a limitation:our approximation applies only to point interactions which are, roughly speaking, attractiveenough in the sense made precise by the condition (2.16) below; recall that, for instance, a singlepoint-interaction perturbation of the Laplacian in the plane is always attractive [1, Sec. I.5]. Onthe other hand, our proof is of a local nature and there would no problem to extend it to covera finite number of point interactions; each of them will be approximated by an appropriate holewith Robin boundary and all estimates in Theorem 2.1 would remain true, even if the involvedexpression would be pretty cumbersome. It is also possible to consider operators with infinitelymany point interactions provided the mutual distances between their supports have a positivelower bound, but then additional restrictions on the coefficients in the differential expressionwould be needed.What is more important, in contrast to standard treatment as one can find in the monograph This claim applies to the so-called δ potentials, there are more singular point interactions in one dimensionfor which the approximation is a far more complicated matter, see e.g. [2, 9]. B can also describe quasi-periodic boundary conditions, our result covers, in the usual Floquetway, infinite periodic systems of point interactions with a single perturbation in the period cellwithout any additional assumptions.Secondly, our unperturbed operator is not just a Laplacian or a Schr¨odinger operator, buta general second-order elliptic operator; in Sec 4.2 we define a point perturbation of such anoperator properly and show that it is self-adjoint. Our results thus allows us to treat singularperturbations of more general systems such as magnetic Schr¨odinger operator or Hamiltonianswith a weight in the kinetic term, in other words, systems with a position-dependent ‘mass’.This could be of interest in solid state physics, where the effective electron mass depends onthe material and becomes nontrivial in composite structures build, say, from different types ofsemiconductors. In such models, the hole in the perturbed problem can be interpreted as alocalized defect in the material with a particular surface interaction at its boundary. Let x = ( x , x ) be Cartesian coordinates in R and Ω ⊆ R be a domain which can be bothbounded or unbounded, including the particular case of Ω = R . If the boundary of Ω isnonempty, we assume that it is C -smooth.By x we denote an arbitrary fixed point of Ω and consider its neighborhood of which we willspeak as of a hole , defined as ω ε := { x : ( x − x ) ε − ∈ ω } , where ω ⊂ R is a bounded simplyconnected set the boundary of which is C -smooth. The hole is supposed to be small, its sizebeing controlled by the positive parameter ε , and we assume that ω contains the origin of thecoordinates so that x ∈ ω ε for all ε > .The main object of our interest is the family of self-adjoint scalar second-order differentialoperators H ε = − X i,j =1 ∂∂x i A ij ∂∂x j + i X j =1 (cid:18) A j ∂∂x j + ∂∂x j A j (cid:19) + A in Ω ε := Ω \ ω ε (2.1)subject to one of the classical, ε -independent boundary conditions on ∂ Ω , B u = 0 on ∂ Ω , (2.2)and to the Robin condition on ∂ω ε that scales singularly with respect to ε as follows, ∂u∂ n = α (cid:0) ε − s ε , ln − ε (cid:1) ε ln ε u on ∂ω ε , (2.3) α ( s, µ ) := α ( s ) + µα ( s ) . (2.4)The operator B in (2.2) can be arbitrary. For instance, B u = u refers to Dirichlet condition and B u = ∂u∂ n + b u describes Robin condition with the parameter b . Another option is representedby quasi-periodic boundary conditions, and any combination of these conditions on differentsubsets of ∂ Ω is also admissible. 3he coefficients A ij = A ij ( x ) , A j = A j ( x ) , and A = A ( x ) in (2.1) are real functions onthe closure Ω . We assume that A ij , A j ∈ C (Ω) , A ∈ C (Ω) , and the functions A ij satisfy thestandard ellipticity condition A ij = A ji , X i,j =1 A ij ( x ) ξ i ξ j > c ( ξ + ξ ) , ξ i ∈ R , x ∈ Ω , (2.5)where c is a fixed positive constant independent of x and ξ . Furthermore, by ∂∂ n we denote theconormal derivative, ∂∂ n := X i,j =1 A ij ν i ∂∂x i − i X j =1 ν j A j , (2.6)where ν = ( ν , ν ) is the unit normal on ∂ω ε pointing inside ω ε , and i is the imaginary unit. Thesymbols α = α ( s ) , α = α ( s ) stand for real functions on ∂ω continuous with respect to thearc length s ∈ [0 , | ∂ω | ] . Similarly s ε denotes the arc length of ∂ω ε for which s ε = εs naturallyholds. If ∂ Ω is empty, then condition (2.2) is simply omitted, and the same applies hereafter toall the conditions imposed on ∂ Ω .The aim of the present paper is to investigate the resolvent convergence of the operators H ε as the scaling parameter ε tends to zero.Before stating our main result, we need to introduce some more notations. By H Ω we denotethe operator in L (Ω) with the differential expression ˆ H given by the right hand side in (2.1)and subject to boundary condition (2.2). Furthermore, it follows from the definition of the hole ω ε that there exist positive constants R , R independent of ε such that ω ε ⊂ B R ε ( x ) ⊂ B R ε ( x ) ⊂ B R ( x ) ⊂ B R ( x ) ⊂ Ω ⊂ Ω , (2.7)where B r ( a ) denotes conventionally the disc of radius r centered at the point a and Ω is the setspecified in the following paragraph.We adopt the following assumptions on the coefficients A ij , A j , A in (2.1), on those specifyingthe operator B in (2.2), and on the operator H Ω . The latter is supposed to be self-adjoint in L (Ω) and semibounded from below, the associated closed symmetric sesquilinear form beingdenoted by h Ω . The domain D ( h Ω ) is a subspace in W (Ω) , and moreover, there exists a domain Ω ⊂ Ω containing x such that the restriction of each function from the domain D ( H Ω ) on Ω belongs to W (Ω ) . The form h Ω satisfies the following lower bound h Ω [ u ] − h Ω [ u ] + c k u k L (Ω \ Ω ) > c k u k W (Ω \ Ω ) (2.8)for all u ∈ D ( h Ω ) , where c , c are positive constants independent of u . More generally, for anarbitrary subdomain ˜Ω ⊂ Ω and vectors u, v ∈ D ( ˜Ω) we denote h ˜Ω ( u, v ) := X i,j =1 (cid:18) A ij ∂u∂x j , ∂v∂x i (cid:19) L (˜Ω) + i X j =1 (cid:18) ∂u∂x j , A j v (cid:19) L (˜Ω) − i X j =1 (cid:18) A j u, ∂v∂x j (cid:19) L (˜Ω) + ( A u, v ) L (˜Ω) . (2.9)4f ˜Ω has a positive distance from ∂ Ω , this form satisfies the lower bound h ˜Ω [ u ] + c k u k L (˜Ω) > c k u k W (˜Ω) (2.10)with the same constants c , c as in (2.8).To define the operator H ε rigorously, we use an infinitely differentiable cut-off function χ Ω taking values in [0 , , equal to one in B R ( x ) , and vanishing outside Ω . Then H ε is theoperator in L (Ω ε ) with the differential expression ˆ H and the domain D ( H ε ) consisting of thefunctions u satisfying condition (2.3) and such that (1 − χ Ω ) u ∈ D ( H Ω ) , χ Ω u ∈ W (Ω \ ω ε ) ; (2.11)the action of H ε is then given by the formula H ε u := H Ω (1 − χ Ω ) u + ˆ H χ Ω u. (2.12)Next we have to specify the limit of the operator family {H ε } ε> . Referring to Section 3below, in Lemma 3.2 we will establish the existence of a unique solution G ∈ W (Ω \ B δ ( x )) ∩ C ( B δ \ { x } ) , δ > , to the boundary-value problem ( ˆ H + c ) G = 0 in Ω \ { x } , B G = 0 on ∂ Ω , (2.13)where c is the constant from (2.8) and (2.10), that behaves in the vicinity of x as follows, G ( x ) = ln | A − ( x − x ) | + a + O (cid:0) | x − x | ln | x − x | (cid:1) , x → x , (2.14)with a ∈ R being a fixed number, E is the unit × matrix and A := (cid:18) A ( x ) A ( x ) A ( x ) A ( x ) (cid:19) . By x = x( s ) we denote the vector equation of the boundary, that is, the curve x : [0 , | ∂ω | ] → Ω coincides with ∂ω . We put α ( s ) = ν · A x( s ) | A − x( s ) | , (2.15)suppose that α is such that K := − Z ∂ω (cid:0) α ( s ) ln | A − x( s ) | + α ( s ) (cid:1) d s > − c k G k L (Ω) − πa tr A (2.16)holds, and denote β := − Kπ tr A , (2.17)assuming in addition that β = a .The limiting operator of the family {H ε } ε> turns out to be the operator with the differentialexpression ˆ H in Ω and a point interaction at the point x . We denote it H ,β ; it is an operatorin L (Ω) with the domain D ( H ,β ) := (cid:8) u = u ( x ) : u ( x ) = v ( x ) + ( β − a ) − v ( x ) G ( x ) , v ∈ D ( H Ω ) (cid:9) (2.18)5cting as H ,β u = H Ω v − c ( β − a ) − v ( x ) G, (2.19)where c is again the constant from (2.8) and (2.10)By k · k X → Y we denote the norm of a bounded operator acting from a Hilbert space X intoa Hilbert space Y . Now we are in position to state our main result: Theorem 2.1.
