Strong coupling asymptotics for a singular Schrödinger operator with an interaction supported by an open arc

Transcription

1 Strong coupling asymptotics for a singular Schrödinger operator with an interaction supported by an open arc Pavel Exner and Konstantin Pankrashkin Department of Theoretical Physics, Nuclear Physics Institute Czech Academy of Sciences, 568 Řež near Prague, Czechia Doppler Institute for Mathematical Physics and Applied Mathematics Czech Technical University, Břehová 7, 59 Prague, Czechia exner@ujf.cas.cz Laboratoire de mathématiques d Orsay, UMR 868 Université Paris-Sud, Bâtiment 45, 94 Orsay, France konstantin.pankrashkin@math.u-psud.fr Abstract We consider a singular Schrödinger operator in L R written formally as δx where is a C 4 smooth open arc in R of length L with regular ends. It is shown that the jth negative eigenvalue of this operator behaves in the strong-coupling limit, +, asymptotically as E j = 4 + µ j + O log where µ j is the jth Dirichlet eigenvalue of the operator d ds κs 4 on L, L with κs being the signed curvature of at the point s, L., Introduction Singular Schrödinger operators with interaction supported by manifolds of a lower dimension have been a subject of investigation in numerous papers,

2 particularly in the last decade. One motivation came from physics where operators formally written as δx with >, where is metric graph embedded in a Euclidean space, are used as models of leaky quantum graphs describing motion of particles confined to a graph in a way allowing quantum tunneling between different parts of. At the same time there is a mathematical motivation to study such operators because they exhibit nontrivial and intetesting relations between spectral properties and the geometry of the interaction support. In the informal language, the above operator is the Laplacian with the boundary conditions on, [ f] = f, where [ f] denotes the jump of the normal derivative of f on ; the rigorous definition is given by the associated sesquilinear form [3], see below, and the boundary conditions should be understood in a certain weak sense. An overview of known results concerning leaky quantum graphs is given in [5] which also offers a number of open problems. Some of them concern the strong-coupling behavior of such operators. For large one expects the eigenfunctions corresponding to eigenvalues at the bottom of the spectrum to be strongly concentrated around which suggests the asymptotic spectral behaviour might be determined by a one-dimensional problem. In the simplest case when we consider the indicated operator in L R and is a sufficiently smooth curve without self-intersections and endpoints either an infinite one with a suitable asymptotic behaviour or a loop such result is indeed known [5,6]: the eigenvalues at the bottom of the spectrum diverge as 4 but the next term in the expansion is the respective eigenvalue of a one-dimensional Schrödinger operator with a potential determined by the curvature of. We note that the smoothness hypothesis is essential; the asymptotics is expected to be completely different, e.g., if has corners, cf. [8]. One asks naturally how such an asymptotics could look like if the curve has endpoints and one has to impose boundary conditions to make the corresponding one-dimensional Schrödinger operator self-adjoint. Note that the Hamiltonian in question can be viewed as a special type mixed problem, cf. e.g. [, 7]. A conjecture was made in Sec. 7. of [5] that under proper regularity assumptions it is the Dirichlet condition which gives the asymptotics. The aim of the present paper is to prove this conjecture in the case when is a C 4 smooth arc in R with regular endpoints. A precise formulation of this result is stated in the next section and the rest of the paper is devoted to the proof. As in the case of a curve without endpoints we employ a bracketing argument imposing Dirichlet and Neumann condition at the boundary of a tubular

3 neighbourhood of. In the present case, however, we need a neighbourhood extending beyond the endpoints and we loose the asymptotic separation of variables employed in [6]. Instead we have to establish the decay of eigenfunctions away of which is technically the main part of the proof. Main result Let be an open C 4 arc in R of length L > and with regular ends. More precisely, we assume that, for some l >, there is an injective C 4 function Γ : [ l, L + l ] s Γ s, Γ s R satisfying at any point Γ s =, and the arc is identified with Γ, L. Denote by κs the signed curvature of at Γs, i.e. κs := Γ sγ s Γ sγ s. Let >. Consider the sesquilinear form h defined on H R by h f, f = f dx fx ds, R and let H be the self-adjoint operator in L R associated with h. Since has a finite length, it is easy to see that the essential spectrum of H is [, +. Denote by E E... E j... the negative eigenvalues of H with their multiplicities taken into account. Our main result reads as follows: Theorem. For any j N, the asymptotic expansion E j = 4 + µ j + O log holds for strong coupling, +, where µ j is the jth Dirichlet eigenvalue of the Schrödinger operator d ds κs 4 with curvature-induced potential on [, L]., 3 Scheme of the proof We put τs := Γ s Γ s, ns := Γ s Γ s ; 3

