Lectures on Quantum Graph Models

Transcription

1 Lectures on Quantum Graph Models Pavel Exner Doppler Institute for Mathematical Physics and Applied Mathematics Prague Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 1/11

2 Course overview The aim to review some recent results in the theory of quantum graphs, standard as well as non-standard Lecture I Ideal graphs their nontrivial aspect, or what is the meaning of the vertex coupling Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 2/11

3 Course overview The aim to review some recent results in the theory of quantum graphs, standard as well as non-standard Lecture I Ideal graphs their nontrivial aspect, or what is the meaning of the vertex coupling Lecture II Leaky graphs what they are, and their spectral and resonance properties Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 2/11

4 Course overview The aim to review some recent results in the theory of quantum graphs, standard as well as non-standard Lecture I Ideal graphs their nontrivial aspect, or what is the meaning of the vertex coupling Lecture II Leaky graphs what they are, and their spectral and resonance properties Lecture III Generalized graphs or what happens if a quantum particle has to change its dimension Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 2/11

5 Quantum graphs The idea of investigating quantum particles confined to a graph is rather old. It was first suggested by L. Pauling and worked out by Ruedenberg and Scherr in 1953 in a model of aromatic hydrocarbons Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 3/11

6 Quantum graphs The idea of investigating quantum particles confined to a graph is rather old. It was first suggested by L. Pauling and worked out by Ruedenberg and Scherr in 1953 in a model of aromatic hydrocarbons Using textbook graphs such as with Kirchhoff b.c. in combination with Pauli principle, they reproduced the actual spectra with a 10% accuracy A caveat: later naive generalizations were less successful Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 3/11

7 Ideal quantum graph concept The beauty of theoretical physics resides in permanent oscillation between physical anchoring in reality and mathematical freedom of creating concepts As a mathematically minded person you can imagine quantum particles confined to a graph of arbitrary shape Hamiltonian: 2 + v(x x 2 j ) j on graph edges, boundary conditions at vertices and, lo and behold, this turns out to be a practically important concept after experimentalists learned in the last years to fabricate tiny graph-like structure for which this is a good model Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 4/11

8 Remarks Most often one deals with semiconductor graphs produced by combination of ion litography and chemical itching. In a similar way metallic graphs are prepared Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 5/11

9 Remarks Most often one deals with semiconductor graphs produced by combination of ion litography and chemical itching. In a similar way metallic graphs are prepared Recently carbon nanotubes became a building material, after branchings were fabricated cca five years ago: see [Papadopoulos et al. 00], [Andriotis et al. 01], etc. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 5/11

10 Remarks Most often one deals with semiconductor graphs produced by combination of ion litography and chemical itching. In a similar way metallic graphs are prepared Recently carbon nanotubes became a building material, after branchings were fabricated cca five years ago: see [Papadopoulos et al. 00], [Andriotis et al. 01], etc. Moreover, from the stationary point of view a quantum graph is also equivalent to a microwave network built of optical cables see [Hul et al. 04] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 5/11

11 Remarks Most often one deals with semiconductor graphs produced by combination of ion litography and chemical itching. In a similar way metallic graphs are prepared Recently carbon nanotubes became a building material, after branchings were fabricated cca five years ago: see [Papadopoulos et al. 00], [Andriotis et al. 01], etc. Moreover, from the stationary point of view a quantum graph is also equivalent to a microwave network built of optical cables see [Hul et al. 04] In addition to graphs one can consider generalized graphs which consist of components of different dimensions, modelling things as different as combinations of nanotubes with fullerenes, scanning tunneling microscopy, etc. we will do that in Lecture III Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 5/11

12 More remarks The vertex coupling is chosen to make the Hamiltonian self-adjoint, or in physical terms, to ensure probability current conservation. This is achieved by the method based on s-a extensions which everybody in this audience knows (at least I suppose so) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 6/11

13 More remarks The vertex coupling is chosen to make the Hamiltonian self-adjoint, or in physical terms, to ensure probability current conservation. This is achieved by the method based on s-a extensions which everybody in this audience knows (at least I suppose so) Here we consider Schrödinger operators on graphs, most often free, v j = 0. Naturally one can external electric and magnetic fields, spin, etc. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 6/11

14 More remarks The vertex coupling is chosen to make the Hamiltonian self-adjoint, or in physical terms, to ensure probability current conservation. This is achieved by the method based on s-a extensions which everybody in this audience knows (at least I suppose so) Here we consider Schrödinger operators on graphs, most often free, v j = 0. Naturally one can external electric and magnetic fields, spin, etc. Graphs can support also Dirac operators, see [Bulla-Trenckler 90], [Bolte-Harrison 03], and also recent applications to graphene and its derivates Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 6/11

15 More remarks The vertex coupling is chosen to make the Hamiltonian self-adjoint, or in physical terms, to ensure probability current conservation. This is achieved by the method based on s-a extensions which everybody in this audience knows (at least I suppose so) Here we consider Schrödinger operators on graphs, most often free, v j = 0. Naturally one can external electric and magnetic fields, spin, etc. Graphs can support also Dirac operators, see [Bulla-Trenckler 90], [Bolte-Harrison 03], and also recent applications to graphene and its derivates The graph literature is extensive; recall just a review [Kuchment 04], proceedings of Snowbird 05 conference, and the recent AGA Programme at INI Cambridge Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 6/11