The operators H ε and H ,β are self-adjoint and H ε converges to H ,β in thenorm resolvent sense as ε → +0 . Namely, the following estimates hold, k ( H ε − λ ) − − ( H ,β − λ ) − k L (Ω) → L (Ω ε ) C | ln ε | − , (2.20) (cid:13)(cid:13) ∇ (cid:0) ( H ε − λ ) − − ( H ,β − λ ) − (cid:1)(cid:13)(cid:13) L (Ω) → L (Ω ε ) C | ln ε | − , (2.21) (cid:13)(cid:13) χ ˜Ω (cid:0) ( H ε − λ ) − − ( H ,β − λ ) − (cid:1)(cid:13)(cid:13) L (Ω) → D ( h Ω ) C | ln ε | − , (2.22) where ˜Ω is an arbitrary fixed subdomain of Ω such that x / ∈ ˜Ω and χ ˜Ω is an infinitely differ-entiable cut-off function equal to one on ˜Ω and vanishing outside some fixed domain containing ˜Ω , still separated from the point x by a positive distance. These estimates are order-sharp; thepositive constants C are independent of ε , the constant in estimate (2.22) may in general dependon the choice of ˜Ω . Our second main results describes the spectral convergence of the operators H ε ; the spectrumof an operator is denoted by σ ( · ) . Theorem 2.2.
The spectrum of the operator H ε converges to that of H ,β as ε → +0 . Morespecifically, if λ / ∈ σ ( H ,β ) , then λ / ∈ σ ( H ε ) provided ε is small enough, while if λ ∈ σ ( H ,β ) ,then there exists a point λ ε ∈ σ ( H ε ) such that λ ε → λ as ε → +0 . For any ̺ , ̺ / ∈ σ ( H ,β ) , ̺ < ̺ , the spectral projection of H ε corresponding to the segment [ ̺ , ̺ ] converges to thespectral projection of H ,β referring to the same segment in the sense of the norm k·k L (Ω) → L (Ω ε ) .For each fixed segment Q := [ ̺ , ̺ ] of the real line the inclusion σ ( H ε ) ∩ Q ⊂ (cid:8) λ ∈ Q : dist( λ, σ ( H ,β ) ∩ Q ) C | ln ε | − } (2.23) holds, where C is a fixed constant independent of ε but depending of Q . If λ is an isolatedeigenvalue of H ,β of a multiplicity n , there exist exactly n eigenvalues of the operator H ε ,counting multiplicities, which converge to λ as ε → +0 . The total projection P ε referring tothese perturbed eigenvalues and the projection P ,β onto the eigenspace associated with λ satisfyestimates analogous to (2.20), (2.21), and (2.22). Before proceeding to the theorems, let us add a few comments. The convergence of H ε to H ,β is expressed in terms of several norms for the corresponding difference of the resolvents,namely those of operators acting from L (Ω) into L (Ω ε ) or W (Ω ε ) , see (2.20), (2.21). One moreestimate is given in (2.22), where the norm involves a cut-off function χ ˜Ω . The presence of thiscut-off function means that the difference of the resolvents is considered on a fixed subdomainseparated from the point x ; this difference is estimated in the norm defined by the form of theoperator H Ω . The convergence rates in (2.20), (2.22) are same being O ( | ln ε | − ) , while the ratein (2.21) is just O ( | ln ε | − ) . The reason is that the norm in (2.21) is stronger than in (2.20)since it involves the gradient; note that in (2.22) its presence plays no role, because the norm isconsidered on the domain separated from the point x .6s indicated in the introduction, the constant β defined by (2.17) can not take all values onthe real line in view of (2.16). This condition obviously fixes an upper bound for the admissiblevalues of β and, at the same time, it is essential for our technique; should (2.17) fail, theconvergence of our operator families could fail as well.Our second result, Theorem 2.2, states the convergence of the spectrum and the associatedspectral projections. This result is based essentially on standard theorems about the convergenceof the spectra with respect to the resolvent norm, however, they can not be applied directly heresince the operators H ε and H ,b act on different spaces. One more problem is that the functionsin the domain of the limiting operator exhibit a logarithmic singularity at x . Nevertheless, wesucceed to overcome these obstacles. Moreover, inclusion (2.23) provides, in fact, an estimatefor the convergence rate of the spectrum, which turns out to be the same as in inequality (2.20).Indeed, this inclusion means that once we consider compact parts of the spectra of H ,β and H ε , the distance between the perturbed spectrum and the limiting one is of order O ( | ln ε | − ) .Considering then how the isolated eigenvalues of the operator H ,β bifurcate into the eigenvaluesof H ε , we are able also to estimate the convergence rate for the associated spectral projectionsarriving at estimates that are the same as (2.20), (2.21), (2.22). Here we collect several auxiliary results, which will help us to prove Theorem 2.1 in the nextsection.
Lemma 3.1.
The identity Z ∂ω α ( s ) d s = Z ∂ω ν · A x( s ) | A − x( s ) | d s = − π tr A (3.1) holds true.Proof. Let us express the integral on the left-hand side of (3.1). We observe that X i,j =1 A ij ( x ) ∂ ∂x i ∂x j ln | A − ( x − x ) | = 0 holds in the vicinity of x . To see that this the case, one can introduce local coordinates, y := A − ( x − x ) , in which the expression in question is nothing else than ∆ ln | y | . Integratingit over ω with a small disc centered at x deleted, using Green’s formula, we get Z ω \{ x : | y | <δ } X i,j =1 A ij ( x ) ∂ ∂x i ∂x j ln | A − ( x − x ) | d x = − Z ∂ω ∂∂ n ln | A − x( s ) | d s − Z { x : | y | = δ } d s | y | Evaluating the integrals on the right-hand side and taking the limit δ → +0 in the second one,we find − Z ∂ω ν · A x( s ) ds | A − x( s ) | d s − π tr A , in other words, the sought identity (3.1). 7n view of the assumptions made about the operator H Ω , in particular, of the estimates (2.8)and (2.10), the spectrum of this operator is contained in the interval [ c − c , ∞ ) , and since c > , the inverse operator ( H Ω + c ) − is well-defined and bounded. In the following lemmawe employ the polar coordinates ( r, θ ) associated with the variables y . Lemma 3.2.
