2D Schrödinger operators with singular potentials concentrated near curves
22D SCHR ¨ODINGER OPERATORS WITH SINGULARPOTENTIALS CONCENTRATED NEAR CURVES
YURIY GOLOVATY
Abstract.
We investigate the Schr¨odinger operators H ε = − ∆+ W + V ε in R with the short-range potentials V ε which are localized around a smooth closedcurve γ . The operators H ε can be viewed as an approximation of the heuris-tic Hamiltonian H = − ∆ + W + a∂ ν δ γ + bδ γ , where δ γ is Dirac’s δ -functionsupported on γ and ∂ ν δ γ is its normal derivative on γ . Assuming that theoperator − ∆ + W has only discrete spectrum, we analyze the asymptotic be-haviour of eigenvalues and eigenfunctions of H ε . The transmission conditionson γ for the eigenfunctions u + = αu − , α ∂ ν u + − ∂ ν u − = βu − , which arise inthe limit as ε →
0, reveal a nontrivial connection between spectral propertiesof H ε and the geometry of γ . Introduction
Solvable type operators with interactions supported by manifolds of a lowerdimension have attracted considerable attention both in the physical and mathe-matical literature in recent years. Such operators are of interest in applications ofmathematics in different fields of science and engineering because they reveal un-questioned effectiveness whenever the exact solvability together with a nontrivialdescription of an actual physical phenomenon is required. The Schr¨odinger ope-rators with pseudo-potentials that are distributions supported by curves, surfaces,metric graphs are used successfully for modelling quantum systems with chargedinclusions, leaky quantum graphs, quantum waveguides etc. There exists a largebody of results on this subject, but we conne ourselves to the case of hypersurfaces(i.e., manifolds of codimension one) as interaction supports. This case is a naturalgeneralization to higher dimensions of the one-dimensional Hamiltonians with pointinteractions such as δ - or δ (cid:48) -interactions.The Schr¨odinger operators formally written as − ∆ + αδ S (1.1)with δ potentials supported by compact or non-compact orientable hypersurfaces S have attracted special attention in the last decade. The pseudo-potential αδ S is adistribution in D (cid:48) ( R n ) acting as (cid:104) αδ S , φ (cid:105) = (cid:90) S αφ dσ for any φ ∈ C ∞ ( R n ), where α is a locally integrable function on S and σ is thenatural measure on S induced by the Riemannian metric. The physical motivation Mathematics Subject Classification.
Primary 35P05; Secondary 81Q10, 81Q15.
Key words and phrases.
Schr¨odinger operator, singular interaction, δ potential, δ (cid:48) -interaction,interaction on curve, asymptotics of eigenvalues. a r X i v : . [ m a t h . SP ] J u l YURIY GOLOVATY comes from nuclear, molecular and solid-state physics [1–4], where the so-calledSDI model (surface delta interaction) has been used since 1965.If the hypersurface S is smooth enough, one can give meaning to the heuristicexpression (1.1) in different ways. We can suppose that H α,S is the Laplacian actingon the functions f ∈ W ( R n \ S ) satisfying the transmission conditions f + = f − , ∂ ν f + − ∂ ν f − = αf on S. (1.2)Here f − and f + denote the one-side traces of f on S and ν is the normal vector fieldon S . The formal expression (1.1) can also be defined rigorously via the symmetricsesquilinear form a ( f, g ) = ( ∇ f, ∇ g ) L ( R n ; C n ) + (cid:90) S αf | S ¯ g | S dσ, dom a = W ( R n ) , where f | S denote the trace of function f ∈ W ( R n ) on S . The form a is a denselydefined, closed, and semibounded in L ( R n ), and hence there exists a self-adjointoperator A α,S in L ( R n ) such that ( A α,S f, g ) L ( R n ) = a ( f, g ) for all f ∈ dom A α,S and g ∈ dom a .An essential advantage of this model is its “stability” with respect to regula-rizations by the Schr¨odinger operators with short-range potentials. If a family ofpotentials U ε with compact supports converges to αδ S in the space of distributions,then the operators − ∆ + U ε converge to H α,S in the norm resolvent sense, as ε → δ S -interactions were investigated in numerous articlesin the recent past; we mention here [5, 6, 8–12] for interactions on finite or infinitefamilies of concentric spheres, [7, 13–20] on closed hypersurfaces and hypersurfaceswith boundary, and [21, 22] for interactions concentrated near conical surfaces.Similarly, one can also generalize to higher dimensions the four-parameter familyof singular point interactions on the line, see [23]. However, for some reason a verypopular interaction in the literature, of course, in addition to the δ S -interaction, isthe so-called δ (cid:48) -interaction supported on hypersurfaces [20,24–26]. This interactionis characterized by the transmission conditions ∂ ν f + = ∂ ν f − , f + − f − = α∂ ν f on S. Despite all advantages of the solvable models, they give rise to many mathe-matical difficulties. One of them deals with the multiplication of distributions; manySchr¨odinger operators with singular potentials are often only formal expressionswithout a precise or unambiguous mathematical meaning. The aim of the presentpaper is to find proper solvable models, i.e., proper transmission conditions on aninteraction support, for the pseudo-Hamiltonians H = − ∆ + W + a∂ ν δ γ + bδ γ (1.3)in R , where W is a regular potential, a and b are some functions on the closed curve γ , and δ γ is Dirac’s δ -function supported on γ . The pseudo-potential a∂ ν δ γ + bδ γ is a distribution in D (cid:48) ( R ) acting as (cid:104) a∂ ν δ γ + bδ γ , φ (cid:105) = (cid:90) γ ( − ∂ ν ( aφ ) + bφ ) dγ for all test functions φ ∈ C ∞ ( R ). It should be noted that this problem is notrelated to the above-mentioned δ (cid:48) -interactions. D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 3
In contrast to (1.1), heuristic expression (1.3) is more singular, and the problemof giving its strictly mathematical meaning is more subtle. First of all, H is “un-stable” under regularizations by Hamiltonians H ε = − ∆ + W + V ε with short-rangepotentials V ε . From a physical viewpoint, it means that the quantum systems with( a∂ ν δ γ + bδ γ )-like localized potentials of different shapes possess slightly differentproperties. Our purpose is to find solvable models describing with admissible fi-delity the real quantum processes governed by H ε for given V ε suitably scaled in thenormal direction to γ . The mathematical motivation for studying such operators isalso that they exhibit nontrivial relations between some spectral properties and thegeometry of γ which arise (upon passing to the limit) in the family of transmissionconditions f + = αf − , α ∂ ν f + − ∂ ν f − = βf − on γ. Assuming that the unperturbed operator − ∆ + W has only point spectrum, weanalyze the asymptotic behaviour of eigenvalues and eigenfunctions of H ε as ε → αδ (cid:48) + βδ )-like potentials, the norm resolvent convergence wasestablished and a family of exactly solvable models was obtained (see also [32–34]for regularizations of αδ (cid:48) + βδ by piecewise constant potentials). Figure 1.