16 Wavefunction coupling at vertices The most simple example is a star graph with the state Hilbert space H = n j=1 L2 (R + ) and the particle Hamiltonian acting on H as ψ j ψ j Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 7/11

17 Wavefunction coupling at vertices The most simple example is a star graph with the state Hilbert space H = n j=1 L2 (R + ) and the particle Hamiltonian acting on H as ψ j ψ j Since it is second-order, the boundary condition involve Ψ(0) := {ψ j (0)} and Ψ (0) := {ψ j (0)} being of the form AΨ(0) + BΨ (0) = 0 ; by [Kostrykin-Schrader 99] the n n matrices A, B give rise to a self-adjoint operator if they satisfy the conditions rank (A,B) = n AB is self-adjoint Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 7/11

18 Unique boundary conditions The non-uniqueness of the above b.c. can be removed: Proposition [Harmer 00, K-S 00]: Vertex couplings are uniquely characterized by unitary n n matrices U such that A = U I, B = i(u + I) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 8/11

19 Unique boundary conditions The non-uniqueness of the above b.c. can be removed: Proposition [Harmer 00, K-S 00]: Vertex couplings are uniquely characterized by unitary n n matrices U such that A = U I, B = i(u + I) One can derive them modifying the argument used in [Fülöp-Tsutsui 00] for generalized point interactions, n = 2 Self-adjointness requires vanishing of the boundary form, n ( ψ j ψ j ψ jψ j )(0) = 0, j=1 which occurs iff the norms Ψ(0) ± ilψ (0) C n with a fixed l 0 coincide, so the vectors must be related by an n n unitary matrix; this gives (U I)Ψ(0) + il(u + I)Ψ (0) = 0 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 8/11

20 Remarks The length parameter is not important because matrices corresponding to two different values are related by U = (l + l )U + l l (l l )U + l + l The choice l = 1 just fixes the length scale Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 9/11

21 Remarks The length parameter is not important because matrices corresponding to two different values are related by U = (l + l )U + l l (l l )U + l + l The choice l = 1 just fixes the length scale The unique b.c. help to simplify the analysis done in [Kostrykin-Schrader 99], [Kuchment 04] and other previous work. It concerns, for instance, the null spaces of the matrices A,B Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 9/11

22 Remarks The length parameter is not important because matrices corresponding to two different values are related by U = (l + l )U + l l (l l )U + l + l The choice l = 1 just fixes the length scale The unique b.c. help to simplify the analysis done in [Kostrykin-Schrader 99], [Kuchment 04] and other previous work. It concerns, for instance, the null spaces of the matrices A,B or the on-shell scattering matrix for a star graph of n halflines with the considered coupling which equals S U (k) = (k 1)I + (k + 1)U (k + 1)I + (k 1)U Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 9/11

23 Examples of vertex coupling Denote by J the n n matrix whose all entries are equal to one; then U = n+iα 2 J I corresponds to the standard δ coupling, n ψ j (0) = ψ k (0) =: ψ(0), j,k = 1,...,n, ψ j(0) = αψ(0) j=1 with coupling strength α R; α = gives U = I Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 10/11

24 Examples of vertex coupling Denote by J the n n matrix whose all entries are equal to one; then U = n+iα 2 J I corresponds to the standard δ coupling, n ψ j (0) = ψ k (0) =: ψ(0), j,k = 1,...,n, ψ j(0) = αψ(0) j=1 with coupling strength α R; α = gives U = I α = 0 corresponds to the free motion, the so-called free boundary conditions (better name than Kirchhoff) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 10/11

25 Examples of vertex coupling Denote by J the n n matrix whose all entries are equal to one; then U = n+iα 2 J I corresponds to the standard δ coupling, n ψ j (0) = ψ k (0) =: ψ(0), j,k = 1,...,n, ψ j(0) = αψ(0) j=1 with coupling strength α R; α = gives U = I α = 0 corresponds to the free motion, the so-called free boundary conditions (better name than Kirchhoff) Similarly, U = I n iβ 2 J describes the δ s coupling n ψ j(0) = ψ k(0) =: ψ (0), j,k = 1,...,n, ψ j (0) = βψ (0) j=1 with β R; for β = we get Neumann decoupling Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 10/11

26 Further examples Another generalization of 1D δ is the δ coupling: n ψ j(0) = 0, ψ j (0) ψ k (0) = β n (ψ j(0) ψ k(0)), 1 j,k n j=1 with β R and U = n+iα n iαi n+iα 2 J ; the infinite value of β refers again to Neumann decoupling of the edges Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 11/11

27 Further examples Another generalization of 1D δ is the δ coupling: n ψ j(0) = 0, ψ j (0) ψ k (0) = β n (ψ j(0) ψ k(0)), 1 j,k n j=1 with β R and U = n+iα n iαi n+iα 2 J ; the infinite value of β refers again to Neumann decoupling of the edges Due to permutation symmetry the U s are combinations of I and J in the examples. In general, interactions with this property form a two-parameter family described by U = ui + vj s.t. u = 1 and u + nv = 1 giving the b.c. (u 1)(ψ j (0) ψ k (0)) + i(u 1)(ψ j(0) ψ k(0)) = 0 n n (u 1 + nv) ψ k (0) + i(u 1 + nv) ψ k(0) = 0 k=1 k=1 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 11/11