The boundary-value problem (2.13), (2.14) has a unique solution which belongs to W (Ω \ B δ ( x )) ∩ C ( B δ \ { x } ) for all sufficiently small δ > and has the following asymptoticbehavior in the vicinity of x , G ( x ) = ln r + a + r (cid:0) ( a sin θ + a cos θ ) ln r + P (sin θ, cos θ ) (cid:1) + O ( r ln r ) , x → x , (3.2) where a is a real number, a , a ∈ C , and P is a polynomial.Proof. The differential expression (2.1) can be rewritten as ˆ H = − X i,j =1 A ij ∂ ∂x i ∂x j + X j =1 A j − X i =1 ∂A ij ∂x i ! ∂∂x j + i X j =1 ∂A j ∂x j + A ! . (3.3)Using this representation and passing to the local variables y in the vicinity of the point x introduced in the proof of Lemma 3.1, it is straightforward to confirm that there exists a function G ( x ) = ln r + r (cid:0) ( a sin θ + a cos θ ) ln r + P (sin θ, cos θ ) (cid:1) + r (cid:0) P (sin θ, cos θ ) ln r + P (sin θ, cos θ ) ln r + P (sin θ, cos θ ) (cid:1) , (3.4)where P and P i , = 1 , , , are some polynomials, such that the function F ( x ) := ( ˆ H + c ) G ( x ) is continuously differentiable in the punctured neighborhood of the point x and exhibits therethe following asymptotics, F ( x ) = O ( r ln r ) , x → x . (3.5)We seek the solution to the boundary-value problem (2.13), (2.14) in the form G ( x ) = G ( x ) + G ( x ) , G := χ Ω G , (3.6)where for the unknown function G we obtain the operator equation ( H Ω + c ) G = F, F := − χ Ω F + F . (3.7)Here F is a linear combination of the products of the derivatives of G and χ Ω up to the secondorder. If δ > is chosen small enough to ensure that B δ ( x ) ⊂ Ω , the above indicated propertiesof the function F imply that F belongs to L (Ω) ∩ C γ ( B δ ( x )) for all γ ∈ (0 , .Since the resolvent ( H Ω + c ) − is well-defined, equation (3.7) has a unique solution whichbelongs to D ( H Ω ) . Moreover, using the standard Schauder estimates [11], we infer that it alsobelongs to C γ ( B δ ) , which means, in particular, that the function G has the Taylor expansion, G ( x ) = a + a y + a y + O ( | y | ) , x → x , a , a ∈ C , (3.8)where y , y are the components of the vector y = A − ( x − x ) . Returning to the function G ,we conclude that problem (2.13), (2.14) is uniquely solvable and identity (3.2) holds true.8t remains to check that the number a is real. According (3.7) and the definition of thefunctions G and F we have the identity h Ω [ G ] + c k G k L (Ω) = ( F, G ) L (Ω) = − (cid:0) ( ˆ H + c ) G , G (cid:1) L (Ω) , (3.9)which can be rewritten as follows, h Ω [ G ] + c k G k L (Ω) − (cid:0) ( ˆ H + c ) G , G (cid:1) L (Ω) = − (cid:0) ( ˆ H + c ) G , G (cid:1) L (Ω) . (3.10)We denote Ω ˜ δ := Ω \ { x : | y | < ˜ δ } . In the last term on the left-hand side of (3.10) we integrateby parts once bearing in mind the asymptotics (3.2), (3.8), the identity (3.1), and the fact that G = G holds in the vicinity of the point x , obtaining (( H Ω + c ) G , G ) L (Ω) = lim ˜ δ → +0 X i,j =1 (cid:18) A ij ∂G ∂x j , ∂G ∂x i (cid:19) L (Ω ˜ δ ) − X j =1 (cid:18) A j ∂G ∂x j , G (cid:19) L (Ω ˜ δ ) + (( A + c ) G , G ) L (Ω ˜ δ ) − Z { x : | y | =˜ δ } G ∂G ∂ n ds ! = lim ˜ δ → +0 X i,j =1 (cid:18) A ij ∂G ∂x j , ∂G ∂x i (cid:19) L (Ω ˜ δ ) − X j =1 (cid:18) A j ∂G ∂x j , G (cid:19) L (Ω ˜ δ ) + (( A + c ) G , G ) L (Ω ˜ δ ) + π tr A ln ˜ δ ! . (3.11)In the same way we integrate by parts twice on the right-hand side of (3.10), (cid:0) ( ˆ H + c ) G , G (cid:1) L (Ω) = lim ˜ δ → +0 Z { x : | y | =˜ δ } (cid:18) G ∂G ∂ n − G ∂G∂ n (cid:19) ds = − πa tr A . (3.12)Substituting this identity together with (3.11) into (3.10), we obtain a formula for the constant a showing that it is real. This concludes the proof.Denote next Π ε := B R ( x ) \ ω ε , then we have the following result. Lemma 3.3.
For all v ∈ W (Π ε ) the estimate k v k L ( ∂ω ε ) Cε (cid:16) | ln ε |k∇ v k L (Π ε ) + k v k L (Π ε ) (cid:17) (3.13) is valid, where C is a fixed constant independent of ε and v . If, in addition, the function v isdefined on entire ball B R ( x ) and belongs to W ( B R ( x )) , then the estimate k v k L ( B R ( x ) Cε (cid:16) | ln ε |k∇ v k L ( B R ( x ) + k v k L ( B R ( x ) (cid:17) , (3.14) holds, where C is a fixed constant independent of ε and v . roof. We denote by χ : R + → [0 , an infinitely differentiable cut-off function, equal to one if t < and vanishing for t > . It is clear that v ( x ) = v ( x ) χ (cid:18) | x − x | R ε (cid:19) =: v ε on ∂ω ε , and v ε = 0 on ∂B R ε ( x ) . (3.15)We rescale variables, x ( x − x ) ε − , and by standard embedding theorems we get k v k L ( ∂ω ε ) = ε k v ε ( x + ε · ) k L ( ∂ω ) Cε k∇ v ε ( x + ε · ) k L ( B R (0) \ ω ) = Cε k∇ v ε k L ( B R ε ( x ) \ ω ε ) C (cid:16) ε k∇ v k L ( B R ε ( x ) \ ω ε ) + ε − k v k L ( B R ε ( x ) \ B R ε ( x )) (cid:17) , (3.16)where the symbol C stands for various inessential constants independent of ε and v . Let usestimate the term k u k L ( B R ε ( x ) \ B R ε ( x )) .It follows from (2.7) that v ( x ) = v ( x ) χ (cid:18) | x − x | R (cid:19) in B R ε ( x ) \ ω ε . We denote r := | x − x | for x ∈ B R ε ( x ) \ B R ε ( x ) , and furthermore, we put x ′ := x + r ′ r ( x − x ) ,then we have | v ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r Z R ∂∂r (cid:18) v ( x ′ ) χ (cid:18) r ′ R (cid:19)(cid:19) d r ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z r |∇ v ( x ′ ) | χ (cid:18) r ′ R (cid:19) d r ′ + 1 R R Z r | v ( x ′ ) | (cid:12)(cid:12)(cid:12)(cid:12) χ ′ (cid:18) r ′ R (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d r ′ . Using next Cauchy-Schwarz inequality together with the properties of the cut-off function, wearrive at the estimate | v ( x ) | R Z r |∇ v ( x ′ ) | χ (cid:18) r ′ R (cid:19) d r ′ + 2 R R Z r | v ( x ′ ) | (cid:12)(cid:12)(cid:12)(cid:12) χ ′ (cid:18) r ′ R (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d r ′ R r R Z r |∇ v ( x ′ ) | r ′ d r ′ + 2 ln 2 R R (cid:16) sup t ∈ [1 , | χ ′ ( t ) | (cid:17) R Z r | v ( x ′ ) | r ′ d r ′ (3.17)Integrating this inequality over B R ε ( x ) \ B R ε ( x ) we find k v k L ( B R ε ( x ) \ B R ε ( x )) Cε (cid:16) | ln ε |k∇ v k L ( B R ( x ) \ B R ε ( x )) + k v k L ( B R ( x ) \ B R ( x )) (cid:17) , and substituting finally from here into the right-hand side of (3.16) we obtain the sought estimate(3.13). Finally, if v ∈ W ( B R ( x ) ) , we integrate estimate (3.17) over B R ε ( x ) and arriveimmediately at estimate (3.14) which concludes the proof.10 emma 3.4. For all v ∈ W (Π ε ) satisfying the condition Z ∂ω ε v ds = 0 (3.18) the inequality k v k L ( ∂ω ε ) Cε k∇ v k L (Π ε ) (3.19) holds, where C is a constant independent of ε and v . If, in addition, the function v is definedon the entire ball B R ( x ) and v ∈ W ( B R ( x )) , then k v k L ( ∂ω ε ) Cε | ln ε |k v k W ( B R ( x )) , (3.20) where C is a constant independent of ε and v .Proof. Throughout the proof the symbol C stands for various inessential constants independentof ε and v . The function v ⊥ := v − h v i ω , h v i ω := 1 ε | B R (0) \ ω | Z B R ε ( x ) \ ω ε v d x, (3.21)obviously satisfies the identities Z B R ε ( x ) \ ω ε v ⊥ ( x ) d x = 0 , Z B R (0) \ ω v ⊥ ( x + ε · ) d x = 0 , (3.22)which allow us to apply the Poincar´e inequality in the following chain of estimates, k v ⊥ k L ( ∂ω ε ) = ε k v ⊥ ( x + ε · ) k L ( ∂ω ) Cε k∇ v ⊥ ( x + ε · ) k L ( B R (0) \ ω ) Cε k∇ v k L ( B R ε ( x ) \ ω ε ) . From this inequality in combination with (3.21) we infer that k v k L ( ∂ω ε ) = k v ⊥ + h v i ω k L ( ∂ω ε ) Cε (cid:16) k∇ v k L ( B R ε ( x ) \ ω ε ) + |h v i ω | (cid:17) . (3.23)Let us assess h v i ω . In the domain B R (0) \ ω we consider the boundary-value problem ∆ X = | ∂ω || B R (0) \ ω | in B R (0) \ ω,∂X∂ν = 1 on ∂ω, ∂X∂ν = 0 on ∂B R (0) , (3.24)where ν is the unit outward normal to the boundary of B R (0) \ ω . This problem is solvablebecause we have | ∂ω | = Z ∂ω d s = Z ∂ω ∂X∂ν d s = Z B R (0) \ ω ∆ X d x = | B R (0) \ ω | .