The ∂ ν δ γ -like potential (part of the plot is cut out forbetter visualization).The distribution − ∂ ν δ γ can be interpreted as the Laplacian of the indicator (orthe characteristic function) x ∈ Ω of the bounded domain Ω enclosed by γ . Namely, ∂ ν δ γ = − ∆ x ∈ Ω in the sense of distributions. Suppose that χ ε is a sequence of smooth functionswhich are identically one on Ω, vanish arbitrarily close to Ω, and χ ε → x ∈ Ω in L ( R ) as ε →
0. Then V ε = − ∆ χ ε is an example of ∂ ν δ γ -like potentials, seeFig. 1. Obviously, V ε converges to ∂ ν δ γ in D (cid:48) ( R ) as ε →
0. Such potentials areinteresting not only in the context of Schr¨odinger operators, they also arise in theNavier-Stokes equations, free boundary problems, and in the potential theory forparabolic and elliptic PDE in bounded domains. The Laplacian of the indicatorfunction with its grid adaptations is the base of the front-tracking method. Thisnumerical method allows simulating unsteady multi-fluid flows in which a sharpinterface separates incompressible fluids of different density and viscosity [35], atime dependent two-dimensional dendritic solidification of pure substances [36],flow-flexible body interactions with large deformation [37]. The Laplacian of the
YURIY GOLOVATY indicator and its regularizations have been used to establish some relationshipsbetween the Dirichlet and Neumann boundary value problems for the heat andLaplace equations and the theory of the Feynman path integrals [38].2.
Statement of Problem and Main Results
We study the family of Schr¨odinger operator H ε = − ∆ + W + V ε in L ( R ), where the potential W belongs to L ∞ loc ( R ) and increases as | x | → + ∞ .Let γ be a closed smooth curve in R without self-intersection points. Assume alsothat W is smooth in a neighbourhood of γ . We define the short-range potentials V ε as follows. Let ω ε be the ε -neighborhood of γ , i.e., the union of all open ballsof radius ε around a point on γ . For ε small enough, ω ε is a domain with smoothboundary. To specify explicit dependence of V ε on ε we introduce local coordinatesin ω ε (see Fig. 2). Let α : [0 , | γ | ) → R be the unit-speed smooth parametrizationof γ with the natural parameter s , and | γ | is the length of γ . Then the vector ν = ( − ˙ α , ˙ α ) is a unit normal on γ . Set x = α ( s )+ rν ( s ) for ( s, r ) ∈ [0 , | γ | ) × ( − ε, ε ),where r is the signed distance from x to γ . Suppose that V ε has the form V ε (cid:0) α ( s ) + rν ( s ) (cid:1) = ε − V (cid:0) ε − r (cid:1) + ε − U (cid:0) s, ε − r (cid:1) , (2.1)where V and U are smooth functions such that the supports of V and U ( s, · ) liein the interval [ − ,
1] for all s . Hence, supp V ε ⊂ ω ε . In general, the potentials V ε diverge in the space of distributions D ( R ); V ε converge only if V is a zero meanfunction, as we will show below. In this case, V ε → a∂ ν δ γ + bδ γ in D (cid:48) ( R ), where a and b are some functions on γ . The unperturbed operator H = − ∆ + W isself-adjoint in L ( R ) and its spectrum is discrete. Obviously, the operators H ε are also self-adjoint with discrete spectrum and dom H ε = dom H . The main taskis to describe the limiting behaviour of the spectrum of H ε by constructing theasymptotics of eigenvalues λ ε and eigenfunctions u ε of the problem − ∆ u ε + ( W + V ε ) u ε = λ ε u ε in R . (2.2)We say that the one-dimensional Schr¨odinger operator − d dr + V in L ( R ) pos-sesses a zero-energy resonance if there exists a nontrivial solution h of the equation e r s w e e - gw e x x Figure 2.
The local coordinates in ω ε . D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 5 − h (cid:48)(cid:48) + V h = 0 that is bounded on the whole line. We call h the half-bound state .The half-bound state is unique up to a scalar factor and has nonzero limits h ( −∞ ) = lim r →−∞ h ( r ) , h (+ ∞ ) = lim r → + ∞ h ( r ) . The closed curve γ divides the plane into two domains Ω − and Ω + , R = Ω − ∪ γ ∪ Ω + .Suppose Ω + is unbounded. Let us introduce the space W + ⊂ L (Ω + ) as follows.We say that f belongs to W + if there exists a functions f belonging to dom H suchthat f = f in Ω + . Also, we set W = { f ∈ L ( R ) : f | Ω − ∈ W (Ω − ) , f | Ω + ∈ W + } .Recall that v ± denote the one-side traces of v on γ .Let E ⊂ (0 ,
1) be an infinite set, for which zero is an accumulation point. Ourmain result reads as follows.
Theorem 1.
Assume that the operator − d dr + V in L ( R ) possesses a zero-energyresonance with the half-bound state h .(i) Suppose that { λ ε } ε ∈E is a sequence of eigenvalues of H ε and { u ε } ε ∈E is thecorresponding sequence of eigenfunctions such that (cid:107) u ε (cid:107) L ( R ) = 1 . If λ ε → λ, u ε → u in L ( R ) weakly (2.3) as E (cid:51) ε → , and u is a non-zero function, then λ is an eigenvalue with theeigenfunction u of the operator H = − ∆ + W in L ( R ) acting on the domain dom H = (cid:8) v ∈ W : v + = θv − , θ∂ ν v + − ∂ ν v − = (cid:0) ( θ − κ + µ (cid:1) v − on γ (cid:9) . Here θ = h (+ ∞ ) /h ( −∞ ) , κ = κ ( s ) is the curvature of γ , and µ ( s ) = 1 h ( −∞ ) (cid:90) R U ( s, r ) h ( r ) dr. (ii) If (2.3) holds and λ is not a point of σ ( H ) , then the sequence of eigenfunc-tions u ε converges to zero as E (cid:51) ε → in the weak topology of L ( R ) .(iii) For each eigenvalue λ of H and all ε small enough there exists an eigenvalue λ ε of H ε such that | λ ε − λ | ≤ cε with the constant c depending only on λ . Remark 1.
The operator − d dr with the trivial potential V = 0 possesses a zero-energy resonance with the half-bound state h = 1. Then V ε ( x ) = ε − U (cid:0) s, ε − r (cid:1) and V ε → µ δ γ in the space of distributions, where µ ( s ) = (cid:90) R U ( s, r ) dr. (2.4)Since θ = 1, the interface conditions on γv + = θv − , θ∂ ν v + − ∂ ν v − = (cid:0) ( θ − κ + µ (cid:1) v − (2.5)become v + = v − , ∂ ν v + − ∂ ν v − = µ v , which correspond to the δ -interactionsupported on the curve [7]. Remark 2.
Note that the potentials V for which the operator − d dr + V has azero-energy resonance are not something exotic. For any V of compact support,there exists a discrete infinite set of real coupling constants α such that − d dr + αV has a zero-energy resonance [27, 29].Let us introduce two operators D − = − ∆ + W in L (Ω − ) , dom D − = { v ∈ W (Ω − ) : v = 0 on γ } , D + = − ∆ + W in L (Ω + ) , dom D + = { v ∈ W + : v = 0 on γ } . YURIY GOLOVATY
Theorem 2.