28 Why are vertices interesting? While usually conductivity of graph structures is controlled by external fields, vertex coupling can serve the same purpose Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 12/11

29 Why are vertices interesting? While usually conductivity of graph structures is controlled by external fields, vertex coupling can serve the same purpose It is an interesting problem in itself, recall that for the generalized point interaction, i.e. graph with n = 2, the spectrum has nontrivial topological structure [Tsutsui-Fülöp-Cheon 01] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 12/11

30 Why are vertices interesting? While usually conductivity of graph structures is controlled by external fields, vertex coupling can serve the same purpose It is an interesting problem in itself, recall that for the generalized point interaction, i.e. graph with n = 2, the spectrum has nontrivial topological structure [Tsutsui-Fülöp-Cheon 01] More recently, the same system has been proposed as a way to realize a qubit, with obvious consequences: cf. quantum abacus in [Cheon-Tsutsui-Fülöp 04] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 12/11

31 Why are vertices interesting? While usually conductivity of graph structures is controlled by external fields, vertex coupling can serve the same purpose It is an interesting problem in itself, recall that for the generalized point interaction, i.e. graph with n = 2, the spectrum has nontrivial topological structure [Tsutsui-Fülöp-Cheon 01] More recently, the same system has been proposed as a way to realize a qubit, with obvious consequences: cf. quantum abacus in [Cheon-Tsutsui-Fülöp 04] Recall also that in a rectangular lattice with δ coupling of nonzero α spectrum depends on number theoretic properties of model parameters [E. 95] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 12/11

32 More on the lattice example Basic cell is a rectangle of sides l 1, l 2, the δ coupling with parameter α is assumed at every vertex l 2 f m+1 y f m l 1 g n g n+1 x Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 13/11

33 More on the lattice example Basic cell is a rectangle of sides l 1, l 2, the δ coupling with parameter α is assumed at every vertex l 2 f m+1 y f m l 1 g n g n+1 Spectral condition for quasimomentum (θ 1,θ 2 ) reads x 2 j=1 cos θ j l j cos kl j sin kl j = α 2k Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 13/11

34 Lattice band spectrum Recall a continued-fraction classification, α = [a 0,a 1,...]: good irrationals have lim sup j a j = (and full Lebesgue measure) bad irrationals have lim sup j a j < (and lim j a j 0, of course) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 14/11

35 Lattice band spectrum Recall a continued-fraction classification, α = [a 0,a 1,...]: good irrationals have lim sup j a j = (and full Lebesgue measure) bad irrationals have lim sup j a j < (and lim j a j 0, of course) Theorem [E. 95]: Call θ := l 2 /l 1 and L := max{l 1,l 2 }. (a) If θ is rational or good irrational, there are infinitely many gaps for any nonzero α (b) For a bad irrational θ there is α 0 > 0 such no gaps open above threshold for α < α 0 (c) There are infinitely many gaps if α L > π2 5 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 14/11

36 Lattice band spectrum Recall a continued-fraction classification, α = [a 0,a 1,...]: good irrationals have lim sup j a j = (and full Lebesgue measure) bad irrationals have lim sup j a j < (and lim j a j 0, of course) Theorem [E. 95]: Call θ := l 2 /l 1 and L := max{l 1,l 2 }. (a) If θ is rational or good irrational, there are infinitely many gaps for any nonzero α (b) For a bad irrational θ there is α 0 > 0 such no gaps open above threshold for α < α 0 (c) There are infinitely many gaps if α L > π2 5 This all illustrates why it is desirable to understand vertex couplings. This will be our main task in Lecture I Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 14/11

37 A natural approach: fat graphs Take a more realistic situation without ambiguities, such as a network of branching tubes and analyze its squeezing limit: Looking simple, it is a nontrivial problem. What is known? Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 15/11

38 A natural approach: fat graphs Take a more realistic situation without ambiguities, such as a network of branching tubes and analyze its squeezing limit: Looking simple, it is a nontrivial problem. What is known? after a long effort the Neumann-like case is understood, see [Freidlin-Wentzell 93], [Freidlin 96], [Saito 01], [Kuchment-Zeng 01], [Rubinstein-Schatzmann 01], [E.-Post 05, 07], [Post 06], giving basically free b.c. only recently a progress achieved in the Dirichlet case [Post 05], [Molchanov-Vainberg 07], [Grieser 07]; we will comment on it briefly below Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 15/11

39 Neumann case survey: first, the graphs The simplest situation in [KZ 01, EP 05] (weights left out) Let M 0 be a finite connected graph with vertices v k, k K and edges e j I j := [0,l j ], j J; the state Hilbert space is L 2 (M 0 ) := j J L 2 (I j ) and in a similar way Sobolev spaces on M 0 are introduced Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 16/11

40 Neumann case survey: first, the graphs The simplest situation in [KZ 01, EP 05] (weights left out) Let M 0 be a finite connected graph with vertices v k, k K and edges e j I j := [0,l j ], j J; the state Hilbert space is L 2 (M 0 ) := j J L 2 (I j ) and in a similar way Sobolev spaces on M 0 are introduced The form u u 2 M 0 := j J u 2 I j with u H 1 (M 0 ) is associated with the operator which acts as M0 u = u j and satisfies free b.c., u j(v k ) = 0 j, e j meets v k Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 16/11

