Resonance asymptotics on quantum graphs and hedgehog manifolds

Transcription

1 Resonance asymptotics on quantum graphs and hedgehog manifolds Jiří Lipovský University of Hradec Králové, Faculty of Science joint work with P. Exner and E.B. Davies Warsaw, 3rd December 2014 Jiří Lipovský Quantum graphs and hedgehog manifolds 1/42

2 Outline 1 Quantum graphs description of the model 2 Resolvent and scattering resonances 3 Asymptotics of resonances for non-magnetic graphs 4 Asymptotics of resonances for magnetic graphs 5 Hedgehog manifolds Jiří Lipovský Quantum graphs and hedgehog manifolds 2/42

3 Quantum graphs scattering system consisting of a compact metric graph with attached halflines equipped with second order differential operator first used for describing diamagnetic properties (Pauling 1936) and π-electrons behaviour in aromatic carbohydrates (Ruedenberg, Scherr 1953) Jiří Lipovský Quantum graphs and hedgehog manifolds 3/42

4 description of mesoscopic systems: quantum wires, nanotubes, photonic crystals, etc. measurements by Hull at all: microwave networks using coaxial cables design of elementary gates in quantum computing (Kostrykin, Schrader 2000) simple model for study of quantum chaos (Kottos, Smilanski 1997; Berkolaiko, Bogomolny, Keating 2000) model of leaky quantum graphs (Exner 2007) operator α(x)δ(x Γ) with Γ R ν, α > 0 Dirac operators on graphs (Bolte, Harrison 2003) Jiří Lipovský Quantum graphs and hedgehog manifolds 4/42

5 Description of the model set of ordinary differential equations graph consists of set of vertices V, set of not oriented edges (both finite E and infinite E ). Hilbert space of the problem H = L 2 ([0, l jn ]) L 2 ([0, )). (j,n) I L j I C states described by columns ψ = (f jn : E jn E, f j : E j E ) T. the Hamiltonian acting as d2 dx 2 + V (x), where V (x) is bounded and supported only on the internal edges corresponds to the Hamiltonian of a quantum particle for the choice = 1, m = 1/2 Jiří Lipovský Quantum graphs and hedgehog manifolds 5/42

6 Domain of the Hamiltonian domain consisting of functions in W 2,2 (Γ) satisfying coupling conditions at each vertex coupling conditions given by A v Ψ v + B v Ψ v = 0, where Ψ v = (ψ 1 (0),..., ψ d (0)) T and Ψ v = (ψ 1 (0),..., ψ d (0) ) T are the vectors of limits of functional values and outgoing derivatives where d is the number edges emanating from the vertex v A and B square d d matrices; selfadjointness assured by AB = A B and Rank (A, B) = d another (unique) description of the coupling conditions: since vectors Ψ v ± iψ v have the same norm, one has U v (Ψ v + iψ v ) = (Ψ v iψ v ), where U v is d d unitary matrix (U v I v )Ψ v + i(u v + I v )Ψ v = 0. Jiří Lipovský Quantum graphs and hedgehog manifolds 6/42

7 Particular choice of coupling conditions δ-conditions f (v) = f n (v), d n=1 f n(v) = α v f (v), α v R corresponding unitary coupling matrix U = 2 d+iα J I, where J is d d matrix with all entries equal to 1 and I is unit matrix in particular α = 0: standard (Kirchhoff s, free, Neumann) coupling conditions δ s-conditions f (v) = f n(v), d f n (v) = β v f (v), n=1 β v R Jiří Lipovský Quantum graphs and hedgehog manifolds 7/42

8 Flower-like model description of coupling conditions using one-vertex graphs suppose that Γ has N internal and M external edges the coupling condition (U I )Ψ + i(u + I )Ψ = 0 describes coupling on the whole graph; U is (2N + M) (2N + M) unitary matrix consisting of blocks U v the above equation decouples into conditions for particular vertices U encodes not only coupling at the vertices, but also the topology of the graph l3 l 2 l 1 l 4 l N Jiří Lipovský Quantum graphs and hedgehog manifolds 8/42

9 Resolvent resonances poles of the meromorphic continuation of the resolvent (H λid) 1 external complex scaling transformation external components of the wavefunction g j (x) U θ g j (x) = e θ/2 g j (xe θ ) with a nontrivial imaginary part of θ non-selfadjoint operator H θ with the domain f jn W 2,2 ([0, l jn ]) and g jθ = U θ g j with g j W 2,2 (L j ) ( ) {gjθ } H θ = {f jn } ( ) ( Uθ 0 U 1 H 0 I θ 0 0 I ) ( ) {gjθ } = {f jn } = ( { e 2θ g jθ } { f jn + V jnf jn } ) Jiří Lipovský Quantum graphs and hedgehog manifolds 9/42

10 0 2 Im θ essential spectrum of H θ is e 2θ [0, ) resonances of H become eigenvalues of H θ for Im θ large enough halfline components of corresponding eigenfunctions of H θ are g jθ (x) = g jθ (0)e ikxeθ Jiří Lipovský Quantum graphs and hedgehog manifolds 10/42