Suppose that the operator − d dr + V in L ( R ) has no zero-energyresonance and { λ ε } ε ∈E is a sequence of eigenvalues of H ε and { u ε } ε ∈E is the cor-responding sequence of eigenfunctions such that (cid:107) u ε (cid:107) L ( R ) = 1 .(i) If λ ε → λ and u ε → u in L ( R ) weakly, as E (cid:51) ε → , and the limit function u is different from zero, then λ is an eigenvalue of the direct sum D − ⊕ D + and u is the corresponding eigenfunction.(ii) In the case when λ ε → λ , as E (cid:51) ε → , and λ (cid:54)∈ σ ( D − ⊕ D + ) , theeigenfunctions u ε converge to zero in L ( R ) weakly.(iii) If λ ∈ σ ( D − ⊕ D + ) , then for all ε small enough we can find an eigenvalue λ ε of H ε such that | λ ε − λ | ≤ cε , where the constant c does not depend on ε . The thin structure that is the support of V ε can produce a infinite series ofeigenvalues that go to the negative infinity as ε →
0. Although for each ε > V ε it can increaseinfinitely as ε →
0. In particular, this means that the family of operators H ε is notgenerally uniformly bounded from below with respect to ε . In this case, any realnumber can be an accumulation point of the eigenvalues λ ε of H ε . Theorems 1 and2 point out the principal difference between the eigenvalues of the limit operatorsand all other real points. This difference is that only the points of σ ( H ) in theresonant case or σ ( D − ⊕ D + ) in the non-resonant case can be approximated bythe eigenvalues λ ε of H ε so that the corresponding eigenfunctions u ε converge tonontrivial limits in L ( R ). The asymptotics of the negative low-lying eigenvaluesremains an open problem for the time being.3. Preliminaries
Returning to the local coordinates ( s, r ), we see that the couple of vectors α = ( ˙ α , ˙ α ), ν = ( − ˙ α , ˙ α ) gives the Frenet frame for γ . The Jacobian of trans-formation x = ˙ α ( s ) − r ˙ α ( s ), x = ˙ α ( s ) + r ˙ α ( s ) has the form J ( s, r ) = (cid:12)(cid:12)(cid:12)(cid:12) ˙ α ( s ) − r ¨ α ( s ) − ˙ α ( s )˙ α ( s ) + r ¨ α ( s ) ˙ α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) = ˙ α ( s ) + ˙ α ( s ) − r (cid:0) ˙ α ( s )¨ α ( s ) − ˙ α ( s )¨ α ( s ) (cid:1) = 1 − r κ ( s ) , where κ = det( ˙ α, ¨ α ) is the signed curvature of γ . We see that J is positive for suffi-ciently small r , because the curvature κ is bounded on γ . Namely, the coordinatesare defined correctly on ω ε for all ε < ε ∗ , where ε ∗ = min γ | κ | − . (3.1)Note also that κ is defined uniquely up to the reparametrization s (cid:55)→ − s . Interfaceconditions (2.5) contain the parameters θ , κ and µ which depend on the particularparametrization chosen for curve γ . The parameters change along with the changeof the Frenet frame. Proposition 1.
The operator H in Theorem 1 does not depend upon the choice ofthe Frenet frame for γ .Proof. Every smooth curve admits two possible orientations of the arc-length pa-rameter and consequently two possible Frenet frames. Let us change the Frenetframe { α, ν } to the frame {− α, − ν } and prove that conditions (2.5) will remain the D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 7 same. The change leads to the following transformations: h ( ±∞ ) (cid:55)→ h ( ∓∞ ) , θ (cid:55)→ θ − , κ (cid:55)→ − κ , µ (cid:55)→ θ − µ, u ± (cid:55)→ u ∓ , ∂ ν u ± (cid:55)→ − ∂ ν u ∓ . The first condition u + − θu − = 0 in (2.5) transforms into u − − θ − u + = 0 andtherefore remains unchanged. For the second condition, we obtain − θ − ∂ ν u − + ∂ ν u + − (cid:0) − ( θ − − κ + θ − µ (cid:1) u + = 0 . Multiplying the equality by θ yields θ∂ ν u + − ∂ ν u − − (cid:0) ( θ − κ + µ (cid:1) θ − u + = 0 , since − θ ( θ − −
1) = θ − ( θ − u − in place of θ − u + , in viewof the first interface condition. (cid:3) The metric tensor g = ( g ij ) in the orthogonal coordinates ( s, r ) has the form g = (cid:18) J
00 1 (cid:19) . In fact, we have g = | x s | = | ˙ α + r ˙ ν | = | (1 − r κ ) ˙ α | = J , by the Frenet-Serret formula ˙ ν = − κ ˙ α , and g = | x r | = | ν | = 1. Then the gradient and theLaplace-Beltrami operator in the local coordinates become ∇ φ = J − ∂ s φ α + ∂ r φ ν, ∆ φ = J − (cid:0) ∂ s ( J − ∂ s φ ) + ∂ r ( J∂ r φ ) (cid:1) . (3.2)All the results presented in Theorems 1 and 2 concern arbitrary potentials V ε of the form (2.1) that generally diverge in the distributional sense. However, thespectra of H ε converge to the spectra of the limit operators without reference tothe convergence of potentials. The following statement shows that the convergenceconditions for V ε and for the spectra of H ε are quite different. Proposition 2.
The family of potentials V ε converges in the space of distributionsif and only if (cid:82) R V dr = 0 . In this case, V ε → µ ∂ ν δ γ + ( µ κ + µ ) δ γ in D (cid:48) ( R ) , where µ = − (cid:82) R rV ( r ) dr and µ is given by (2.4) .Proof. It is evident that the sequence ε − U (cid:0) s, ε − r (cid:1) converges to µ δ γ in D (cid:48) ( R ).Write g ε = ε − V (cid:0) ε − r (cid:1) and n = ε − r . Then we have (cid:90) R g ε φ dx = (cid:90) ω ε g ε φ dx = ε − (cid:90) ε − ε (cid:90) | γ | V ( ε − r ) φ ( s, r )(1 − r κ ( s )) ds dr = ε − (cid:90) − (cid:90) | γ | V ( n ) φ ( s, εn )(1 − εn κ ( s )) ds dn = ε − (cid:90) − V ( n ) dn (cid:90) | γ | φ ( s, ds + (cid:90) − nV ( n ) dn (cid:90) | γ | (cid:0) ∂ n φ ( s, − κ ( s ) φ ( s, (cid:1) ds + O ( ε )as ε → φ ∈ C ∞ ( R ). The sequence g ε has a finite limit in D (cid:48) ( R ) iff (cid:82) R V dn = 0. In this case, we have (cid:90) R g ε φ dx → µ (cid:90) γ ( ∂ ν δ γ + κ δ γ ) φ dγ, which completes the proof. (cid:3) YURIY GOLOVATY Formal Asymptotics
Now we will show how interface conditions (2.5) can be found by direct calcu-lations. Here we use the asymptotic methods similar to those in [39–41]. In thesequel, the normal vector field ν on γ will be outward to the domain Ω − . Hencethe local coordinate r increases in the direction from Ω − to Ω + . Also, it will beconvenient to parameterize the curve γ by points of a circle. It will allow us not toindicate every time that functions on γ are periodic on s . Let S be the circle of thelength | γ | . Then ω ε is diffeomorphic to the cylinder Q ε = S × ( − ε, ε ). We denoteby γ t the curve that is obtained from γ by flowing for “time” t along the normalvector field, i.e., γ t = { x ∈ R : x = α ( s ) + tν ( s ) , s ∈ S } . Then the boundary of ω ε consists of two curves γ − ε and γ ε .We look for the approximation to the eigenvalue λ ε and the corresponding eigen-function u ε of (2.2) in the form λ ε ≈ λ, u ε ( x ) ≈ (cid:40) u ( x ) in R \ ω ε ,v (cid:0) s, rε (cid:1) + εv (cid:0) s, rε (cid:1) + ε v (cid:0) s, rε (cid:1) in ω ε . (4.1)To match the approximations in ω ε and R \ ω ε , we hereafter assume that[ u ε ] ± ε = 0 , [ ∂ r u ε ] ± ε = 0 , (4.2)where [ w ] t stands for the jump of w across γ t in the positive direction of the localcoordinate r . Since u ε solves (2.2) and the domain ω ε shrinks to γ , the function u must be a solution of the equation − ∆ u + W u = λu in R \ γ (4.3)subject to appropriate transmission conditions on γ . To find these conditions, weconsider equation (2.2) in the local coordinates ( s, n ), where n = r/ε . By (3.2), theLaplacian can be written as∆ = 11 − εn κ (cid:18) ε − ∂ n (1 − εn κ ) ∂ n + ∂ s (cid:16) − εn κ ∂ s (cid:17)(cid:19) , in the cylinder Q = S × ( − , ε − ∂ n − ε − κ ( s ) ∂ n − n κ ( s ) ∂ n + ∂ s + εP ε , (4.4)where P ε is a PDE of the second order on s and the first one on n whose coefficientsare uniformly bounded in Q with respect to ε .Substituting v + εv + ε v and (4.4) into equation (2.2) in particular yields − ∂ n v + V v = 0 , − ∂ n v + V v = − κ ∂ n v − U v , − ∂ n v + V v = − ( κ ∂ n + U ) v + ( ∂ s − n κ ∂ n − W ( · ,
0) + λ ) v (4.5)in Q . From (4.2) we see that necessarily u − ( s ) = v ( s, − , u + ( s ) = v ( s, , (4.6) ∂ n v ( s, −
1) = 0 , ∂ n v ( s,
1) = 0 , (4.7) ∂ n v ( s, −
1) = ∂ r u − ( s ) , ∂ n v ( s,
1) = ∂ r u + ( s ) . (4.8) D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 9
Combining (4.7)–(4.8), we conclude that v and v solve the problems (cid:40) − ∂ n v + V ( n ) v = 0 in Q,∂ n v ( s, −
1) = 0 , ∂ n v ( s,
1) = 0 , s ∈ S ; (4.9) (cid:40) − ∂ n v + V ( n ) v = − κ ( s ) ∂ n v − U ( s, n ) v in Q,∂ n v ( s, −
1) = ∂ r u − ( s ) , ∂ n v ( s,
1) = ∂ r u + ( s ) , s ∈ S (4.10)respectively. Hence we have the boundary value problems in Q including the “non-elliptic” partial differential operator − ∂ n + V . These problems can also be regardedas the boundary value problems on I for ordinary differential equations whichdepend on the parameter s ∈ S .4.1. Case of zero-energy resonance.