41 n the other hand, Laplacian on manifold Consider a Riemannian manifold X of dimension d 2 and the corresponding space L 2 (X) w.r.t. volume dx equal to (detg) 1/2 dx in a fixed chart. For u Ccomp(X) we set q X (u) := du 2 X = du 2 dx, du 2 = g ij i u j u X i,j The closure of this form is associated with the s-a operator X which acts in fixed chart coordinates as X u = (detg) 1/2 i,j i ((detg) 1/2 g ij j u) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 17/11

42 n the other hand, Laplacian on manifold Consider a Riemannian manifold X of dimension d 2 and the corresponding space L 2 (X) w.r.t. volume dx equal to (detg) 1/2 dx in a fixed chart. For u Ccomp(X) we set q X (u) := du 2 X = du 2 dx, du 2 = g ij i u j u X i,j The closure of this form is associated with the s-a operator X which acts in fixed chart coordinates as X u = (detg) 1/2 i,j i ((detg) 1/2 g ij j u) If X is compact with piecewise smooth boundary, one starts from the form defined on C (X). This yields X as the Neumann Laplacian on X and allows us to treat fat graphs and sleeves on the same footing Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 17/11

43 Relating the two together We associate with the graph M 0 a family of manifolds M ε e j v k U ε,j V ε,k M 0 M ε We suppose that M ε is a union of compact edge and vertex components U ε,j and V ε,k such that their interiors are mutually disjoint for all possible j J and k K Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 18/11

44 Manifold building blocks U ε,j ε ε V ε,k e j v k Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 19/11

45 Manifold building blocks U ε,j ε ε V ε,k e j v k However, M ε need not be embedded in some R d. It is convenient to assume that U ε,j and V ε,k depend on ε only through their metric: for edge regions we assume that U ε,j is diffeomorphic to I j F where F is a compact and connected manifold (with or without a boundary) of dimension m := d 1 for vertex regions we assume that the manifold V ε,k is diffeomorphic to an ε-independent manifold V k Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 19/11

46 Eigenvalue convergence Let thus U = I j F with metric g ε, where cross section F is a compact connected Riemannian manifold of dimension m = d 1 with metric h; we assume that volf = 1. We define another metric g ε on U ε,j by g ε := dx 2 + ε 2 h(y) ; the two metrics coincide up to an O(ε) error This property allows us to treat manifolds embedded in R d (with metric g ε ) using product metric g ε on the edges Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 20/11

47 Eigenvalue convergence Let thus U = I j F with metric g ε, where cross section F is a compact connected Riemannian manifold of dimension m = d 1 with metric h; we assume that volf = 1. We define another metric g ε on U ε,j by g ε := dx 2 + ε 2 h(y) ; the two metrics coincide up to an O(ε) error This property allows us to treat manifolds embedded in R d (with metric g ε ) using product metric g ε on the edges The sought result now looks as follows. Theorem [KZ 01, EP 05]: Under the stated assumptions λ k (M ε ) λ k (M 0 ) as ε 0 (giving thus free b.c.!) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 20/11

48 A stronger convergence The b.c. are not the only problem. The ev convergence for finite graphs is rather weak. Fortunately, one can do better. Theorem [Post 06]: Let M ε be graphlike manifolds associated with a metric graph M 0, not necessarily finite. Under some natural uniformity conditions, Mε M0 as ε 0+ in the norm-resolvent sense (with suitable identification), in particular, the σ disc and σ ess converge uniformly in an bounded interval, and ef s converge as well. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 21/11

49 A stronger convergence The b.c. are not the only problem. The ev convergence for finite graphs is rather weak. Fortunately, one can do better. Theorem [Post 06]: Let M ε be graphlike manifolds associated with a metric graph M 0, not necessarily finite. Under some natural uniformity conditions, Mε M0 as ε 0+ in the norm-resolvent sense (with suitable identification), in particular, the σ disc and σ ess converge uniformly in an bounded interval, and ef s converge as well. The natural uniformity conditions mean (i) existence of nontrivial bounds on vertex degrees and volumes, edge lengths, and the second Neumann eigenvalues at vertices, (ii) appropriate scaling (analogous to the described above) of the metrics at the edges and vertices. Proof is based on an abstract convergence result. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 21/11

50 Convergence of resonances In a similar way we can treat convergence of resonances. As a motivating example one can think of a fat lasso graph, with the ε-squeezing setting the same as before: Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 22/11

51 Convergence of resonances In a similar way we can treat convergence of resonances. As a motivating example one can think of a fat lasso graph, with the ε-squeezing setting the same as before: U ε,int X ε e ε U ε,v ε v U ε,ext e int X 0 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 22/11

52 Convergence of resonances, continued Let H 0, with free b.c., and H ε will be as above. We use an exterior complex scaling extending to complex θ the map U θ f := (detdφ θ ) 1/2 (f Φ θ ) where Φ θ e(x) := e θ x on external edges, and (detdφ θ ) 1/2 equals one and e θ/2, respectively, on X 0,int and X 0,ext. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 23/11