11n view of the assumed smoothness of the boundary ∂ω and the standard Schauder estimate,we can conclude that X ∈ C (2+ γ ) ( B R (0) \ ω ) for all γ ∈ (0 , . A solution to problem (3.24) isdefined up to an additive constant which we fix it by the requirement Z B R (0) \ ω X ( x ) d x = 0 . (3.25)Combining problem (3.24) and assumption (3.18), we can rewrite h v i ω using integration by parts, h v i ω = 1 | ∂ω | Z B R ε ( x \ ω ε v ( x )∆ X (cid:18) x − x ε (cid:19) d x = 1 ε | ∂ω | Z ∂ω ε v ( x ) d s − ε | ∂ω | Z B R ε ( x ) \ ω ε ∇ v ( x ) · ( ∇ X ) (cid:18) x − x ε (cid:19) d x = − ε | ∂ω | Z B R ε ( x ) \ ω ε ∇ v ( x ) · ( ∇ X ) (cid:18) x − x ε (cid:19) d x, (3.26)and consequently, by Cauchy-Schwarz inequality we can infer that |h v i ω | ε | ∂ω | k∇ v k L ( B R ε ( x ) \ ω ε ) (cid:13)(cid:13)(cid:13)(cid:13) ( ∇ X ) (cid:18) x − x ε (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( B R ε ( x ) \ ω ε ) C k∇ v k L ( B R ε ( x ) \ ω ε ) . (3.27)This estimate together with (3.23) yields inequality (3.19).Assume finally that v ∈ W ( B R ( x )) . Then we can replace v in (3.17) with ∂v∂x i , i = 1 , ,and integrate such an estimate over B R ε ( x ) . This gives k∇ v k L (2 B R ε ( x )) k∇ v k L (2 B R ε ( x )) Cε | ln ε |k v k W ( B R ( x )) , (3.28)which in combination with (3.23), (3.27) implies (3.20) concluding thus the proof.Next we consider for any v ∈ W (Π ε ) the mean value over the boundary of ω ε , h v i ∂ω ε := 1 ε | ∂ω | Z ∂ω ε v ds. (3.29) Lemma 3.5.
For all ϕ ∈ C ( ∂ω ) and all v ∈ W ( B R ( x )) the inequality (cid:12)(cid:12)(cid:12)(cid:12) ε − Z ∂ω ε ϕ (cid:16) s ε ε (cid:17) v ( x ) d s − c ( ϕ ) v ( x ) (cid:12)(cid:12)(cid:12)(cid:12) Cε | ln ε | k v k W ( B R ( x )) , c ( ϕ ) := Z ∂ω ϕ ( s ) d s, (3.30) holds true, where C is a constant independent of ε and v . roof. We put v ⊥ := v − h v i ∂ω ε , Z ∂ω ε v ⊥ d s = 0 , and note the following obvious identity, ε − Z ∂ω ε ϕ (cid:16) s ε ε (cid:17) v ( x ) d s = ε − h v i ∂ω ε Z ∂ω ε ϕ (cid:16) s ε ε (cid:17) d s + ε − Z ∂ω ε ϕ (cid:16) s ε ε (cid:17) v ⊥ ( x ) d s = c ( ϕ ) h v i ∂ω ε + ε − ( v ⊥ , ϕ ) L ( ∂ω ε ) (3.31)and from Lemma 3.4 we get (cid:12)(cid:12)(cid:12) ε − ( v ⊥ , ϕ ) L ( ∂ω ε ) (cid:12)(cid:12)(cid:12) Cε | ln ε | k v k W ( B R ( x )) . (3.32)Let us assess the difference h v i ∂ω ε − v ( x ) . To this aim, we consider the boundary-value problem ∆ Y = 0 in ω \ { } , ∂Y∂ν = 1 on ∂ω, Y ( x ) = | ∂ω | π ln | x − x | + O (1) , x → x , (3.33)where ν is the unit outward normal to the boundary of ω . This problem has a unique solutionup to a constant which can be chose in such a way that Z ω Y ( x ) d x = 0 . (3.34)in view of the assumed smoothness of the boundary ∂ω and the standard Schauder estimate, wehave Y ∈ C (2+ γ ) ( ω \ B δ (0)) for any γ ∈ (0 , and all δ > .Let v ∈ C ( ω ε ) . Using integration by parts and taking into account the indicated propertiesof the function Y we get Z ω ε v ∆ Y (cid:18) x − x ε (cid:19) d x = ε − Z ∂ω ε v ( x ) d s − Z ∂ω ε Y (cid:18) x − x ε (cid:19) ∂v∂ν ( x ) d s − | ∂ω | v ( x ) . Since the space C ( ω ε ) is dense in W ( ω ε ) , the above identity holds for all v ∈ W ( ω ε ) as well,and by Cauchy-Schwarz inequality and Lemma 3.3 it implies (cid:12)(cid:12) h v i ∂ω ε − v ( x ) (cid:12)(cid:12) = 1 | ∂ω | (cid:12)(cid:12)(cid:12)(cid:12) Z ∂ω ε Y (cid:18) x − x ε (cid:19) ∂v∂ν ( x ) d s (cid:12)(cid:12)(cid:12)(cid:12) Cε k∇ v k L ( ∂ω ε ) Cε | ln ε | k v k W ( ω ε ) . which together with (3.31), (3.32) yields the sought result. The goal of this section is to prove Theorems 2.1 and 2.2. The argument consists of two mainparts. In the first we establish the self-adjointness of the operators H ε and H ,β , while thesecond part is devoted to the verification of the norm resolvent convergence and the spectralconvergence. 13 .1 Self-adjointness of the operator H ε We start by introducing a sesquilinear form h ε in L (Ω ε ) defined by the identity h ε ( u, v ) := h Ω (cid:0) (1 − χ Ω ) u, (1 − χ Ω ) v (cid:1) + h Ω \ ω ε (cid:0) χ Ω u, (1 − χ Ω ) v (cid:1) + h Ω \ ω ε (cid:0) (1 − χ Ω ) u, χ Ω v (cid:1) + h Ω \ ω ε ( χ Ω u, χ Ω v ) − ε − ( αu, v ) L ( ∂ω ε ) (4.1)on the domain D ( h ε ) := n u : (1 − χ Ω ) u ∈ D ( h Ω ) , χ Ω u ∈ W (Ω \ ω ε ) o (4.2)It is clear that this form is symmetric; let us check that it is associated with the operator H ε , inother words, that we have h ε ( u, v ) = ( H ε u, v ) L (Ω ε ) for all u ∈ D ( H ε ) , v ∈ D ( h ε ) . (4.3)Indeed, since u ∈ W (Ω \ ω ε ) , v ∈ W (Ω \ ω ε ) , according the definition of H ε , h and χ Ω , we canuse integration by parts to rewrite the last four terms on the right-hand side of (4.1) as follows, h Ω \ ω ε (cid:0) χ Ω u, (1 − χ Ω ) v (cid:1) + h Ω \ ω ε (cid:0) (1 − χ Ω ) u, χ Ω v (cid:1) + h Ω \ ω ε ( χ Ω u, χ Ω v ) − ε − ( αu, v ) L ( ∂ω ε ) = (cid:0) H Ω χ Ω u, (1 − χ Ω ) v (cid:1) L (Ω \ ω ε ) + (cid:0) H Ω (1 − χ Ω ) u, χ Ω v (cid:1) L (Ω \ ω ε ) + ( H Ω χ Ω u, χ Ω v ) L (Ω \ ω ε ) = (cid:0) H Ω χ Ω u, v (cid:1) L (Ω ε ) + (cid:0) H Ω (1 − χ Ω ) u, χ Ω v (cid:1) L (Ω ε ) . (4.4)As for the remaining term, since by assumptions made about the cut-off function χ Ω we have (1 − χ Ω ) u ∈ D ( H Ω ) and (1 − χ Ω ) v ∈ D ( h Ω ) , we infer that h Ω (cid:0) (1 − χ Ω ) u, (1 − χ Ω ) v (cid:1) = (cid:0) H Ω (1 − χ Ω ) u, (1 − χ Ω ) v (cid:1) L (Ω) = (cid:0) H Ω (1 − χ Ω ) u, (1 − χ Ω ) v (cid:1) L (Ω ε ) . (4.5)We also have (1 − χ Ω ) u ∈ W (Ω ) , and therefore H Ω (1 − χ Ω ) u = ˆ H (1 − χ Ω ) u on Ω \ B R ( x ) . (4.6)Substituting this identity together with (4.5), (4.4) into definition (4.1) we get h ε ( u, v ) = (cid:0) H Ω (1 − χ Ω ) u, (1 − χ Ω ) v (cid:1) L (Ω ε ) + (cid:0) ˆ H χ Ω u, v (cid:1) L (Ω \ ω ε ) + (cid:0) ˆ H (1 − χ Ω ) u, χ Ω v (cid:1) L (Ω \ ω ε ) = (cid:0) H Ω (1 − χ Ω ) u, v (cid:1) L (Ω ε ) − (cid:0) H Ω (1 − χ Ω ) u, χ Ω ) v (cid:1) L (Ω ε ) + (cid:0) ˆ H χ Ω u, v (cid:1) L (Ω \ ω ε ) + (cid:0) ˆ H (1 − χ Ω ) u, χ Ω v (cid:1) L (Ω \ ω ε ) = (cid:0) H Ω (1 − χ Ω ) u, v (cid:1) L (Ω ε ) + (cid:0) ˆ H χ Ω u, v (cid:1) L (Ω ε ) (4.7)which proves relation (4.3).Our next step is to check that the form h ε is semibounded from below. Here we shall makeuse of the following two auxiliary results concerning the function G introduced in Lemma 3.2. Lemma 4.1.