4 in other words τs is a unit tangent vector and ns is a unit normal vector to at the point Γs, by assumption both continuously depending on the arc-length parameter, not only on the arc itself but also on the extensions beyond its endpoints, i.e. for s [ l, L + l ]. In what follows we denote K := κ L l,l+l. For any α, l let us introduce the following subdomains in R : and the prolonged arc P α := α, L + α α, α, Ωα := { Γs + tns : s, t, L α, α }, Πα := { Γs + tns : s, t P α } 3 a := Γ a, L + a Πa. Clearly, a for any a >. Furthermore, one can check in a similar way as in [6] that there is a, such that the map K P a s, t Φs, t = Γs + tns Πa 4 is a diffeomorphism for any fixed a, a ]. Throughout the rest of the paper we will always use a = 6 log. 5 Let us introduce the following sesquilinear forms: h,a = f dx f ds, f H Πa, 6 h,a = Πa Ωa ĥ,a = Πa f dx f ds, f H Ωa, 7 f dx f ds, f H Πa, 8 a and denote the associated self-adjoint operators, acting respectively in L Πa, L Ωa and L Πa, by L, L, L. We consider their eigenvalues Λ j, Λ j, Λ j enumerated in the non-decreasing order taking their multiplicities into account; by the max-min principle we have E j Λ j. 9 The asymptotic behavior of the right-hand side can be found easily: 4

5 Proposition. For all sufficiently large one has Λ j = 4 + µ j + O log Proof. Due to the max-min principle for any j N we have Λ j Λ j Λ j. Furthermore, the asymptotics of the estimating eigenvalues Λ j and Λ j can be obtained using the technique introduced in [6], that is, an asymptotic separation of variables: Λ j = 4 + µ j + O log., Λj = 4 + µ j + O log where µ j and µ j are the jth Dirichlet eigenvalues of the operators acting as d ds κs 4 on [, L] and [ a, L + a], respectively; recall that a depends on. As the Dirichlet eigenvalues are C functions of the interval edges, see e.g. [4], we log have µ j = µ j + Oa = µ j + O, which proves the result. Hence the claim of Theorem will be a consequence of the following asymptotic relation: Proposition 3. For any j N one has log Λ j E j = O as the coupling parameter tends to +. This is our main estimate and the rest of the paper will dedicated to the proof of Proposition 3., 4 Technical estimates We denote by dx, the distance between a point x R and the arc. In the present section we give some expressions of dx, for x Πa which we need in the following. Some the formulæ are known, but we prefer to collect all the necessary information in this section for the sake of completeness. Recall first the Frenet formulæ τ s = κsns, n s = κsτs. 5

6 In particular, for all s, t, s, t P a one has the representations Γs = Γs + s sτs + s s ρ s, s, ns = ns s sκsτs + s s ρ s, s, τs = τs + s sκsns + s s ρ 3 s, s, 3 Φs, t = Φs, t + s s t κs τs + t tns + s s ρ s, s + t ρ s, s = Φs, t + s s t κs τs + t tns + s s ρ 4 s, t, s, t, 4 with ρ, ρ, ρ 3 L a, L + a, ρ 4 L P a. Lemma 4. Let α, a. Then there are C, C > such that C s s +t t Φs, t Φs, t C s s +t t 5 holds for all s, t, s, t P α. Proof. We have P α P a and Πα Πa. The upper bound in 5 follows then from the boundedness of the partial derivatives of Φ on P α. Let us prove the lower one. Suppose that the inequality is not valid, then one can find sequences s n, t n, s n, t n P α such that, for all n N, Φs n, t n Φs n, t n < r n n, r n := s n s n + t n t n. 6 As P α is compact, without loss of generality we may assume that both the sequences converge, s n, t n s, t and s n, t n s, t as n with some s, t, s, t P α, and by 6 one has Φs, t = Φs, t. As Φ is a diffeomorphism between P a P α and Πa, one has s, t = s, t, and consequently lim n r n =. On the other hand, using the representation 4 and the fact that τs and ns are unit vectors, we get Φs n, t n Φs n, t n = t nκs n s n s n + t n t n + Or 3/ n. We have t n < α < a < for any n, and choosing n large enough hence K having r n small, we obtain Φs n, t n Φs n, t n r n 4 which contradicts, however, to relation 6. 6