53 Convergence of resonances, continued Let H 0, with free b.c., and H ε will be as above. We use an exterior complex scaling extending to complex θ the map U θ f := (detdφ θ ) 1/2 (f Φ θ ) where Φ θ e(x) := e θ x on external edges, and (detdφ θ ) 1/2 equals one and e θ/2, respectively, on X 0,int and X 0,ext. Theorem [E.-Post 07]: Let λ(0) be a resonance of H 0 of multiplicity m > 0, then for small enough ε > 0 there is m resonances λ 1 (ε),...,λ m (ε) of H ε, not necessarily distinct, which all converge to λ(0) as ε 0. The same is true for embedded ev s of H 0, when Im λ j (ε) 0 holds in general. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 23/11

54 Convergence of resonances, continued Let H 0, with free b.c., and H ε will be as above. We use an exterior complex scaling extending to complex θ the map U θ f := (detdφ θ ) 1/2 (f Φ θ ) where Φ θ e(x) := e θ x on external edges, and (detdφ θ ) 1/2 equals one and e θ/2, respectively, on X 0,int and X 0,ext. Theorem [E.-Post 07]: Let λ(0) be a resonance of H 0 of multiplicity m > 0, then for small enough ε > 0 there is m resonances λ 1 (ε),...,λ m (ε) of H ε, not necessarily distinct, which all converge to λ(0) as ε 0. The same is true for embedded ev s of H 0, when Im λ j (ε) 0 holds in general. Remarks: (i) The above Φ θ can have a shifted discontinuity, or be replaced by a smooth flow, with the same result (ii) The result persists if a magnetic field is added Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 23/11

55 Briefly about the Dirichlet case Generically it is expected that that the limit with the energy around the threshold gives Dirichlet decoupling, but there may be exceptional cases Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 24/11

56 Briefly about the Dirichlet case Generically it is expected that that the limit with the energy around the threshold gives Dirichlet decoupling, but there may be exceptional cases if the vertex regions squeeze faster than the tubes one gets Dirichlet decoupling [Post 05] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 24/11

57 Briefly about the Dirichlet case Generically it is expected that that the limit with the energy around the threshold gives Dirichlet decoupling, but there may be exceptional cases if the vertex regions squeeze faster than the tubes one gets Dirichlet decoupling [Post 05] on the other hand, if you blow up the spectrum for a fixed point separated from thresholds, i.e. 0 λ 1 λ λ 2 one gets a nontrivial limit with b.c. fixed by scattering on the fat star [Molchanov-Vainberg 07] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 24/11

58 More on the threshold case As said above, generically the squeezing limit with renormalization w.r.t. the spectral threshold leads to Dirichlet decoupled graph see [Molchanov-Vainberg 07] or [Dell Antonio-Tenuta 07] for a particular case of a broken line which is not very interesting Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 25/11

59 More on the threshold case As said above, generically the squeezing limit with renormalization w.r.t. the spectral threshold leads to Dirichlet decoupled graph see [Molchanov-Vainberg 07] or [Dell Antonio-Tenuta 07] for a particular case of a broken line which is not very interesting A nontrivial limit may arise in the (non-generic) case when the system has a threshold resonance. For compact graphs this was proved in [Grieser 08] On the other hand, [Albeverio-Cacciapuoti-Finco 07] provide a worked out example, the star graph with n = 2. In the limit they get the family of scale-invariant point interactions on the line. Using a nonlinear scaling one can get in this example a two-parameter family of point interactions including the δ coupling, cf. [Cacciapuoti-E 07] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 25/11

60 Potential approximations A more modest goal: let us look what we can achieve with potential families on the graph alone Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 26/11

61 Potential approximations A more modest goal: let us look what we can achieve with potential families on the graph alone Consider once more star graph with H = n j=1 L2 (R + ) and Schrödinger operator acting on H as ψ j ψ j + V jψ j Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 26/11

62 Potential approximations A more modest goal: let us look what we can achieve with potential families on the graph alone Consider once more star graph with H = n j=1 L2 (R + ) and Schrödinger operator acting on H as ψ j ψ j + V jψ j We make the following assumptions: V j L 1 loc (R +), j = 1,...,n δ coupling with a parameter α in the vertex Then the operator, denoted as H α (V ), is self-adjoint Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 26/11

63 Potential approximation of δ coupling Suppose that the potential has a shrinking component, W ε,j := 1 ( x ε W j, j = 1,...,n ε) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 27/11

64 Potential approximation of δ coupling Suppose that the potential has a shrinking component, W ε,j := 1 ( x ε W j, j = 1,...,n ε) Theorem [E. 96]: Suppose that V j L 1 loc (R +) are below bounded and W j L 1 (R + ) for j = 1,...,n. Then H 0 (V + W ε ) H α (V ) as ε 0+ in the norm resolvent sense, with the parameter α := n j=1 0 W j (x)dx Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 27/11

65 Potential approximation of δ coupling Suppose that the potential has a shrinking component, W ε,j := 1 ( x ε W j, j = 1,...,n ε) Theorem [E. 96]: Suppose that V j L 1 loc (R +) are below bounded and W j L 1 (R + ) for j = 1,...,n. Then H 0 (V + W ε ) H α (V ) as ε 0+ in the norm resolvent sense, with the parameter α := n j=1 0 W j (x)dx Proof: Analogous to that for δ interaction on the line. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 27/11

66 More singular couplings The above scheme does not work for graph Hamiltonians with discontinuous wavefunctions such as δ s Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 28/11