Assume that − d dr + V has a zero energyresonance with the half-bound state h . Set I = ( − , V lies in I , the function h is constant outside I as a bounded solution of the equation h (cid:48)(cid:48) = 0. Therefore the restriction of h to I is a nonzero solution of the Neumannboundary value problem − h (cid:48)(cid:48) + V h = 0 in I , h (cid:48) ( −
1) = 0 , h (cid:48) (1) = 0 . (4.11)Hereafter, we fix h by the additional condition h ( −
1) = 1. Then h ( ±∞ ) = h ( ± θ = h (1).In this case, (4.9) admits the infinitely many solutions v ( s, n ) = a ( s ) h ( n ),where a is an arbitrary function on S . From (4.6) we deduce that u − = a , u + = h (1) a = θa and hence that v ( s, n ) = u − ( s ) h ( n ) and u + = θu − on γ. (4.12)Next, problem (4.10) is in general unsolvable, since (4.9) admits nontrivial so-lutions. To find solvability conditions, we rewrite the equation in (4.10) in theform − ∂ n v + V ( n ) v = − (cid:0) κ ( s ) h (cid:48) ( n ) + U ( s, n ) h ( n ) (cid:1) u − ( s ), multiply by a ( s ) h ( n ), a ∈ L ( S ), and then integrate over Q (cid:90) Q (cid:0) − ∂ n v + V ( n ) v (cid:1) a ( s ) h ( n ) dn ds = − (cid:90) Q (cid:0) κ ( s ) h (cid:48) ( n ) + U ( s, n ) h ( n ) (cid:1) u − ( s ) a ( s ) h ( n ) dn ds. (4.13)Since h is a solution of (4.11), integrating by parts twice on the left-hand side yields (cid:90) S (cid:90) I (cid:0) − ∂ n v + V v (cid:1) ah dn ds = − (cid:90) S ( ∂ n v h − v h (cid:48) ) (cid:12)(cid:12) n =1 n = − a ds − (cid:90) S (cid:90) I av ( − h (cid:48)(cid:48) + V h ) dn ds = − (cid:90) S (cid:0) θ∂ r u + − ∂ r u − (cid:1) a ds, in view of the boundary conditions for v . Hence (4.13) becomes (cid:90) S (cid:0) θ∂ r u + − ∂ r u − (cid:1) a ds = (cid:90) S u − a (cid:90) I (cid:0) κ hh (cid:48) + U h (cid:1) dn ds. The equality hh (cid:48) = ( h ) (cid:48) implies (cid:90) I hh (cid:48) dn = ( h (1) − h ( − ( θ − . (4.14)Therefore we obtain (cid:90) S (cid:0) θ∂ r u + − ∂ r u − (cid:1) a ds = (cid:90) S (cid:0) ( θ − κ + µ (cid:1) u − a ds for all a ∈ L ( S ), where µ ( s ) = (cid:82) I U ( s, n ) h ( n ) dn . From this we deduce θ∂ r u + − ∂ r u − = (cid:0) ( θ − κ + µ (cid:1) u − on γ, which is necessary for solvability of (4.10). In view of the Fredholm alternative,this condition is also sufficient. Moreover it is a jump condition for the normalderivative of u at the interface γ , since ∂ ν u ± = ∂ r u ± on γ . Therefore λ and u in(4.1) must solve the problem − ∆ u + W u = λu in R \ γ, (4.15) u + − θu − = 0 , θ∂ ν u + − ∂ ν u − = (cid:0) ( θ − κ + µ (cid:1) u − on γ, (4.16)which can be equivalently rewritten as the spectral equation H u = λu .Assume that λ is an eigenvalue of H and u is an eigenfunction for this eigenvalue.Now we can calculated the trace u − on γ and finally determine v ( s, n ) = u − ( s ) h ( n ).Since the second condition in (4.16) holds, problem (4.10) is solvable and v isdefined up to the term a ( s ) h ( n ). Let us fix a solution of (4.10) so that v ( s, −
1) = 0 , s ∈ S. (4.17)Finally equation (4.5) admits a unique solution v satisfying the conditions v ( s, −
1) = 0 , ∂ n v ( s, −
1) = 0 , s ∈ S. (4.18)The functions v k are smooth in Q due to the smoothness of V , U and κ . Recallalso that W is smooth in an neighbourhood of γ . So we have constructed all termsin asymptotics (4.1).4.2. Non-resonant case.