For all u ∈ D ( h ε ) we have the identity h ε ( G, u ) + c ( G, u ) L (Ω ε ) = (cid:16) ∂G∂ n − ε − αG, u (cid:17) L ( ∂ω ε ) . (4.8)14 roof. To begin with, we observe that G ∈ D ( H ε ) , and therefore the quantity h ε ( u, G ) is welldefined. It also follows from the definition of the function G and (4.6) that h Ω ((1 − χ Ω ) G, (1 − χ Ω ) u ) + c (cid:0) G, (1 − χ Ω ) u (cid:1) L (Ω ε ) = − h Ω \ ω ε ( χ Ω G, (1 − χ Ω ) u ) . (4.9)Thus we have h ε ( G, u ) + c ( G, u ) L (Ω ε ) = h Ω \ ω ε (cid:0) (1 − χ Ω ) G, χ Ω u (cid:1) + h Ω \ ω ε ( χ Ω G, χ Ω u ) − ε − ( αG, u ) L ( ∂ω ε ) + c (cid:0) G, χ Ω u (cid:1) L (Ω ε ) = h Ω \ ω ε ( G, χ Ω u ) + c ( G, χ Ω u ) L (Ω ε ) − ε − ( αG, u ) L ( ∂ω ε ) ; integrating then by parts and using the equation that G satisfies, we arrive at (4.8). Lemma 4.2.
The identities (cid:18) ∂G∂ n − ε − αG, G (cid:19) L ( ∂ω ε ) = K + πa tr A + O (ln − ε ) , (4.10) (cid:18) ∂G∂ n − ε − αG (cid:19) ( ε − s ε ) = ε − (cid:0) ln − ε Φ ( ε − s ε )+ ln − ε Φ ( ε − s ε , ε ) (cid:1) on ∂ω ε (4.11) hold true, where where K is defined by formula (2.16) and Φ ( s ) := − α ( s )(ln | A − x( s ) | + a ) − α ( s ) , (4.12) and Φ = Φ ( s, ε ) is a function uniformly bounded in ε and s .Proof. The stated asymptotic expansions (4.11), (4.12) can be easily confirmed by straightfor-ward calculations using relations (3.2). These three formul æ in combination with the identities(3.1), (2.15) yield in turn (4.10) which concludes the proof.Given an arbitrary u ∈ W (Ω ε ) we denote u ⊥ := u − h u i G G, h u i G := h u i ∂ω ε h G i ∂ω ε , (4.13)recalling that the averaging h·i ∂ω ε was introduced in (3.29). Then in view of Lemma 4.1 we have h ε ( u, u ) + c k u k L (Ω ε ) = h ε (cid:0) u ⊥ + h u i G G, u (cid:1) + c (cid:0) u ⊥ + h u i G G, u (cid:1) L (Ω ε ) = h ε (cid:0) u ⊥ , u (cid:1) + c ( u ⊥ , u ) L (Ω ε ) + h u i G (cid:18) ∂G∂ n − ε − αG, u (cid:19) L ( ∂ω ε ) = h ε (cid:2) u ⊥ ] + c k u ⊥ k L (Ω ε ) + |h u i G | (cid:18) ∂G∂ n − ε − αG, G (cid:19) L ( ∂ω ε ) + 2 Re h u i G (cid:18) ∂G∂ n − ε − αG, u ⊥ (cid:19) L ( ∂ω ε ) . (4.14)15n view of the asymptotics (4.11) we have (cid:18) ∂G∂ n − ε − αG, u ⊥ (cid:19) L ( ∂ω ε ) = ε − ln − ε (cid:0) Φ + ln − Φ , u ⊥ (cid:1) L ( ∂ω ε ) and therefore, by virtue of Lemma 3.4, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂G∂ n − ε − αG, u ⊥ (cid:19) L ( ∂ω ε ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C | ln ε | − k u ⊥ k W ( B R ( x ) \ ω ε ) , (4.15)where C is a constant independent of ε and u .To proceed we have to analyze the term h ε ( u ⊥ , u ⊥ ) in (4.14). The estimate (2.8) implies h Ω (cid:2) (1 − χ Ω ) u ⊥ ] + c k u ⊥ k L (Ω \ Ω ) > c k u ⊥ k W (Ω \ Ω ) + h Ω (cid:2) (1 − χ Ω ) u ⊥ (cid:3) = c k u ⊥ k W (Ω \ Ω ) + h Ω \ ω ε (cid:2) (1 − χ Ω ) u ⊥ (cid:3) . (4.16)At the same time, it is straightforward to confirm that h Ω \ ω ε (cid:2) (1 − χ Ω ) u ⊥ (cid:3) + h Ω \ ω ε (cid:0) (1 − χ Ω ) u ⊥ , χ Ω u ⊥ (cid:1) + h Ω \ ω ε (cid:0) χ Ω u ⊥ , (1 − χ Ω ) u ⊥ (cid:1) + h Ω \ ω ε [ χ Ω u ⊥ ] = h Ω \ ω ε [ u ⊥ ] , (4.17)hence by the definition of the form h ε and estimates (4.16), (2.10) we get h ε [ u ⊥ ] + c k u ⊥ k L (Ω ε ) > c k u ⊥ k W (Ω \ Ω ) + h Ω \ ω ε [ u ⊥ ]+ c k u ⊥ k L (Ω \ ω ε ) − ε − ( αu ⊥ , u ⊥ ) L ( ∂ω ε ) > c k u ⊥ k W (Ω ε ) − ε − ( αu ⊥ , u ⊥ ) L ( ∂ω ε ) . (4.18)By Lemma 3.4 and the definition of α by (2.4) and (2.15) we also have (cid:12)(cid:12) ε − ( αu ⊥ , u ⊥ ) L ( ∂ω ε ) (cid:12)(cid:12) C | ln ε | − k∇ u ⊥ k L (Ω \ ω ε ) , (4.19)where C is a fixed constant independent of ε and u , hence in view of (4.18) we finally obtain h ε [ u ⊥ ] + c k u ⊥ k L (Ω ε ) > (cid:0) c − C | ln ε | − (cid:1) k u ⊥ k L (Ω ε ) . (4.20)This estimate together with (4.15), (4.14), and (4.10) implies that h ε ( u, u ) + c k u k L (Ω ε ) > ( c − C | ln ε | − ) k u ⊥ k W (Ω ε ) + ( K + πa tr A − C | ln ε | − ) |h u i G | , (4.21)where C is again a fixed constant independent of ε and u . Furthermore, using Cauchy-Schwarzinequality it is easy to check that k u k L (Ω ε ) = k u ⊥ k L (Ω ε ) + 2 Re h u i G ( G, u ⊥ ) L (Ω ε ) + |h u i G | k G k L (Ω ε ) > − η k u ⊥ k L (Ω ε ) + η k G k L (Ω ε ) η |h u i G | (4.22)16olds for an arbitrary η ∈ (0 , , and this identity in turn implies ( c − C | ln ε | − ) k u ⊥ k W (Ω ε ) + ( K + πa tr A − C | ln ε | − ) |h u i G | + c k u k L (Ω ε ) > ( c − C | ln ε | − ) k∇ u ⊥ k L (Ω ε ) + ( c − C | ln ε | − − c η ) k u ⊥ k L (Ω ε ) + (cid:18) K + πa tr A − C | ln ε | − + c η k G k L (Ω ε ) η (cid:19) |h u i G | (4.23)for any c > . Having in mind that k G k L (Ω ε ) = k G k L (Ω) + o (1) , we choose c and η in such away that c η is less than c and η is small enough. In view of (2.16) we can achieve that c − C | ln ε | − − c η > c , K + πa tr A − C | ln ε | − + c η k G k L (Ω ε ) η > c (4.24)holds for all sufficiently small ε , where c is a fixed positive constant independent of ε , and c isindependent of ε as well. By means of (4.21), (4.23) we then have h ε [ u ] + c k u k L (Ω ε ) > c (cid:0) k u ⊥ k W (Ω ε ) + |h