67 More singular couplings The above scheme does not work for graph Hamiltonians with discontinuous wavefunctions such as δ s Inspiration: Recall that δ on the line can be approximated by δ s scaled in a nonlinear way [Cheon-Shigehara 98] Moreover, the convergence is norm resolvent and gives rise to approximations by regular potentials [Albeverio-Nizhnik 00], [E.-Neidhardt-Zagrebnov 01] Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 28/11

68 More singular couplings The above scheme does not work for graph Hamiltonians with discontinuous wavefunctions such as δ s Inspiration: Recall that δ on the line can be approximated by δ s scaled in a nonlinear way [Cheon-Shigehara 98] Moreover, the convergence is norm resolvent and gives rise to approximations by regular potentials [Albeverio-Nizhnik 00], [E.-Neidhardt-Zagrebnov 01] This suggests the following scheme: H b,c a c(a) b(a) a 0 H β β Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 28/11

69 δ s approximation Theorem [Cheon-E. 04]: H b,c (a) H β as a 0+ in the norm-resolvent sense provided b, c are chosen as b(a) := β a 2, c(a) := 1 a Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 29/11

70 δ s approximation Theorem [Cheon-E. 04]: H b,c (a) H β as a 0+ in the norm-resolvent sense provided b, c are chosen as b(a) := β a 2, c(a) := 1 a Proof : Green s functions of both operators are found explicitly be Krein s formula, so the convergence can be established by straightforward computation Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 29/11

71 δ s approximation Theorem [Cheon-E. 04]: H b,c (a) H β as a 0+ in the norm-resolvent sense provided b, c are chosen as b(a) := β a 2, c(a) := 1 a Proof : Green s functions of both operators are found explicitly be Krein s formula, so the convergence can be established by straightforward computation Remark: Similar approximation can be worked out also for the other couplings. Using two δ s at each edge one can get 2n-parameter families [E-Turek 07] Moreover, using extra edges and vertices properly scaled, see [E-Turek 07], one can get the ( n+1) 2 -parameter family of time-reversal-invariant couplings Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 29/11

72 Summarizing Lecture I The (ideal) graph model is easy to handle and useful in describing a host of physical phenomena Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 30/11

73 Summarizing Lecture I The (ideal) graph model is easy to handle and useful in describing a host of physical phenomena Vertex coupling: to employ the full potential of the graph model, it is vital to understand the physical meaning of the corresponding boundary conditions Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 30/11

74 Summarizing Lecture I The (ideal) graph model is easy to handle and useful in describing a host of physical phenomena Vertex coupling: to employ the full potential of the graph model, it is vital to understand the physical meaning of the corresponding boundary conditions Fat manifold approximations: using the simplest geometry only we get free b.c. in the Neumann-like case, partial results in the Dirichlet case Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 30/11

75 Summarizing Lecture I The (ideal) graph model is easy to handle and useful in describing a host of physical phenomena Vertex coupling: to employ the full potential of the graph model, it is vital to understand the physical meaning of the corresponding boundary conditions Fat manifold approximations: using the simplest geometry only we get free b.c. in the Neumann-like case, partial results in the Dirichlet case Potential approximation to δ: well understood Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 30/11

76 Summarizing Lecture I The (ideal) graph model is easy to handle and useful in describing a host of physical phenomena Vertex coupling: to employ the full potential of the graph model, it is vital to understand the physical meaning of the corresponding boundary conditions Fat manifold approximations: using the simplest geometry only we get free b.c. in the Neumann-like case, partial results in the Dirichlet case Potential approximation to δ: well understood Potential approximation to more singular coupling: the results exist for generic time-reversal-invariant couplings Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 30/11

77 Some literature to Lecture I [CE07] C. Cacciapuoti, P.E.: Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide, J. Phys. A: Math. Gen. A40 (2007), F [CE04] T. Cheon, P.E.: An approximation to δ couplings on graphs, J. Phys. A: Math. Gen. A37 (2004), L [E96] P.E.: Weakly coupled states on branching graphs, Lett. Math. Phys. 38 (1996), [E97] P.E.: A duality between Schrödinger operators on graphs and certain Jacobi matrices, Ann. Inst. H. Poincaré: Phys. Théor. 66 (1997), [ENZ01] P.E., H. Neidhardt, V.A. Zagrebnov: Potential approximations to δ : an inverse Klauder phenomenon with norm-resolvent convergence, CMP 224 (2001), [EP05, 07] P.E., O. Post: Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005), ; Convergence of resonances on thin branched quantum wave guides, J. Math. Phys. 48 (2007), [ET07] P.E., O. Turek: Approximations of singular vertex couplings in quantum graphs, Rev. Math. Phys. 19 (2007), and references therein, see also exner Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 31/11

78 Lecture II Leaky graphs what they are, and their spectral and resonance properties Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 32/11

79 Lecture overview Why we might want something better than the ideal graph model of the previous lecture Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 33/11

80 Lecture overview Why we might want something better than the ideal graph model of the previous lecture A model of leaky quantum wires and graphs, with Hamiltonians of the type H α,γ = αδ(x Γ) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 33/11

81 Lecture overview Why we might want something better than the ideal graph model of the previous lecture A model of leaky quantum wires and graphs, with Hamiltonians of the type H α,γ = αδ(x Γ) Geometrically induced spectral properties of leaky wires and graphs Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 33/11

82 Lecture overview Why we might want something better than the ideal graph model of the previous lecture A model of leaky quantum wires and graphs, with Hamiltonians of the type H α,γ = αδ(x Γ) Geometrically induced spectral properties of leaky wires and graphs How to find spectrum numerically: an approximation by point interaction Hamiltonians Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 33/11