7 Lemma 5. There exists α, a such that d Φs, t, = t holds for all s, t, L α, α. Proof. Let us pick s, t, L α, α and consider the function f :, L R + defined by fσ = Φs, t Φσ, = Γσ Γs tns Γσ Γs + t. Using again the fact that τσ = we find f σ = τσ Γσ Γs tns τσ, f σ = kσnσ Γσ Γs + tkσns nσ, in particular, fs = t, f s = and f s = tκs. Hence one can choose α sufficiently small to have f s > for all s, t, L α, α, in which case s is a local minimum of f. Note also that f is bounded. Therefore, using the Taylor expansion, we see that one can find δ > such that Φs, t Φσ, t s σ + 4 for all s, t, L α, α and all σ with < s σ < δ. > t 7 On the other hand, we infer from Lemma 4 that there are α > and c > such that Φs, t Φσ, c t + s σ c s σ 8 holds for all s, t, L α, α and all σ, L. Choosing now α < mincδ, α, we get for any s, t, L α, α the following alternatives cδ > α > t, s σ δ by 8, Φs, t Φσ, > t, < s σ < δ by 7, = t, σ = s, which concludes the proof. Lemma 6. There exists α, a such that d Φs, t, { Φs, t Φ,, s, t α, α, α, = Φs, t ΦL,, s, t L, L + α α, α. Proof. We will prove the first equality only, the second one can be demonstrated in a similar way. Pick s, t a, a, a and consider the function f s,t :, L R +, f s,t σ = Φs, t Φσ, Γs Γσ + tns. 7

8 Using,, 3 and denoting κ := κ, τ = τ, n := n we have which gives Γs = Γ + sτ + s ρ s,, Γσ = Γ + στ + σ ρ σ,, ns = n sκ τ + s ρ s,, τσ = τ + σk n + σ ρ 3 σ,, f s,tσ = τσ Γs Γσ + tns = τ + σk n + σ ρ 3 σ tκ s σ τ + tn + s ρ s, σ ρ σ, + ts ρ s,. Using the orthogonality of τ and n, this can be rewritten in the form f s,tσ = tk σ s + s As, t, σ + σ Bs, t, σ, where A and B are certain bounded functions. Hence one can choose a, a such that for all s, t a, a, a and all σ, a one has f s,tσ >, and consequently f s,t = Next one can find a, a such that inf f s,tσ. σ,a B Φ,, a = B Φ,, a Γ, a, and finally we take α, a such that Φ α, α, α B Φ,, a For any s, t α, α, α we infer now, using the monotonicity of the associated function f s,t, that d Φs, t, = inf Φs, t x x = inf Φs, t x x BΦ,,α Γ,a = inf Φs, t Φσ, σ,a : Γσ BΦ,,α = inf σ,a. Φs, t Φσ, = Φs, t Φ,. 8

9 Lemma 7. For s < we have in the limit s, t the relation d Φs, t, = s + t + Os + t. 9 Similarly, for s > L and s, t L, we have d Φs, t, = s L + t + O s L + t. Proof. We again limit ourselves to checking the first relation; the proof of the second one is analogous. By Lemma 6, for s, t close to, with s < one has d Φs, t, = Γs Γ + tns = sτ + s ρ s, + tn tsκ τ + ts ρ s, = s + t + ts As, t + s 3 Bs, t with some bounded functions A and B, where we have again employed the orthogonality of τ and n. Hence we have d Φs, t, = s + t + O s + t, which yields the relation 9. Applying Lemmata 5 and 7 to the boundary of Πa we obtain Corollary 8. There are α, a and C > such that dx, α Cα holds for all α, α and x Πα. For a fixed b > we introduce the set W b = {x : dx, < b}. and derive an integral estimate on the complement of such a neighborhood: Lemma 9. Let k, c >. In the limit + we have e log dx, dx = O k+. R \W k log c Proof. During the demonstration we denote by C j various fixed positive k numbers. Pick p, with p >. Then by Lemmata 5 and 7 one k can find α > such that 9