Now we suppose that − d dr + V has no zero energy res-onance. Then problem (4.11), and hence problem (4.9), admit the trivial solutions h = 0 and v = 0 only, and (4.6) imply u − = 0 and u + = 0 on γ . We thus get − ∆ u + W u = λu in R \ γ, u | γ = 0 . Let us suppose that λ is an eigenvalue of the direct sum D − ⊕ D + and u is thecorresponding eigenfunction. In this case, problem (4.10) has the form (cid:40) − ∂ n v + V ( n ) v = 0 in Q,∂ n v ( s, −
1) = ∂ ν u − , ∂ n v ( s,
1) = ∂ ν u + and admits a unique solution. Let us substitute v = 0 into equation (4.5) andassume that v is a solution of the Cauchy problem (cid:40) − ∂ n v + V ( n ) v = − κ ( s ) ∂ n v + U ( s, n ) v in Q,v ( s, −
1) = 0 , ∂ n v ( s, −
1) = 0 , s ∈ S. D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 11
Quasimodes of H ε . To prove that λ belonging to either σ ( H ) or σ ( D − ⊕D + )is an accumulation point for a sequence of eigenvalues λ ε of H ε , we will apply themethod of quasimodes. Let A be a self-adjoint operator in a Hilbert space L . Wesay a pair ( a, φ ) ∈ R × dom A is a quasimode of A with the accuracy δ , if φ (cid:54) = 0and (cid:107) ( A − a ) φ (cid:107) L ≤ δ (cid:107) φ (cid:107) L . Proposition 3 ([42, p.139]) . Assume ( a, φ ) is a quasimode of A with accuracy δ > and the spectrum of A is discrete in the interval [ a − δ, a + δ ] . Then thereexists an eigenvalue λ ∗ of A such that | λ ∗ − a | ≤ δ . Since its proof is so simple, we reproduce it here for the reader’s convenience. If a ∈ σ ( A ), then λ ∗ = a . Otherwise the distance d a from a to the spectrum of A canbe computed as d a = (cid:107) ( A − a ) − (cid:107) − = inf ψ (cid:54) =0 (cid:107) ψ (cid:107) L (cid:107) ( A − a ) − ψ (cid:107) L , where ψ is an arbitrary vector of L . Taking ψ = ( A − a ) φ , we deduce d a ≤ (cid:107) ( A − a ) φ (cid:107) L (cid:107) φ (cid:107) L ≤ δ, from which the assertion follows.In order to construct the quasimodes of H ε , we must modify the approximationˆ v ε ( x ) = (cid:40) u ( x ) in R \ ω ε ,v (cid:0) s, rε (cid:1) + εv (cid:0) s, rε (cid:1) + ε v (cid:0) s, rε (cid:1) in ω ε obtained above. The approximation does not in general belong to dom H ε , becauseˆ v ε has jump discontinuities on ∂ω ε . Let us define the function ζ plotted in Fig. 3.This function is smooth outside the origin, ζ ( r ) = 1 for r ∈ [0 , β/
2] and ζ ( r ) = 0in the set R \ [0 , β ). We assume that 2 β < ε ∗ , where ε ∗ is given by (3.1). Set η ε = (cid:0) [ˆ v ε ] ε + [ ∂ ν ˆ v ε ] ε ( r − ε ) (cid:1) ζ ( r − ε ) + (cid:0) [ˆ v ε ] − ε + [ ∂ ν ˆ v ε ] − ε ( r + ε ) (cid:1) ζ ( − r − ε ) . (4.19)It is easy to check that η ε and ∂ r η ε have the same jumps across the boundary of ω ε as ˆ v ε and ∂ ν ˆ v ε respectively. In addition, η ε is different from zero in the set ω β + ε \ ω ε only. Therefore the function v ε ( x ) = (cid:40) u ( x ) − η ε ( x ) in R \ ω ε ,v (cid:0) s, rε (cid:1) + εv (cid:0) s, rε (cid:1) + ε v (cid:0) s, rε (cid:1) in ω ε belongs to the domain of H ε . We have not changed ˆ v ε too much, sincesup x ∈ R \ ω ε (cid:0) | η ε ( x ) | + | ∆ η ε ( x ) | (cid:1) ≤ cε. (4.20) Figure 3.
Plot of the function ζ . It follows from explicit formula (4.19) and the smallness of jumps of ˆ v ε and ∂ ν ˆ v ε across ∂ω ε . Indeed, using (4.6)–(4.8), (4.17) and (4.18) for the case of resonancewe deduce[ˆ v ε ] − ε = v ( s, − − u ( s, − ε ) = u − ( s ) − u ( s, − ε ) = O ( ε ) , [ˆ v ε ] ε = u ( s, ε ) − v ( s, − εv ( s, − ε v ( s, u ( s, ε ) − θu − ( s ) + O ( ε ) = u ( s, ε ) − u + ( s ) + O ( ε ) = O ( ε ) , [ ∂ ν ˆ v ε ] − ε = ε − ∂ n v ( s, −
1) + ∂ n v ( s, − − ∂ r u ( s, − ε )= ∂ r u − ( s ) − ∂ r u ( s, − ε ) = O ( ε ) , [ ∂ ν ˆ v ε ] ε = ∂ r u ( s, ε ) − ε − ∂ n v ( s, − ∂ n v ( s, − ε∂ n v ( s,
1) + O ( ε )= ∂ r u + ( s ) − ∂ r u ( s, ε ) + O ( ε ) = O ( ε ) , as ε →
0. Here we also have utilized condition (4.12) and the inequality | u ( s, ± ε ) − u ± ( s ) | + | ∂ r u ( s, ± ε ) − ∂ r u ± ( s ) | ≤ cε. Note that the eigenfunction u is smooth in a neighbourhood of γ . Obviously thejumps are also of order O ( ε ) in the non-resonant case, when v = 0 and u ± = 0. Lemma 1.
The pairs ( λ, v ε ) constructed above are quasimodes of H ε with theaccuracy O ( ε ) as ε → .Proof. Write (cid:37) ε = ( H ε − λ ) v ε . Thus (4.3) implies (cid:37) ε = ( − ∆ + W − λ )( u − η ε ) = ( − ∆ + W − λ ) η ε outside ω ε . Therefore sup x ∈ R \ ω ε | (cid:37) ε ( x ) | ≤ c ε , because of (4.20). Recall η ε is afunction of compact support. Applying representation (4.4) of the Laplace operatorin the local coordinates, we deduce − ∆ + W ( x ) + V ε ( x ) = − ε − ∂ n + ε − κ ∂ n + n κ ∂ n − ∂ s − εP ε + W ( s, εn )+ ε − V ( n ) + ε − U ( s, n ) = ε − (cid:96) + ε − (cid:96) + (cid:96) + W ( s, εn ) − εP ε for x ∈ ω ε , where (cid:96) = − ∂ n + V , (cid:96) = κ ∂ n + U and (cid:96) = n κ ∂ n − ∂ s . Then (cid:37) ε = ( − ∆+ W + V ε − λ ) v ε = (cid:0) ε − (cid:96) + ε − (cid:96) + (cid:96) + W ( s, εn ) − εP ε − λ (cid:1)(cid:0) v + εv + ε v (cid:1) = ε − (cid:96) v + ε − ( (cid:96) v + (cid:96) v ) + (cid:0) (cid:96) v + (cid:96) v + ( (cid:96) + W ( s, − λ ) v (cid:1) + ( W ( s, εn ) − W ( s, v + ε (cid:0) (cid:96) v + ( (cid:96) + W ( s, εn ) − λ )( v + εv ) − P ε v ε (cid:1) for x ∈ ω ε . From our choice of v k , we derive that the first three terms of theright-hand side vanish. The potential W is a C ∞ -function in a neighbourhood of γ , then we have sup x ∈ ω ε | (cid:37) ε ( x ) | ≤ c ε . Hence (cid:107) ( H ε − λ ) v ε (cid:107) L ( R ) = (cid:107) (cid:37) ε (cid:107) L ( R ) ≤ | ω β | / sup R | (cid:37) ε | ≤ c ε, since supp (cid:37) ε ⊂ ω β + ε ⊂ ω β for ε small enough. On the other hand, the maincontribution to the L ( R )-norm of v ε is given by the eigenfunction u . Therefore (cid:107) v ε (cid:107) L ( R ) ≥ (cid:107) u (cid:107) L ( R ) for ε small enough. Finally, we obtain (cid:107) ( H ε − λ ) v ε (cid:107) L ( R ) ≤ c ε ≤ c ε (cid:107) u (cid:107) − L ( R ) (cid:107) v ε (cid:107) L ( R ) ≤ c ε (cid:107) v ε (cid:107) L ( R ) , and this is precisely the assertion of the lemma. (cid:3) D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 13 Proof of Main Results
Let { λ ε } ε ∈E be a sequence of eigenvalues of H ε and { u ε } ε ∈E be the sequence ofthe corresponding eigenfunctions and (cid:107) u ε (cid:107) L ( R ) = 1. Let χ ε be the characteristicfunction of R \ ω ε . Lemma 2.