u i G | (cid:1) , (4.25)where c and c are fixed constants independent of ε and u .We also observe that we if we replace estimate (4.16) by the identity h Ω (cid:2) (1 − χ Ω ) u ⊥ ] + c k u ⊥ k L (Ω \ Ω ) = h Ω (cid:2) (1 − χ Ω ) u ⊥ ] + c k u ⊥ k L (Ω \ Ω ) − h Ω (cid:2) (1 − χ Ω ) u ⊥ (cid:3) + h Ω (cid:2) (1 − χ Ω ) u ⊥ (cid:3) and proceed as in (4.17)–(4.24), taking in addition (4.25) into account, we get one more estimate,namely h ε [ u ] + c k u k L (Ω ε ) > h Ω (cid:2) (1 − χ Ω ) u ⊥ ] + c k u ⊥ k L (Ω \ Ω ) − h Ω (cid:2) (1 − χ Ω ) u ⊥ (cid:3) + c (cid:0) k u ⊥ k W (Ω ε ) + |h u i G | (cid:1) , (4.26)Finally, let us show that the form h ε is closed. We recall that the domain D ( h Ω ) is byassumption a subspace in W (Ω) and take an arbitrary sequence u n ∈ D ( h ε ) such that k u n − u k L (Ω ε ) → , h ε [ u n − u m ] → as n, m → ∞ (4.27)for some u ∈ L (Ω ε ) . In view of (4.25), this immediately implies that k u ⊥ n − u ⊥ m k W (Ω ε ) → , h u n − u m i G → as n, m → ∞ . (4.28)and taking (4.13) and (4.27) into account, we then conclude that k u n − u m k W (Ω ε ) → as n, m → ∞ . (4.29)This u n converges in W (Ω ε ) and due to the first claim in (4.27), the limiting function is u whichmeans that u ∈ W (Ω ε ) . By definition (4.1) of the form h ε together with (4.27), (4.29) thisimplies h Ω [(1 − χ Ω )( u n − u m )] → , k (1 − χ Ω ) u n − (1 − χ Ω ) u k L (Ω ε ) → as n, m → ∞ . Since the form h Ω is closed, it follows that (1 − χ Ω ) u n converges to (1 − χ Ω ) u with respect tothe norm in the subspace D ( h Ω ) of the Sobolev space W (Ω) . Consquently, (1 − χ Ω ) u ∈ D ( h Ω ) ,and in view of (4.2) we may conclude that u ∈ D ( h Ω ) , and also h ε [ u n − u ] → as n → ∞ . Thismeans that the form h ε is closed.This brings us to the desired conclusion: the operator H ε is associated with a closed sym-metric sesquilinear form semibounded from below, and therefore it is self-adjoint.17 .2 Self-adjointness of the operator H ,β By definition, the domain of the adjoint operator H ∗ ,β consists of all v ∈ L (Ω) such that thereexists a function g ∈ L (Ω) obeying the identity ( H ,β u, v ) L (Ω) = ( u, g ) L (Ω) for all u ∈ D ( H ,β ) , H ∗ ,β v = g. (4.30)Since u = u + ( β − a ) − u ( x ) G , u ∈ D ( H Ω ) , we can rewrite the above identity as (cid:0) H Ω u − c ( β − a ) − u ( x ) G, v (cid:1) L (Ω) = ( u , g ) L (Ω) + ( β − a ) − u ( x )( G, g ) L (Ω) and hence, ( H Ω u , v ) L (Ω) − ( β − a ) − u ( x )( G, c v + g ) L (Ω) = ( u , g ) L (Ω) . (4.31)Proceeding as in the proof of Lemma 3.2, cf. (3.9)–(3.12), it is straightforward to check that ( f, G ) L (Ω δ ) + ( β − a ) − ( λ + c ) u ( x ) k G k L (Ω δ ) + ( λ + c )( u , G ) L (Ω δ ) = − Z { x : | y | = δ } (cid:18) G ∂u ∂ n − u ∂G∂ n (cid:19) d s. Passing to the limit as δ → +0 in the above identity, the left-hand side converges to the analogousexpression with the scalar product referring to L (Ω) . In view of Lemmata 3.3, 3.5 with ω ε replaced by { x : | y | < δ } together with the asymptotics (3.2) and identity (3.1), we also get lim δ → +0 Z { x : | y | = δ } G ∂u ∂ n d s = 0 , lim δ → +0 Z { x : | y | = δ } u ∂G∂ n d s = − πv ( x ) tr A . Recalling the definition of u , the limit δ → +0 thus yields − πu ( x ) tr A = (( H Ω + c ) u , G ) L (Ω) (4.32)which allows us to rewrite (4.31) as ( H Ω u , v ) L (Ω) − ( β − a ) − κ (( H Ω + c ) u , G ) L (Ω) = ( u , g ) L (Ω) , κ := − ( G, c v + g ) L (Ω) π tr A , or equivalently as ( H Ω u , v − ( β − a ) − κG ) L (Ω) = ( u , g + ( β − a ) − c κG ) L (Ω) . Since the operator H Ω is self-adjoint, the above identity implies that w := v − ( β − a ) − kG ∈ D ( H Ω ) , H ∗ ,β w = g + ( β − a ) − c κG. (4.33)Using then the identity (4.32) with u replaced by w , we get − πw ( x ) tr A = (( H Ω + c ) w, G ) L (Ω) = ( g + c v, G ) L (Ω) = − πκ tr A , and therefore, by virtue of (4.33), v = w + ( β − a ) − w ( x ) G, H ∗ ,β w = g + c ( β − a ) − w ( x ) G, w ∈ D ( H Ω ) , which means that H ∗ ,β = H ,β . 18 .3 Resolvent convergence Since both the operators H ε and H ,β are self-adjoint, their resolvents are well defined for λ awayfrom the real axis, Im λ = 0 . We choose an arbitrary f ∈ L (Ω) and denote u := ( H ,β − λ ) − f , u ε := ( H ε − λ ) − f , where in the latter definition the resolvent is applied to the restriction of thefunction f to Ω ε ; with an abuse of notation we keep the same symbol for it. We put v ε := u ε − u .This function obviously belongs to W (Ω ε ) and solves the boundary-value problem ( ˆ H − λ ) v ε = 0 in Ω ε , B v ε = 0 on ∂ Ω , ∂v ε ∂ n = ε − αv ε + g ε on ∂ω ε , where g ε := (cid:18) ∂∂ n − ε − α (cid:19) u . (4.34)The corresponding integral equation reads h ε [ v ε ] − λ k v ε k L (Ω ε ) = − ( g ε , v ε ) L ( ∂ω ε ) . (4.35)The next step is to estimate the right-hand side in (4.35). Since u ∈ D ( H ,β ) according(2.18), it can be represented as u ( x ) = v ( x ) + ( β − a ) − v ( x ) G ( x ) , v ∈ W (Ω) , (4.36)and f = ( H ,β − λ ) u = ( H Ω − λ ) v − ( β − a ) − ( λ + c ) v ( x ) G. Lemma 4.3.