10 d Φs, t, = t holds for all s, L and t α, α, p s + t d Φs, t, p s + t holds for all s α, and t α, α, and similarly, p s L + t d Φs, t, p s L + t holds for all s L, L + α and t α, α. One can represent the integration domain as follows: k log c [ k log c ] R \ W = W α \ W [ ] W L \ W α [ R \ W L ]. Let us estimate the contribution to the integral from each of these three components. Using the diffeomorphism Φ one easily reduces the integration on W α\w k log to the integration on two rectangles and two half-discs: this yields the estimate W α\w k log c C p k log c x p α e log dx, dx C 3 e p log x dx + C p α L re p log r dr + C 4 α k log c α e log t dt ds e log t dt. k log c p k log c We have α k log c e log t dt = k e c+ k log c log e α log log = O k+ and similarly, p α k log c p log re p log r dr = O = O p k+ k+.

11 Putting these estimates together we find W α\w k log c e log dx, dx = O k+. Furthermore, the measure of the second component, W L\W α, certainly does not exceed 9πL, while for all x in this domain the integrated function is majorized by e log dx, α e α, which gives e log dx, dx 9πL α e α = Oe α/. W L\W α Finally, to estimate the integral over the the complement of W L let us pick a point x and consider x / W L. One has dx, x x L x x x x Hence we have e ln dx, dx R \W L = π L x x >L e ln x x / dx = x x. re ln r/ 4L dr = π log + 4 log L e L and summing up the three terms one obtains the sought result. = Oe L/, 5 Eigenfunctions estimates Let us give first a rough a priori estimate for the eigenvalues E j. Lemma. For any j N one has ln as the coupling parameter +. E j + ln Proof. The upper bound follows from 9 and Proposition. To prove the other inequality, note that one can construct a C 4 loop such that

12 . Denote by N the self-adjoint operator in L R associated with the sesquilinear form n f, f = f dx f ds R and denote by Ẽj its eigenvalues arranged in the ascending order with their multiplicities taken into account. By the max-min principle, we have Ẽ j E j where the left-hand side behaves by [6] asymptotically as Ẽ j = 4 + κ j + O log κ j being the eigenvalues of the Schrödinger operator with the curvatureinduced potential on. This gives E j + O, and thus the sought result. Let u j, be now an L -normalized eigenfunction of H corresponding to the eigenvalue E j, j N. By [3, ] one can represent it as u j, x = G x, y; EF j, y ds y, where F j, L is an appropriate solution to the integral equation G x, y; Ej F j, yds y = F j,x, x, coming from the corresponding Krein s formula, and G is the Green function of the two-dimensional free Laplacian given explicitly by, G x, y; z = π K z x y ; here and in the following K ν denotes the modified Bessel function of order ν, see [, Section 9.6]. The following estimate will be of crucial importance for our result. Lemma. F j, L = O log holds as +. Proof. Throughout the proof again C j will denote various positive constants. To avoid using cumbersome notation we identify the function F j, with F j, Φ, Fj, Γ and write simply E instead of Ej. We will employ the following well-know relation [, Eqs and 9.6.7]: π K ν w = w e w + o, w +, ν =,, K = K. 3

13 According to and, one has u j, = F j,, 4 and moreover, using and 3 we can write u j, x = Ey x K E x y Fj, y ds y. π x y Another property to use [, Eqs and 9.6.] is the representation K t = t + Mt, Mt = tg t log t + g t, 5 where g and g are analytic functions. It yields u j, x = π y x x y F j,y ds y + π Ey x M E y x F j, y ds y. 6 x y Let us estimate the expression ns u j, Φs, t. In view of the representation 5 and the asymptotics we have a uniform bound Mw πc for all w >, and therefore ns u j, Φs, t π L ns Φσ, Φs, t Φσ, Φs, t F j, σ dσ + C E Fj, L. 7 Furthermore, for large enough Lemma 4 implies the estimate π Φσ, Φs, t C 4 s σ + t. for all s, σ, L and t a, a, recall the assumption 5. Next note that Φσ, Φs, t = tns + Γσ Γs, hence using we get ns Φσ, Φs, t = t + σ s ρσ, s, where ρσ, s = ns ρ σ, s is uniformly bounded on [, L] [, L]. Consequently, there are C 5, C 6 > such that ns Φσ, Φs, t t + C 6 s σ t C5 Φσ, Φs, t s σ + t C 5 s σ + t + C 5C 6 3