83 Lecture overview Why we might want something better than the ideal graph model of the previous lecture A model of leaky quantum wires and graphs, with Hamiltonians of the type H α,γ = αδ(x Γ) Geometrically induced spectral properties of leaky wires and graphs How to find spectrum numerically: an approximation by point interaction Hamiltonians A solvable resonance model: interaction supported by a line and a family of points a caricature but solvable Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 33/11

84 Drawbacks of ideal graphs Presence of ad hoc parameters in the b.c. describing branchings. A natural remedy: fit these using an approximation procedure, e.g. As we have seen in Lecture I it is possible but not quite easy and a lot of work remains to be done Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 34/11

85 Drawbacks of ideal graphs Presence of ad hoc parameters in the b.c. describing branchings. A natural remedy: fit these using an approximation procedure, e.g. As we have seen in Lecture I it is possible but not quite easy and a lot of work remains to be done More important, quantum tunneling is neglected in ideal graph models recall that a true quantum-wire boundary is a finite potential jump hence topology is taken into account but geometric effects may not be Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 34/11

86 Leaky quantum graphs We consider leaky graphs with an attractive interaction supported by graph edges. Formally we have H α,γ = αδ(x Γ), α > 0, in L 2 (R 2 ), where Γ is the graph in question. Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 35/11

87 Leaky quantum graphs We consider leaky graphs with an attractive interaction supported by graph edges. Formally we have H α,γ = αδ(x Γ), α > 0, in L 2 (R 2 ), where Γ is the graph in question. A proper definition of H α,γ : it can be associated naturally with the quadratic form, ψ ψ 2 L 2 (R n ) α ψ(x) 2 dx, which is closed and below bounded in W 2,1 (R n ); the second term makes sense in view of Sobolev embedding. This definition also works for various wilder sets Γ Γ Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 35/11

88 Leaky graph Hamiltonians For Γ with locally finite number of smooth edges and no cusps we can use an alternative definition by boundary conditions: H α,γ acts as on functions from W 2,1 loc (R2 \ Γ), which are continuous and exhibit a normal-derivative jump, ψ n (x) ψ + n (x) = αψ(x) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 36/11

89 Leaky graph Hamiltonians For Γ with locally finite number of smooth edges and no cusps we can use an alternative definition by boundary conditions: H α,γ acts as on functions from W 2,1 loc (R2 \ Γ), which are continuous and exhibit a normal-derivative jump, ψ n (x) ψ + n (x) = αψ(x) Remarks: for graphs in R 3 we use generalized b.c. which define a two-dimensional point interaction in normal plane one can combine edges of different dimensions as long as codim Γ does not exceed three Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 36/11

90 Geometrically induced spectrum (a) Bending means binding, i.e. it may create isolated eigenvalues of H α,γ. Consider a piecewise C 1 -smooth Γ : R R 2 parameterized by its arc length, and assume: Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 37/11

91 Geometrically induced spectrum (a) Bending means binding, i.e. it may create isolated eigenvalues of H α,γ. Consider a piecewise C 1 -smooth Γ : R R 2 parameterized by its arc length, and assume: Γ(s) Γ(s ) c s s holds for some c (0, 1) Γ is asymptotically straight: there are d > 0, µ > 1 2 and ω (0, 1) such that 1 Γ(s) Γ(s ) s s d [ 1 + s + s 2µ] 1/2 in the sector S ω := { (s,s ) : ω < s s < ω 1 } straight line is excluded, i.e. Γ(s) Γ(s ) < s s holds for some s,s R Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 37/11

92 Bending means binding Theorem [E.-Ichinose, 2001]: Under these assumptions, σ ess (H α,γ ) = [ 1 4 α2, ) and H α,γ has at least one eigenvalue below the threshold 1 4 α2 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 38/11

93 Bending means binding Theorem [E.-Ichinose, 2001]: Under these assumptions, σ ess (H α,γ ) = [ 1 4 α2, ) and H α,γ has at least one eigenvalue below the threshold 1 4 α2 The same for curves in R 3, under stronger regularity, with 1 4 α2 is replaced by the corresponding 2D p.i. ev For curved surfaces Γ R 3 such a result is proved in the strong coupling asymptotic regime only Implications for graphs: let Γ Γ in the set sense, then H α, Γ H α,γ. If the essential spectrum threshold is the same for both graphs and Γ fits the above assumptions, we have σ disc (H α,γ ) by minimax principle Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 38/11

94 Proof: generalized BS principle Classical Birman-Schwinger principle based on the identity (H 0 V z) 1 = (H 0 z) 1 + (H 0 z) 1 V 1/2 { I V 1/2 (H 0 z) 1 V 1/2} 1 V 1/2 (H 0 z) 1 can be extended to generalized Schrödinger operators H α,γ [BEKŠ 94]: the multiplication by (H 0 z) 1 V 1/2 etc. is replaced by suitable trace maps. In this way we find that κ 2 is an eigenvalue of H α,γ iff the integral operator R κ α,γ on L 2 (R) with the kernel (s,s ) α 2π K 0 has an eigenvalue equal to one ( κ Γ(s) Γ(s ) ) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 39/11