Assume that λ ε → λ and u ε → u in L ( R ) weakly as E (cid:51) ε → .(i) For any bounded or unbounded domain D in R such that D ∩ γ = ∅ theeigenfunctions u ε converge to u in W ( D ) weakly, and u solves the equation − ∆ u + W u = λu in R \ γ. (5.1) (ii) χ ε ∇ u ε → ∇ u in L (Ω ± ) weakly.(iii) Treating u ε ( x ) as u ε ( s, r ) , we have u ε ( · , − ε ) → u − and u ε ( · , ε ) → u + in L ( S ) weakly . Proof. (i)
Recall that supp V ε lies in ω ε and chose ε so small that D ∩ ω ε = ∅ . Thenfor any φ ∈ C ∞ ( D ) we conclude from (2.2) that (cid:90) D ∆ u ε φ dx = (cid:90) D ( W − λ ε ) u ε φ dx. The right-hand side has a limit as
E (cid:51) ε → φ ∈ C ∞ ( D ), i.e., ∆ u ε → ∆ u in L ( D ) weakly. From thiswe deduce that u ε converges to u in W ( D ) weakly, and hence that (cid:90) D ∆ uφ dx = (cid:90) D ( W − λ ) uφ dx. Since D is an arbitrary domain such that D ∩ γ = ∅ , we have (cid:90) R ∆ uφ dx = (cid:90) R ( W − λ ) uφ dx for all test functions φ ∈ C ∞ ( R ) for which supp φ ∩ γ = ∅ . Therefore u is a solutionof (5.1). (ii) We conclude from (cid:90) R \ ω ε ∆ u ε ψ dx = (cid:90) R \ ω ε ( W − λ ε ) u ε ψ dx, ψ ∈ C ∞ ( R )that the family of functionals χ ε ∆ u ε in L ( R ) is pointwise bounded, since theright-hand side is bounded as ε →
0. In view of the uniform boundedness principle,we have (cid:107) χ ε ∆ u ε (cid:107) L ( R ) ≤ c , from which the estimate (cid:107) u ε (cid:107) W ( R \ ω ε ) ≤ c follows.Now for ψ ∈ C ∞ (Ω + ) and ε small enough, we have (cid:90) Ω + ( χ ε ∇ u ε − ∇ u ) ψ dx = (cid:90) supp ψ ( ∇ u ε − ∇ u ) ψ dx → ε →
0, in view of (i) . Therefore χ ε ∇ u ε → ∇ u in L (Ω + ) weakly, because C ∞ (Ω + ) is dense in L (Ω + ). Similar considerations apply to Ω − . (iii) Choose the cutoff function ζ ε ( r ) = (cid:40) ( r − ε ) ζ ( r ) if r ≥ ε, where ζ is plotted in Fig. 3. Let a be a smooth function on γ . Multiplying equation(2.2) by a ( s ) ζ ε ( r ) and integrating by parts yield (cid:90) γ ε u ε a dγ = (cid:90) ω ε,β ( W − λ ε ) u ε aζ ε dx − (cid:90) ω ε,β u ε ∆( aζ ε ) dx, (5.2)since ζ ε ( ε ) = 0 and ζ (cid:48) ε ( ε + 0) = 1. Here ω ε,β = { x ( s, r ) : s ∈ S, ε < r < β } is thesupport of aζ ε . Similarly, from (5.1) we obtain the equality (cid:90) γ u + a dγ = (cid:90) ω ,β ( W − λ ) uaζ dx − (cid:90) ω ,β u ∆( aζ ) dx, where ζ ( r ) = rζ ( r ). It is evident that (cid:90) ω ε,β ( W − λ ε ) u ε aζ ε dx → (cid:90) ω ,β ( W − λ ) uaζ dx, because ζ ε converges to ζ uniformly on R . Next, we have (cid:90) ω ε,β u ε ∆( aζ ε ) dx = (cid:90) ω β ,β u ε ∆( aζ ε ) dx + (cid:90) ω ε, β u ε ∆( aζ ε ) dx. The first integral of the right hand side converges to (cid:90) ω β ,β u ∆( aζ ) dx, since ∆( aζ ε ) → ∆( aζ ) uniformly on [ β , β ]. Recalling (3.2), we can write∆( aζ ε ) = (cid:0) ζ ε ∂ s ( a (cid:48) J − ) + ∂ r ( aJζ (cid:48) ε ) (cid:1) = J − (cid:0) ( r − ε ) ∂ s ( a (cid:48) J − ) − a κ (cid:1) in the set ω ε, β , since ζ ε ( r ) = r − ε for r ∈ [ ε, β ]. From this we conclude that (cid:90) ω ε, β u ε ∆( aζ ε ) dx = (cid:90) β ε (cid:90) S u ε ( s, r ) (cid:0) ( r − ε ) ∂ s ( a (cid:48) ( s ) J − ( s, r )) − a ( s ) κ ( s ) (cid:1) ds dr → (cid:90) β (cid:90) S u ( s, r ) (cid:0) r∂ s ( a (cid:48) ( s ) J − ( s, r )) − a ( s ) κ ( s ) (cid:1) ds dr = (cid:90) ω , β u ∆( aζ ) dx as E (cid:51) ε →
0, and finally that (cid:90) ω ε,β u ε ∆( aζ ε ) dx → (cid:90) ω ,β u ∆( aζ ) dx. (5.3)Combining now (5.2)–(5.3) we at last deduce (cid:82) γ ε u ε a dγ → (cid:82) γ u + a dγ for all a ∈ C ∞ ( γ ), hence u ε ( · , ε ) → u + in L ( S ) weakly. The proof of the weak convergencefor u ε ( · , − ε ) is similar. (cid:3) Proof of Theorem 1.