The inequality k v k W (Ω) + k v k W ( B R ( x )) + | v ( x ) | C k f k L (Ω) holds, where C is a constant independent of f but in general depending on λ .Proof. Throughout the proof the symbol C stands again for various inessential constants inde-pendent of v . Since the operator H ,β is self-adjoint and λ / ∈ σ ( H ,β ) , we immediately get k v k L (Ω) + 2 Re ( β − a ) − v ( x )( v , G ) L (Ω) + | v ( x ) | k G k L (Ω) = k u k L (Ω) k f k L (Ω) dist( λ, σ ( H ,β )) . (4.37)We observe that the function v solves the operator equation ( H Ω − λ ) v = f + ( β − a ) − ( λ + c ) v ( x ) G in Ω . (4.38)Repeating the steps that led us to identity (4.32), we can confirm that ( f, G ) L (Ω) + ( λ + c )( u , G ) L (Ω) = − πv ( x ) tr A . In view of (4.37), this implies that | v ( x ) | C k f k L (Ω) , (4.39)19nd by Cauchy-Schwarz inequality we then find (cid:12)(cid:12)(cid:12) ( β − a ) − v ( x )( v , G ) L (Ω) (cid:12)(cid:12)(cid:12) | ( β − a ) − || v ( x ) |k v k L (Ω) k G k L (Ω) k v k L (Ω) + C k f k L (Ω) . This estimate in combination with (4.37) yields k v k L (Ω) C k f k L (Ω) , (4.40)thus by (4.39) we infer that the right-hand side in (4.38) can be estimated as k f + ( β − a ) − ( λ + c ) v ( x ) G k L (Ω) C k f k L (Ω) . It follows then from (4.38) that k v k W (Ω) + k v k W ( B R ( x )) + k v k W (Ω) C k f k L (Ω) and his estimate together with (4.40) completes the proof.Recalling (3.29), (4.13), we represent functions v ε and v as v , ⊥ ( x ) := v ( x ) − h v i ∂ω ε , v ε = v ⊥ ε + h v ε i G G, Z ∂ω ε v , ⊥ d s = Z ∂ω v ⊥ ε d s = 0 . (4.41)Furthermore, in view of (4.34) and (4.36) the function g ε has the following representation, g ε = g ε, + g ε, + g ε, , g ε, := ∂v ∂ n ,g ε, := ε − ( v − v ( x )) α, g ε, := v ( x )( β − a ) − (cid:18) ∂G∂ν − ε − αG (cid:19) − ε − αv ( x ) . (4.42)We have ( g ε, , v ε ) L ( ∂ω ε ) = ( g ε, , v ⊥ ε ) L ( ∂ω ε ) + h v ε i G ( g ε, , G ) L ( ∂ω ε ) , and therefore from Lemmata 4.3, 3.3, and 3.4 we infer that (cid:12)(cid:12) ( g ε, , v ε ) L ( ∂ω ε ) (cid:12)(cid:12) Cε | ln ε |k v k W ( B R ( x )) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) Cε | ln ε |k f k L (Ω) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) . (4.43)As before, the symbol C stands for inessential constants independent of ε , f , v , v ε , and x .In view of the decomposition (4.41), the function g ε, can be represented as g ε, = ε − ( v , ⊥ + g ε, ) α, g ε, := h v i ∂ω ε − v ( x ) , and using Lemmata 3.3, 3.4, 3.5, and 4.3, we obtain (cid:12)(cid:12) ( g ε, , v ε ) L ( ∂ω ε ) (cid:12)(cid:12) Cε − | ln ε | − (cid:0) k v , ⊥ k L ( ∂ω ε ) + ε | g ε, | (cid:1) k v ε k L ( ∂ω ε ) Cε k f k L (Ω) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) . (4.44)20et us proceed to assessment of the scalar product ( g ε, , v ε ) L ( ∂ω ε ) . Using representation (4.42)together with (4.10), (2.15), we get ( g ε, , v ε ) L ( ∂ω ε ) = ( g ε, , v ⊥ ε ) L ( ∂ω ε ) + h v ε i G ( g ε, , G ) L ( ∂ω ε ) (4.45)and ( g ε, , G ) L ( ∂ω ε ) = v ( x ) (cid:0) ( β − a ) − ( K + πa tr A) + π tr A + O (ln − ε ) (cid:1) . In view of (2.17) and Lemma 4.3 we thus have (cid:12)(cid:12) h v ε i G ( g ε, , G ) L ( ∂ω ε ) (cid:12)(cid:12) C | ln ε | − k f k L ( ∂ Ω) (cid:12)(cid:12) h v ε i G (cid:12)(cid:12) . (4.46)Next we use identities (4.11), (2.4) and Lemmata 3.4, 4.3 to estimate the first term on theright-hand side of (4.45), (cid:12)(cid:12) ( g ε, , v ⊥ ε ) L ( ∂ω ε ) (cid:12)(cid:12) = ε − | ln ε | − (cid:12)(cid:12)(cid:12)(cid:0) Φ − α + ln − ε (Φ − α ) , v ⊥ ε (cid:1) L ( ∂ω ε ) (cid:12)(cid:12)(cid:12) C | ln ε | − k f k L (Ω) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) . This estimate and (4.46) lead us to a bound for ( g ε, , v ε ) L ( ∂ω ε ) , (cid:12)(cid:12) ( g ε, , v ε ) L ( ∂ω ε ) (cid:12)(cid:12) C | ln ε | − k f k L (Ω) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) . then (4.44), (4.43), and (4.42) imply the final estimate for the right-hand side in (4.35), (cid:12)(cid:12) ( g ε , v ε ) L ( ∂ω ε ) (cid:12)(cid:12) C | ln ε | − k f k L (Ω ε ) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) . (4.47)Now we consider separately the imaginary and real part of the both sides of equation (4.35),then using (4.25) we arrive at k v ε k L (Ω ε ) C | ln ε | − k f k L (Ω ε ) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) , k v ⊥ ε k W (Ω ε ) + |h v ε i G | C | ln ε | − k f k L (Ω ε ) (cid:0) k v ⊥ ε k W (Ω ε ) + |h v ε i G | (cid:1) , (4.48)where the second estimate implies k v ⊥ ε k W (Ω ε ) + |h v ε i G | C | ln ε | − k f k L (Ω ε ) . (4.49)In this way we get the inequality k v ε k L (Ω ε ) k v ⊥ ε k L (Ω ε ) + |h v ε i G |k G k L (Ω ε ) C | ln ε | − k f k L (Ω ε ) (4.50)which proves the convergence (2.20).As for the second claim of Theorem 2.1, using asymptotics (3.2) it is easy to check that k∇ G k L (Ω \ ω ε ) C | ln ε | , k∇ (1 − χ Ω ) G k L (Ω ε ) + h Ω [(1 − χ Ω ) G ] C. (4.51)and consequently, by virtue of (4.49), k∇ v ε k L (Ω ε ) k∇ v ⊥ ε k L (Ω ε ) + 2 |h v ε i G | k∇ G k L (Ω ε ) C | ln ε | − k f k L (Ω) . (4.52)This inequality in combination with (4.50) proves (2.21).21et us pass to the last claim. It follows from the estimate (4.26) and identity (4.35) that h Ω [(1 − χ Ω ) v ⊥ ε ] + c k v ⊥ ε k L (Ω ε ) C k v ε k L + (cid:12)(cid:12) ( g ε , v ε ) L (Ω ε ) (cid:12)(cid:12) + h Ω [(1 − χ Ω ) v ⊥ ε ] . Using now (4.47), (4.49), and (4.50), we obtain h Ω [(1 − χ Ω ) v ⊥ ε ] C | ln ε | − k f k L (Ω) and by (4.51) and (4.50) this implies that h Ω [(1 − χ Ω ) v ε ] + k (1 − χ Ω ) v ε k L (Ω ε ) C | ln ε | − k f k L (Ω) . Together with (4.52) and (4.50), the above inequality leads us to (2.22).Let us finally demonstrate that the estimates (2.20), (2.21), and (2.22) are order sharp. Tothis aim, it is sufficient to consider a suitable particular case, for instance,