95 Sketch of the proof We treat R κ α,γ as a perturbation of the operator Rκ α,γ 0 referring to a straight line. The spectrum of the latter is found easily: it is purely ac and equal to [0,α/2κ) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 40/11

96 Sketch of the proof We treat R κ α,γ as a perturbation of the operator Rκ α,γ 0 referring to a straight line. The spectrum of the latter is found easily: it is purely ac and equal to [0,α/2κ) The curvature-induced ) perturbation is sign-definite: we have (R κ α,γ Rκ (s,s ) 0, and the inequality is sharp α,γ0 somewhere unless Γ is a straight line. Using a variational argument with a suitable trial function we can check the inequality sup σ(r κ α,γ ) > α 2κ Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 40/11

97 Sketch of the proof We treat R κ α,γ as a perturbation of the operator Rκ α,γ 0 referring to a straight line. The spectrum of the latter is found easily: it is purely ac and equal to [0,α/2κ) The curvature-induced ) perturbation is sign-definite: we have (R κ α,γ Rκ (s,s ) 0, and the inequality is sharp α,γ0 somewhere unless Γ is a straight line. Using a variational argument with a suitable trial function we can check the inequality sup σ(r κ α,γ ) > 2κ α Due to the assumed asymptotic straightness of Γ the perturbation R κ α,γ Rκ α,γ 0 is Hilbert-Schmidt, hence the spectrum of R κ α,γ in the interval (α/2κ, ) is discrete Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 40/11

98 Sketch of the proof We treat R κ α,γ as a perturbation of the operator Rκ α,γ 0 referring to a straight line. The spectrum of the latter is found easily: it is purely ac and equal to [0,α/2κ) The curvature-induced ) perturbation is sign-definite: we have (R κ α,γ Rκ (s,s ) 0, and the inequality is sharp α,γ0 somewhere unless Γ is a straight line. Using a variational argument with a suitable trial function we can check the inequality sup σ(r κ α,γ ) > 2κ α Due to the assumed asymptotic straightness of Γ the perturbation R κ α,γ Rκ α,γ 0 is Hilbert-Schmidt, hence the spectrum of R κ α,γ in the interval (α/2κ, ) is discrete To conclude we employ continuity and lim κ R κ α,γ = 0. The argument can be pictorially expressed as follows: Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 40/11

99 Pictorial sketch of the proof σ(r κ α,γ ) 1 α/2 κ Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 41/11

100 More geometrically induced properties (b) Perturbation theory for punctured manifolds: a natural question is what happens with σ disc (H α,γ ) if Γ has a small hole. We will give the answer for a compact, (n 1)-dimensional, C 1+[n/2] -smooth manifold in R n Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 42/11

101 More geometrically induced properties (b) Perturbation theory for punctured manifolds: a natural question is what happens with σ disc (H α,γ ) if Γ has a small hole. We will give the answer for a compact, (n 1)-dimensional, C 1+[n/2] -smooth manifold in R n S ε Γ Consider a family {S ε } 0 ε<η of subsets of Γ such that each S ε is Lebesgue measurable on Γ they shrink to origin, sup x Sε x = O(ε) as ε 0 σ disc (H α,γ ), nontrivial for n 3 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 42/11

102 Punctured manifolds: ev asymptotics Call H ε := H α,γ\sε. For small enough ε these operators have the same finite number of eigenvalues, naturally ordered, which satisfy λ j (ε) λ j (0) as ε 0 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 43/11

103 Punctured manifolds: ev asymptotics Call H ε := H α,γ\sε. For small enough ε these operators have the same finite number of eigenvalues, naturally ordered, which satisfy λ j (ε) λ j (0) as ε 0 Let ϕ j be the eigenfunctions of H 0. By Sobolev trace thm ϕ j (0) makes sense. Put s j := ϕ j (0) 2 if λ j (0) is simple, ( ) otherwise they are ev s of C := ϕ i (0)ϕ j (0) corresponding to a degenerate eigenvalue Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 43/11

104 Punctured manifolds: ev asymptotics Call H ε := H α,γ\sε. For small enough ε these operators have the same finite number of eigenvalues, naturally ordered, which satisfy λ j (ε) λ j (0) as ε 0 Let ϕ j be the eigenfunctions of H 0. By Sobolev trace thm ϕ j (0) makes sense. Put s j := ϕ j (0) 2 if λ j (0) is simple, ( ) otherwise they are ev s of C := ϕ i (0)ϕ j (0) corresponding to a degenerate eigenvalue Theorem [E-Yoshitomi 03]: Under the assumptions made about the family {S ε }, we have λ j (ε) = λ j (0) + αs j m Γ (S ε ) + o(ε n 1 ) as ε 0 Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 43/11

105 Remarks Formally a small-hole perturbation acts as a repulsive δ interaction with the coupling αm Γ (S ε ) Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 44/11

106 Remarks Formally a small-hole perturbation acts as a repulsive δ interaction with the coupling αm Γ (S ε ) No self-similarity of S ε required Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 44/11

107 Remarks Formally a small-hole perturbation acts as a repulsive δ interaction with the coupling αm Γ (S ε ) No self-similarity of S ε required If n = 2, i.e. Γ is a curve, m Γ (S ε ) is the length of the hiatus. In this case the same asymptotic formula holds for bound states of an infinite curved Γ Student Colloquium and School on Mathematical Physics; Stará Lesná, August 23-29, 2008 p. 44/11