14 and π L ns Φσ, Φs, t Φσ, Φs, t F j, σ dσ C 5 Using the Cauchy-Schwarz inequality we obtain L L t s σ + t F j,σ dσ + C 5 C 6 F j, L. L t s σ + t F t / j,σ dσ s σ + t dσ F j, L = t / R R t s σ + t dσ / F j, L = t R dσ / σ + t F j, L dξ / ξ + F j, L = C 7 t / F j, L. 8 Putting everything together and using a rough estimate E = O from Lemma, we get a bound ns u j, Φs, t C8 t / + F j, L 9 with some constant C 8 >. Next we denote δ := log and for large enough we construct a new function v on Ωδ by v j, Φs, t := uj, Φs,, for which the triangle inequality yields s, t, L δ, δ, u L Ωδ v j, L Ωδ u j, v j, L Ωδ. 3 Using 4, one can write now the following estimates: v j, L Ωδ C 9 = C δ 9 δ L δ L δ uj, Φs, ds dt F j, s ds dt = C δ F j, L = C 4 log F j, L. 3 4

15 On the other hand, the second term on the right-hand side of 3 satisfies u j, v j, L Ωδ C L δ δ u j, Φs, t uj, Φs, dt ds. 3 To estimate the integrated function, we employ the relation d dt u j, Φs, t = ns uj, Φs, t, which yields, through 9, the bound u j, Φs, t uj, Φs, = C 8 t t and consequently, t ns u j, Φs, ξ dξ ns u j, Φs, ξ dξ ξ / + dξ F j, L = C 8 t / + t F j, L, L u j, v j, 8C 8 C L Ωδ δ δ t + t dt ds F j, L C δ + δ 3 F j, L C 3 4 log F j, L. 33 Substituting finally 3 and 33 into 3 we obtain = u j, L R u j, L Ωδ C log C 3 log which gives the sought result. F j, L Lemma. For any k, c > one can find a D > such that u j, x D exp u j, x D 3 exp k log c holds whenever x / W. C 4 log F j, L, log dx,, 34 log dx, 35 5

16 Proof. Recall that we have the integral representation for the eignefunction u j,, hence using Lemma and Cauchy-Schwarz inequality we infer that u j, x sup K E j x y Fj, L y C log sup K E j x y. 36 y k log c For x / W and y we have, using Lemma, E j x y E j dx, log k log c = k log + O as +. For fixed x, y the asymptotics and Lemma imply K E j x y / log x y C 3 E j x y exp / log dx, C 3 E j dx, exp C 4 exp log log dx, Combining this inequality with 36 we obtain the bound 34. To estimate u j, we use 3 and write u j, x = E j x x y K E j x y F j, yds y. It is now enough note that x x y and that Ej = O by Lemma, hence estimating the integral again with the help of we arrive at the bound Cut-off functions In this section we introduce a family of cut-off functions that will be used in the following when we will apply the max-min principle in the last step of the argument. An inspiration for this type of constructions came from the paper [9]. 6

17 We choose a mollifying function C function ψ : R [, + such that ψs = for s and ψs = for s. Next we consider the function ρ a : P a R, ρ a s, t = min { a t, a + t, L + a s, s + a }, in other words, ρ a s, t is the distance between the point s, t P a and the boundary of the rectangle P a. We use it to introduce the function R : Πa R by R x = ρ a Φ x. where Φ x means the pre-image of the point x Πa with respect to the map 4 and the parameters are related by 5. Finally, for sufficiently large and we introduce the function g : R R by log R x + log ψ, x Πa, g x = log log 37, otherwise. Note that g belongs to H R and has a compact support since gx = for all x / Πa. In addition, we have gx = for those x Πa that can be represented as x = Φs, t with ρ a s, t. On the other hand, log gx = holds for x Πa with R x. In particular, { supp g Θ := Φs, t : s, t P a, supp g V := { Φs, t : s, t P a, Lemma 3. In the limit + one has g x dx = O, Θ Θ g x dx = O log. log ρ as, t ρ a s, t }. Proof. Let D s,t Φ denote the Jacobian matrix value of the map Φ at s, t. We have g Φs, t = ψ log R Φs, t + log R Φs, t log log R Φs, t log log = ψ log ρ a s, t + log log log ρ a s, t log log ρ as, td s,t Φ. }, 7