Assume first that the operator − d dr + V possessesa zero-energy resonance. Let Ψ θ be the class of functions ψ of compact supportthat are twice differentiable in R \ γ , bounded together with their first and secondderivatives in the closure of Ω + and Ω − and ψ + = θψ − on γ . We also setΦ = { φ ∈ W ( R ) : φ has a compact support } . D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 15 If λ and u are the eigenvalue and the corresponding eigenfunction of H , then (cid:90) Ω + ∇ u ∇ ψ dx + (cid:90) Ω − ∇ u ∇ ψ dx + (cid:90) R ( W − λ ) uψ dx + (cid:90) γ Υ u − ψ − dγ = 0 (5.4)for all ψ ∈ Ψ θ , where Υ = ( θ − κ + µ . We want to take the limit as E (cid:51) ε → (cid:90) R (cid:0) ∇ u ε ∇ φ + ( W + V ε − λ ε ) u ε φ (cid:1) dx = 0 , φ ∈ Φ , (5.5)and to obtain (5.4) for the limiting function u . But identities (5.5) and (5.4) holdfor the different sets of test functions. If θ (cid:54) = 0, the set Ψ θ is not contained in Φ,because the functions from Ψ θ have jump discontinuities on γ .We introduce the family of operators R ε : Ψ θ → Ψ as follows. Let h = h ( n )and h = h ( s, n ) be solutions of the Cauchy problems − h (cid:48)(cid:48) + V h = 0 , h ( −
1) = 0 , h (cid:48) ( −
1) = 1; (5.6) − h (cid:48)(cid:48) + V h = κ h (cid:48) + U h, h ( s, −
1) = 0 , ∂ n h ( s, −
1) = 0 (5.7)on the interval I , where h is a half-bound state of − d dr + V such that h ( −
1) = 1.Given ψ ∈ Ψ θ , we write ψ ε ( s, n ) = ψ ( s, − ε ) h ( n ) , ψ ε ( s, n ) = ∂ r ψ ( s, − ε ) h ( n ) − ψ ( s, − ε ) h ( s, n ) . (5.8)Then we set ˆ ψ ε ( x ) = (cid:40) ψ ( x ) , if x ∈ R \ ω ε ,ψ ε ( s, rε ) + εψ ε (cid:0) s, rε (cid:1) if x ∈ ω ε . The function ˆ ψ ε is continuous on γ − ε by construction. But it does not in generalbelong to W ( R ), because it has a discontinuity on γ ε . Let R ε ψ = ˆ ψ ε + ρ ε , where ρ ε ( x ) = (cid:40) − [ ˆ ψ ε ] ε ζ ( r − ε ) , if x ∈ ω β \ ω ε , , otherwise . The direct calculations show that [ R ε ψ ] ε = 0 and, therefore, R ε ψ belongs to W ( R ). We immediately see that R ε ψ → ψ in L ( R ) as ε →
0, since ψ ε and ψ ε are bounded in the small set ω ε and[ ˆ ψ ε ] ε ( s ) = ψ ( s, ε ) − ψ ε ( s, − εψ ε ( s,
1) = ψ ( s, ε ) − θψ ( s, − ε ) + O ( ε )= ψ ( s, +0) − θψ ( s, −
0) + O ( ε ) = O ( ε ) (5.9)as ε → S . Proposition 4.
For any ψ ∈ Ψ θ , we have (cid:90) Q ( ∂ n u ε ∂ n ψ ε + V u ε ψ ε ) J ε dn ds = ε (cid:90) Q κ u ε ∂ n ψ ε dn ds, (5.10) (cid:90) Q ( ∂ n u ε ∂ n ψ ε + V u ε ψ ε + U u ε ψ ε ) J ε dn ds = (cid:90) S (cid:16) u ε ( s, − ε ) ∂ r ψ ( s, − ε ) (cid:0) ε κ ( s ) (cid:1) − θ − u ε ( s, ε ) (cid:0) ∂ r ψ ( s, − ε ) − Υ( s ) ψ ( s, − ε ) (cid:1) (1 − ε κ ( s )) (cid:17) ds − (cid:90) Q κ u ε ∂ n ψ ε dn ds + ε (cid:90) Q κ u ε ( κ ∂ n ψ ε − ∂ n ψ ε ) dn ds, (5.11) where J ε ( s, n ) = 1 − εn κ ( s ) and ψ εk are given by (5.8) .Proof. The function ψ ε solves − ∂ n v + V v = 0 in Q = S × I and satisfies theconditions h (cid:48) ( −
1) = h (cid:48) (1) = 0. Then0 = (cid:90) Q u ε ( − ∂ n ψ ε + V ψ ε ) J ε dn ds = − (cid:90) S ψ ( s, − ε )( u ε J ε h (cid:48) ) (cid:12)(cid:12) n =1 n = − ds + (cid:90) Q ( ∂ n u ε ∂ n ψ ε + V u ε ψ ε ) J ε dn ds + (cid:90) Q u ε ∂ n J ε ∂ n ψ ε dn ds = (cid:90) Q ( ∂ n u ε ∂ n ψ ε + V u ε ψ ε ) J ε dn ds − ε (cid:90) Q κ u ε ∂ n ψ ε dn ds, (5.12)from which (5.10) follows. Since h (1) = θ , the Lagrange identity ( h h (cid:48) − h (cid:48) h ) | − = 0for (5.6) implies h (cid:48) (1) = θ − . (5.13)Multiplying the equation in (5.7) by h and integrating by parts twice yield( h (cid:48) h − h ∂ n h ) (cid:12)(cid:12) − = κ ( s ) (cid:90) I hh (cid:48) dn + (cid:90) I U ( s, n ) h ( n ) dn. Recalling now (4.14), we derive that θ ∂ n h ( s,
1) = − ( θ − κ ( s ) − µ ( s ) andfinally that ∂ n h ( s,
1) = − θ − Υ( s ) . (5.14)Next, ψ ε is a solution of − ∂ n v + V v = − κ ∂ n ψ ε − U ψ ε , which follows from (5.6)and (5.7). Hence (cid:90) Q u ε ( − ∂ n ψ ε + V ψ ε + U ψ ε ) J ε dn ds = − (cid:90) Q κ u ε ∂ n ψ ε J ε dn ds. (5.15) D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 17
On the other hand, integrating by parts with respect to n , we find − (cid:90) Q u ε ∂ n ψ ε J ε dn ds = (cid:90) Q ( J ε ∂ n u ε − ε κ u ε ) ∂ n ψ ε dn ds − (cid:90) S u ε ( s, εn ) J ε ( s, n ) (cid:0) ∂ r ψ ( s, − ε ) h (cid:48) ( n ) − ψ ( s, − ε ) ∂ n h ( s, n ) (cid:1)(cid:12)(cid:12)(cid:12) n =1 n = − ds = (cid:90) Q ( J ε ∂ n u ε − ε κ u ε ) ∂ n ψ ε dn ds − (cid:90) S (cid:18) θ − u ε ( s, ε ) (cid:0) ∂ r ψ ( s, − ε ) − Υ( s ) ψ ( s, − ε ) (cid:1) (1 − ε κ ( s )) − u ε ( s, − ε ) ∂ r ψ ( s, − ε ) (cid:0) ε κ ( s ) (cid:1)(cid:17) ds, in view of initial conditions (5.6), (5.7) and equalities (5.13), (5.14). Substitutingthe last equality into (5.15), we obtain (5.11). (cid:3) Lemma 3.