Ω = R , x = 0 , ˆ H = − ∆ , c = 1 , A = E , Ω := B (0) . The function G can be then found explicitly, G ( x ) = π H (i | x | ) , where H is the Hankel function of the first kind. For the ‘hole’ we choose the disc of radius b ,that is, ω := B b (0) . Then according to (2.15), the function α is constant, α = − b − , on thehole parimeter, and we choose α being a constant as well. The asymptotics of G is well known, G ( x ) = ln | x | + a + O (cid:0) | x | ln | x | (cid:1) , | x | → , a := γ − ln 2 , γ := lim n → + ∞ (cid:18) n X m =1 m − ln n (cid:19) . The constants K and β defined in (2.16), (2.17) are in this case the following, K = 2 π (cid:0) ln b − bα (cid:1) , β = bα − ln b. We also observe that in terms of the standard definition of the point interaction, the aboveoperator coincides with − ∆ ζ,x introduced in [1, Thm. I.5.3], referring to the coupling constant πζ = − b . The hole radius b is positive by definition, so in this case we are able determineexplicitly the range of the coupling strengths for which our approximation works.Let v ∈ C ∞ ( R ) be a non-vanishing radially symmetric function such that v (0) = 0 . Thenthe function u ( x ) := v ( x ) + ( β − a ) − v (0) G ( x ) is in the domain of the operator H ,β and ( H ,β − λ ) u = f := − ∆ v + ( β − a ) − (1 − λ ) v ( x ) G for each λ = k with Im k > , Im λ = 0 . It follows that the function v ε := ( H ε − λ ) − f − u solves the boundary-value problem ( − ∆ − λ ) v ε = 0 in R \ B bε (0) , − ∂v ε ∂ | x | + 1 ε ln ε (cid:18) b − α ln ε (cid:19) v ε = h ε on ∂B bε (0) ,h ε := (cid:18) ∂u ∂ | x | − ε ln ε (cid:18) b − α ln ε (cid:19) u (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | x | = bε v ε ( x ) = h ε c ε H (i k | x | ) , c ε := (cid:18) − ∂∂r + 1 ε ln ε (cid:18) b − α ln ε (cid:19)(cid:19) H (i k | x | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = bε . With the explicit formul æ for all the considered functions in hand, we can find the asymptoticsof the quotient h ε /c ε , h ε c ε = i π v ( x ) α b ( β − a ) − (1 − ( β − a ) − ln k ) ln ε + O (ln − ε ) , which means that k v ε k L (Ω ε ) > C | ln ε | − , k∇ v ε k L (Ω ε ) > C | ln ε | − , where C is a positive constant independent of ε . Consequently, the estimates (2.20), (2.21), and(2.22) are sharp up to a multiplicative constant. This concludes the proof of Theorem 2.1. In this subsection we prove Theorem 2.2. We employ the ideas proposed in the proof of a similarstatement in [4], see Theorem 2.5 and Section 7 in the cited work.The proof is based on standard results on the convergence of spectra and associated spectralprojectors with respect to the resolvent norm, see, for instance, [14, Thm. VIII.23]. However,we can not apply directly this theorem since our operators H ε and H ,β act in different Hilbertspaces, L (Ω ε ) and L (Ω ) . To overcome this obstacle, we introduce an auxiliary multiplicationoperator in L ( ω ε ) acting as H ω ε u := ε − u . This simple operator is self-adjoint, its spectrumconsists of the only eigenvalue λ = ε of an infinite multiplicity and the resolvent satisfies therelation k ( H ω ε − λ ) − k L (Ω ε ) → L (Ω ε ) = ε | − ελ | , λ = ε. (4.53)In view of Lemma 4.3 and estimate (3.14) in Lemma 3.3 we have an obvious estimate, k ( H ,β − λ ) − k L (Ω) → L ( ω ε ) Cε | ln ε | , (4.54)valid for all λ with a non-zero imaginary part, where C is a constant independent of ε butdepending on λ .We regard the space L (Ω) as the direct sum L (Ω) = L (Ω ε ) ⊕ L ( ω ε ) and consider thedirect sum ˜ H ε := H ε ⊕ H ω ε . Then estimates (2.20) and (4.53), (4.54) imply that k ( ˜ H ε − λ ) − − ( H ,β − λ ) − k L (Ω) → L (Ω) k ( H ε − λ ) − − ( H ,β − λ ) − k L (Ω) → L (Ω ε ) + k ( H ω ε − λ ) − k L ( ω ε ) → L ( ω ε ) + k ( H ,β − λ ) − k L (Ω) → L ( ω ε ) C | ln ε | − (4.55)for Im λ = 0 , where C is a constant independent of ε but depending on Im λ . Now we applyTheorem VIII.23 from [14] to conclude that the spectrum of the operator ˜ H ε converges to that ofthe operator H ,β . Since the spectrum of H ω ε consists of the only point λ = ε − , which escapesto the infinity as ε → +0 , and σ ( ˜ H ε ) = σ ( H ε ) ∪ { ε − } , (4.56)23e obtain the stated convergence of the spectrum of the operator H ε . The convergence of thespectral projections corresponding to any interval [ ̺ , ̺ ] with ̺ and ̺ from the resolvent setof H ,β also follows from Theorem VIII.23 in [14].Let us next prove inclusion (2.23). We choose an arbitrary but fixed segment Q := [ ̺ , ̺ ] and consider λ ∈ C such λ = t + i | ln ε | − with t ∈ Q ∩ σ ( H ,β ) ; the set of such λ is denoted by Q ε . For λ ∈ Q ε we recall the well-known formul æ (cid:13)(cid:13) ( H ,β − λ ) − (cid:13)(cid:13) L (Ω) → L (Ω) = 1dist( λ, σ ( H ,β )) , (cid:13)(cid:13) ( ˜ H ε − λ ) − (cid:13)(cid:13) L (Ω) → L (Ω) = 1dist( λ, σ ( ˜ H ε )) = 1dist( λ, σ ( H ε )) , where in the latter identity we have also employed (4.56). These relations and estimate (4.55)imply that (cid:12)(cid:12)(cid:12)(cid:12) λ, σ ( H ε )) − λ, σ ( H ,β )) (cid:12)(cid:12)(cid:12)(cid:12) C | ln ε | − , and hence, for λ ∈ Q ε , λ, σ ( H ε )) > λ, σ ( H ,β )) − C | ln ε | − > | ln ε | − C | ln ε | − > | ln ε | , in other words, dist( λ, σ ( H ε )) | ln ε | − as λ ∈ Q ε . Hence the distance from the set σ ( H ε ) ∩ Q to the set σ ( H ,b ) ∩ Q does not exceed | ln ε | − andthis proves inclusion (2.23).Finally, let λ be an isolated eigenvalue of the operator H ,β of multiplicity n and P ,β be the projection on the associated eigenspace in L (Ω) . Then the above proven facts implyimmediately that there exist exactly n isolated eigenvalues of the operator H ε converging to λ ,naturally with the multiplicities taken into account; we refer to them as to perturbed eigenvalues.By P ε we denote the total projection associated with them. Inclusion (2.23) ensures that thedistance from the perturbed eigenvalues to λ is estimated by C | ln ε | − with some constant C independent of ε . We fix δ > such that the ball B δ ( λ ) in the complex plane contains no otherpoints of spectra of H ε and H ,β except for λ and the perturbed eigenvalues. Then we knowthat P ε = 12 π i Z ∂B δ ( λ ) ( ˜ H ε − λ ) − d λ = 12 π i Z ∂B δ ( λ ) ( H ε − λ ) − d λ, P ,β = 12 π i Z ∂B δ ( λ ) ( H ,β − λ ) − d λ, and consequently, P ε − P = 12 π i Z ∂B δ ( λ ) (cid:0) ( H ε − λ ) − − ( H ,β − λ ) − (cid:1) d λ. (4.57)Since the contour ∂B δ ( λ ) is separated from the spectra of both operators H ,β and H ε , estimates(2.20), (2.21), (2.22) remain true also for λ ∈ B δ ( λ ) . Indeed, one can reproduce literallythe argumentation in Section 4.3 because the fact that Im λ is non-zero was employed only inLemma 4.3 and in (4.48); both this lemma and the inequalities obviously remain true in our case.Now the desired estimates for the spectral projections follow from identity (4.57) and estimates(2.20), (2.21), (2.22). This completes the proof of Theorem 2.2.24 cknowledgements The work of P.E. was supported by the European Union within the project CZ.02.1.01/0.0/0.0/16019/0000778.
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