18 We have ρa s, t and Ds,t Φ M for some M > and all s, t P a if is sufficiently large. Hence it holds with some C >, and g x ν C ν dx C log log ν Θ g Φs, t C ρ a s, t log log. s,t P a, log ρas,t Since the integration variables run through the set log ρ as, t, ds dt, ν =,. 38 ρ a s, t ν the integral on the right-hand side is the sum of contributions from integration over four rectangles and eight triangles. Using the obvious symmetries, we can rewrite it as I ν := s,t P a log <ρas,t< + log L+a ds dt ρ a s, t ν = a a+ a+ dt ds + 8 sν for ν =, and some C 3 >. Hence log log s log dt ds tν t ν dt ds C 3 log dt t ν. I C 3 log dt t = C 3 log log. and I ν = C 3 log dt t = C 3 log. Finally, by 38 we infer that g x dx = C C I log log C C C 3 log log = O log log Θ 8

19 and Θ g x dx = C C I log log C C C 3 log log log = O log. holds as + which we have set out to prove. Lemma 4. For sufficiently large there is a constant D > such that u j, x D, 39 u j, x D 4 holds for all x V. Proof. By Corollary 8 there exists a C > such that dx, a C a for all x Πa, holds provided is sufficiently large. On the other hand, for any x = Φs, t V one can find s, t P a with ρ a s, t = s s + t t. As Πa = Φ Πa, it follows from Lemma 4 that for all x V d x, Πa Φs, t Φs, t C s s + t t C holds with some C >. Consequently, for sufficiently large we have V R \ W a C 6 log = R C \ W, and Lemma is applicable. For x V and large we can estimate E j dx, log 6 log C = 3 log + O, by Lemma, hence applying 34 and 35 we get the sought bounds. 7 Using the max-min principle Let us fix now an integer N. Consider the first N eigenvalues E j and the associated orthonormal eigenfunctions u j, of H and denote ϕ j, := g u j,, where g is the function 37. As supp g Πa, one has ϕ j, H Πa. Following the usual convention, we denote here and in the following by δ jl the Kronecker delta symbol. 9

20 Lemma 5. In the limit + one has ϕ j,, ϕ l, L Πa = δ jl + O. 4 5 log Proof. Denote for brevity S := W. In a way similar to the proof of Lemma 4 one can show that for all sufficiently large we have S Πa and g S =. Moreover, for x / S one can estimate u j, x with the help of Lemma. Hence using first the boundedness of the function g and applying subsequently Lemma 9, we get u j,, u l, L R ϕ j,, ϕ l, L Πa = u j,, u l, L R ϕ j,, ϕ l, L R = g x u j, xu l, x dx = g x u j, xu l, x dx R C R \S u j, xu l, x dx C 4 R \S R \S e log dx, dx = O with some constants C, C >. As {u j, } is an orthonormal system by assumption, we arrive at the relation 4. Lemma 6. In the limit + one has u j,, u l, L Πa u j, su l, s ds = E j δ jl + O. Proof. Note first that the relations u j,, u l, L R u j, su l, sds = E j δ jl hold by assumption and that a certain neighborhood of is included into Πa, hence it is sufficient to check the estimate u j,, u l, L R \Πa = O. As in the proof of Lemma 4 we can check that the inclusion 6 log C W Πa.

21 holds for some C > and all sufficiently large. Using then the estimate 35 and subsequently Lemma 9, we get u j,, u l, L R \Πa C 6 R \W 6 log C R \W 6 log C u j, x u l, x dx e log dx, dx C = O. Our principal estimate concerns the question what happens if u j, in the above formula is replaced by the moliffied function; our aim is to show that this makes the error term worse but only by a logarithmic factor. Lemma 7. In the limit + one has log ϕ j,, ϕ l, L Πa ϕ j, sϕ l, s ds = E j δ jl + O. Proof. Using ϕ j, = u j, let us write the expression in question as ϕ j,, ϕ l, L Πa ϕ j, sϕ l, s ds = u j,, u l, L Πa + u j, su l, s ds g x u j, x u l, x dx Πa + Πa + Πa + g x u j, xu l, x dx g xu j, x g x u l, x dx g xu l, x u j, x g x dx. Πa The sum of the first two terms on the right-hand side has been already estimated in Lemma 6, hence we just need to show that the sum of the last