Under the assumptions of Lemma 2, we have (cid:90) R \ ω ε ∇ u ε ∇ ( R ε ψ ) dx → (cid:90) Ω + ∇ u ∇ ψ dx + (cid:90) Ω − ∇ u ∇ ψ dx, (5.16) (cid:90) ω ε (cid:0) ∇ u ε ∇ ( R ε ψ ) + V ε u ε R ε ψ (cid:1) dx → (cid:90) γ Υ u − ψ − dγ (5.17) as E (cid:51) ε → for all ψ ∈ Ψ θ .Proof. Set ψ ε = R ε ψ . Recalling (3.2), we write (cid:90) R \ ω ε ∇ u ε ∇ ψ ε dx = (cid:90) Ω + χ ε ∇ u ε ∇ ψ dx + (cid:90) Ω − χ ε ∇ u ε ∇ ψ dx − (cid:90) βε (cid:90) S [ ˆ ψ ε ] ε ∂ r u ε ζ (cid:48) ( r − ε ) J ds dr − (cid:90) βε (cid:90) S ζ ( r − ε ) ∂ s u ε ∂ s [ ˆ ψ ε ] ε J − ds dr. Hence assertion (5.16) follows from Lemma 2 (ii) and (5.9). Next, we have (cid:90) ω ε (cid:0) ∇ u ε ∇ ψ ε + V ε u ε ψ ε (cid:1) dx = ε − (cid:90) Q ( ∂ n u ε ∂ n ψ ε + V u ε ψ ε ) J ε dn ds + (cid:90) Q ( ∂ n u ε ∂ n ψ ε + V u ε ψ ε + U u ε ψ ε ) J ε dn ds + ε (cid:90) Q U u ε ψ ε J ε dn ds + ε (cid:90) Q ∂ s u ε ∂ s ψ ε J − ε dn ds. (5.18)In view of Proposition 4, we deduce (cid:90) ω ε (cid:0) ∇ u ε ∇ ψ ε + V ε u ε ψ ε (cid:1) dx = (cid:90) S (cid:16) u ε ( s, − ε ) ∂ r ψ ( s, − ε ) (cid:0) ε κ ( s ) (cid:1) − θ − u ε ( s, ε ) (cid:0) ∂ r ψ ( s, − ε ) − Υ( s ) ψ ( s, − ε ) (cid:1) (1 − ε κ ( s )) (cid:17) ds + ε (cid:90) Q u ε (cid:0) κ ∂ n ψ ε − κ ∂ n ψ ε + U ψ ε J ε (cid:1) dn ds + ε (cid:90) Q ∂ s u ε ∂ s ψ ε J − ε dn ds. (5.19) For any sequence { w ε } ε> bounded in L ( Q ), the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Q u ε ( s, εn ) w ε ( s, n ) dn ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)(cid:90) Q | u ε ( s, εn ) | dn ds (cid:19) / (cid:107) w ε (cid:107) L ( Q ) ≤ c (cid:18) ε − (cid:90) ω ε | u ε ( x ) | dx (cid:19) / ≤ c ε − / holds, since (cid:107) u ε (cid:107) L ( R ) = 1. Also, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Q ∂ s u ε ∂ s ψ ε J − ε dn ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) Q u ε ∂ s ( J − ε ∂ s ψ ε ) dn ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ε − / , because κ ∈ C ∞ ( γ ) and ψ ∈ Ψ γ ( R ) and, therefore, the function ∂ s ( J − ε ∂ s ψ ε ) isbounded on Q uniformly on ε . Then (5.19) and Lemma 2 (iii) implies (cid:90) ω ε (cid:0) ∇ u ε ∇ ψ ε + V ε u ε ψ ε (cid:1) dx → (cid:90) S (cid:16) u ( s, − ∂ r ψ ( s, − − θ − u ( s, +0) (cid:0) ∂ r ψ ( s, − − Υ( s ) ψ ( s, − (cid:1)(cid:17) ds = (cid:90) γ (cid:16) u − ∂ r ψ − − θ − u + (cid:0) ∂ r ψ − − Υ ψ − (cid:1)(cid:17) dγ = (cid:90) γ Υ u − ψ − dγ, since θ − u + = u − . (cid:3) Now we can finish the proof of Theorem 1. If λ ε → λ and u ε → u in L ( R )weakly as E (cid:51) ε →
0, then for all ψ ∈ Ψ θ (cid:90) R (cid:0) ∇ u ε ∇ ψ ε + ( W + V ε − λ ε ) u ε ψ ε (cid:1) dx = (cid:90) R \ ω ε ∇ u ε ∇ ψ ε dx + (cid:90) ω ε (cid:0) ∇ u ε ∇ ψ ε + V ε u ε ψ ε (cid:1) dx + (cid:90) R ( W − λ ε ) u ε ψ ε dx → (cid:90) Ω + ∇ u ∇ ψ dx + (cid:90) Ω − ∇ u ∇ ψ dx + (cid:90) γ Υ u − ψ − dγ + (cid:90) R ( W − λ ) uψ dx, in view of Lemma 3. Hence the identity (5.4) holds for the pair ( λ, u ). If the limitfunction u is different from zero, then it must be an eigenfunction of the operator H associated with the eigenvalue λ . If λ (cid:54)∈ σ ( H ), then u = 0. In view of Lemma 1and Proposition 3, there exists an eigenvalue λ ε of H ε such that | λ ε − λ | ≤ c ε for all ε small enough, not only for ε ∈ E .5.2. Proof of Theorems 2.
Suppose now that the operator − d dr + V has nozero-energy resonance. First we note that the assertions of Lemmas 1 and 2 areindependent of whether − d dr + V has a zero-energy resonance or not.Set Φ γ = { φ ∈ Φ : φ = 0 on γ } . If λ is an eigenvalue with eigenfunction u of thedirect sum D − ⊕ D + , then u belongs to Φ γ and (cid:90) R (cid:0) ∇ u ∇ φ + ( W − λ ) uφ (cid:1) dx = 0 for all φ ∈ Φ γ . (5.20) D SCHR ¨ODINGER OPERATORS WITH SINGULAR POTENTIALS 19
In this case the proof is much easier, because Φ γ is a subspace of Φ. Setting R ε = I and arguing as in the proof of Theorem 1, we can take the limit as E (cid:51) ε → γ and obtain identity (5.20). It remains to prove that u = 0 on γ . Lemma 4.
Under the assumptions of Theorem 2, we have as
E (cid:51) ε → that u ε ( · , ± ε ) → in L ( S ) weakly . Proof.
Let h be the solution of the Cauchy problem − h (cid:48)(cid:48) + V h = 0 in I , h ( −
1) = 1 , h (cid:48) ( −
1) = 0 . Set θ = h (1) and ψ ε ( x ) s = ψ ( s, − ε ) h ( rε ) for some ψ ∈ Ψ θ . Then the function ψ ε ( x ) = (cid:40) ψ ( x ) + ( θ ψ ( s, − ε ) − ψ ( s, ε )) ζ ( r − ε ) , if x ∈ R \ ω ε ,ψ ε ( x ) if x ∈ ω ε belongs to Φ. Reasoning as in (5.12) we obtain (cid:90) Q ( ∂ n u ε ∂ n ψ ε + V u ε ψ ε ) J ε dn ds = − h (cid:48) (1) (cid:90) S u ε ( s, ε ) ψ ( s, − ε ) ds + εh (cid:48) (1) (cid:90) S κ ( s ) u ε ( s, ε ) ψ ( s, − ε ) ds + ε (cid:90) Q κ u ε ∂ n ψ ε dn ds instead of (5.10) in the resonant case. From (5.18) we deduce (cid:90) ω ε (cid:0) ∇ u ε ∇ ψ ε + V ε u ε ψ ε (cid:1) dx = − ε − h (cid:48) (1) (cid:90) S u ε ( s, ε ) ψ ( s, − ε ) ds + o (1) , as ε →
0. In the non-resonant case, h (cid:48) (1) is always different from zero. However,from identity (5.5) and Lemma 2 (ii) it follows immediately that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ω ε (cid:0) ∇ u ε ∇ ψ ε + V ε u ε ψ ε (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C for all ε ∈ E . Hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) S u ε ( s, ε ) ψ ( s, − ε ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ cε, where c does not depend of ε . By the arbitrariness of ψ , we have that u ε ( · , ε ) → L ( S ) weakly. To prove the weak convergence of u ε ( · , − ε ) to zero, we can choose h as a solution of − h (cid:48)(cid:48) + V h = 0 in I , h (1) = 1, h (cid:48) (1) = 0. (cid:3) Using Lemma 4 and part (iii) of Lemma 2 we get u ± = 0. The rest of the proofruns as before. References [1] Green, I. M., Moszkowski, S. A. Nuclear coupling schemes with a surface delta interaction.Physical Review, 1965, 139(4B), B790.[2] Lloyd, P. Pseudo-potential models in the theory of band structure. 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