22 four terms on the right-hand side is of order O log. By definition of the function g and Lemma 4 there are constants C, C > such that I := g x u j, x u l, xdx Πa = V g x u j, x u l, xdx C V u j, x u l, x dx C V. Furthermore, by definition of V we have { V = ΦU, U = s, t P a : ρ a s, t }, and since the measure U is of order O, we get also V = O, which in turn gives I = O. Using next the inclusion supp g Θ V, Lemma 4 and after that Lemma 3, we have I := g x u j, xu l, x dx Πa = Θ g x u j, xu l, x dx C 3 Θ Using the same reasoning we infer that I j,l := g xu j, x g x u l, x dx Πa = Θ g x g xu j, x g x u l, x dx C 4 Θ Putting the estimates together we find log I + I + I j,l + I l,j = O, which concludes the proof. log dx = O. g x dx = O.

23 Now we are in position to complete the proof of our main result. Proof of Proposition 3. Fix an integer N. By the max-min principle one has f dx f ds Λ N = max G S N min f G Πa f L Πa where S N stands for the family of the subspaces of H Πa the codimension of which in L Πa equals N. In view of Lemma 5, the functions ϕ j,, j =,..., N, are linearly independent in L Πa for all sufficiently large, hence each subspace G S N contains at least one linear combination ϕ of the form ϕ = N b j ϕ j,, b = b,..., b N C N, b C N =. j= Using once more Lemma 5, we find that ϕ C L Πa. holds for large with a constant C >. On the other hand, Lemma 7 yields ϕ dx ϕ ds Πa = = N b j b l ϕ j,, ϕ l, L Πa j,l= N log b j b l E j δ jl + O = j,l= ϕ j, sϕ l, sds N log E j b j + O log E N + O. 4 Using the above estimates, we conclude that there are C, C 3 > such that f dx f ds ϕ dx ϕ ds Πa min f G f L Πa Πa j= ϕ L Πa E N + C log C E N + C 3 log., 3

24 What is important is that the constant C 3 can be chosen independent of the vector b and hence independent of G S N, then we have automatically Λ N E N + C 3 log. Combining this with 9 we obtain Λ N E N = O log. 8 Acknowledgmenets The second named author thanks the Doppler Institute in Prague for the warm hospitality during the stay in May-June. The research was partially supported by ANR NOSEVOL and GDR Dynamique Quantique, and by Czech Science Foundation within the project P3//7. References [] M. Abramowitz, I. A. Stegun eds.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. th printing volume 55 of Applied Mathematics Series, US National Bureau of Standards, 97. Available online at ~cbm/aands/. [] M. S. Agranovich: Strongly elliptic second-order systems with boundary vonditions on a nonclosed Lipschitz surface. Funct. Anal. Appl. 45:,. [3] J. F. Brasche, P. Exner, Yu. A. Kuperin, P. Šeba: Schrödinger operators with singular ineractions. J. Math. Anal. Appl , 39. [4] M. Dauge, B. Helffer: Eigenvalues variation I. Neumann problem for Sturm-Liouville operators. J. Differential Eqs 4 993, [5] P. Exner: Leaky quantum graphs: a review, Proceedings of the Isaac Newton Institute programme Analysis on Graphs and Applications, AMS Proceedings of Symposia in Pure Mathematics Series, vol. 77, Providence, R.I., 8; pp [6] P. Exner, K. Yoshitomi: Asymptotics of eigenvalues of the Schrödinger operator with a strong δ-interaction on a loop. J. Geom. Phys. 4, [7] G. Grubb: The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates. J. Math. Anal. Appl. 38,

25 [8] M. Levitin, L. Parnovski: On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachrichten 8 8, 7 8. [9] S. A. Nazarov: An example of multiple gaps in the spectrum of a periodic waveguide. Sbornik: Mathematics :4, [] A. Posilicano: Boundary triples and Weyl functions for singular perturbations of selfadjoint operator, Meth. Funct. Anal. Topol. 4,