11 Scattering Resonances solutions on the external edges: superposition of incoming and outgoing waves g j (x) = c j e ikx + d j e ikx motivation for incoming and outgoing waves: time-dependent Schrödinger equation ( 2 x i t )u j (x, t) = 0 separating variables we can write: u j (x, t) = e itk2 g j (x), where g j (x) solves time-independent Schrödinger equation hence u j (x, t) = c j e ik(x+kt) + d j e ik(x kt) S = S(k) which maps the vector of amplitudes of the incoming waves c = {c n } into the vector of the amplitudes of the outgoing waves d = {d n } scattering resonances complex energies where S diverges Jiří Lipovský Quantum graphs and hedgehog manifolds 11/42

12 Effective coupling on a finite graph replacing noncompact graph by its compact part instead of halflines there are effective coupling matrices N internal, M external edges U consists of blocks U j ( ) U1 U U = 2, U 3 U 4 where U 1 is 2N 2N matrix corresponding to coupling between internal edges, etc. let Ψ 1 correspond to vector of functional values for internal edges and Ψ 2 for external edges (similarly for vectors outgoing derivatives) by complex scaling of the external edges one obtains Ψ 2 = ikψ 2 Jiří Lipovský Quantum graphs and hedgehog manifolds 12/42

13 by a standard procedure (Schur, etc.) one gets ( ) ( ) U1 I 2N 2N U 2 Ψ1 + U 3 U 4 I M M Ψ 2 ( ) ( ) U1 + I + i 2N 2N U 2 Ψ 1 = 0. U 3 U 4 + I M M ikψ 2 effective coupling matrix Ũ(k) = U 1 (1 k)u 2 [(1 k)u 4 (k + 1)I ] 1 U 3 coupling condition has the same form only with U replaced by Ũ(k) (Ũ(k) I )Ψ 1 + i(ũ(k) + I )Ψ 1 = 0 Jiří Lipovský Quantum graphs and hedgehog manifolds 13/42

14 Asymptotics of resonances on quantum graphs N(R) number of resolvent resonances in the circle of radius R in the k plane expected behaviour of the counting function N(R) = 2V π R + O(1) trivial example of a graph where this asymptotics is not satisfied Jiří Lipovský Quantum graphs and hedgehog manifolds 14/42

15 Asymptotics for standard conditions Theorem (Davies, Pushnitski) Suppose that Γ is the graph with standard (Kirchhoff) coupling conditions at all the vertices. Then N(R) = 2 WR + O(1), as R, π where the coefficient W satisfies 0 W vol (Γ). The behaviour is non-weyl (0 W < vol (Γ)) iff there is a balanced vertex. vol (Γ) is sum of lengths of the internal edges balanced vertex: number of internal edges is equal to the number of external edges Jiří Lipovský Quantum graphs and hedgehog manifolds 15/42

16 Asymptotics for general coupling conditions Theorem (Davies, Exner, J.L.) Let Γ be a graph with coupling described by unitary matrix U. Then it is non-weyl iff the corresponding effective coupling matrix Ũ has eigenvalue 1+k 1 k 1 k or 1+k. explanation: presence of these eigenvalues corresponds to a subspace of the domain of H in which there is standard coupling or antikirchhhoff s coupling idea of the proof: the resonance condition can be expressed in the form of exponential polynomials n P r (k) exp iσ r k = 0 r=1 Jiří Lipovský Quantum graphs and hedgehog manifolds 16/42

17 zeros of these function are in logarithmic strips Im k + m r log( k ) = ±C r and the counting functions behaves as σn σ 0 π R + O(1) Jiří Lipovský Quantum graphs and hedgehog manifolds 17/42

18 Non-Weyl graphs for permutation symmetric coupling two-parameter class of coupling conditions which are symmetric with respect to permutation of any two edges U = aj bi, where complex numbers a and b satisfy b = 1 and b + da = 1 graph is non-weyl iff there is a balanced vertex with standard (U = 2 d J I ) or antikichhoff s (U = 2 d J + I ) coupling idea of the proof: one can straightforwardly obtain effective coupling matrix Ũ(k) = ab(1 k) a(1+k) J + bi ; graph is (aq+b)(1 k) (k+1) ab(1 k) a(1+k) non-weyl iff its eigenvalues d (aq+b)(1 k) (k+1) + b correspond to 1±k 1 k graphs with non-weyl asymptotics are exceptional Jiří Lipovský Quantum graphs and hedgehog manifolds 18/42

19 Example: loop with two leads graph consisting of two leads and two semiinfinite links; standard condition in the middle vertex f 1 (x) g 1 (x) 0 l f 2 (x) g 2 (x) trick: separate the subspace of symmetric functions from the subspace of antisymmetric ones f sym (x) = 1 2 (f 1 (x) + f 2 (x)), f ant (x) = 1 2 (f 1 (x) f 2 (x)), g sym (x) = 1 2 (g 1 (x) + g 2 (x)), g ant (x) = 1 2 (g 1 (x) g 2 (x)). Jiří Lipovský Quantum graphs and hedgehog manifolds 19/42

20 Example: loop with two leads standard coupling conditions at the vertices: f 1 (0) = f 2 (0), f 1(0) = f 2(0), f 1 (l) = f 2 (l) = g 1 (0) = g 2 (0), f 1(l) f 2(l) + g 1(0) + g 2(0) = 0. coupling conditions in terms of f sym and f ant : f sym(0) = 0, f ant (0) = 0, f sym (l) = g sym (0), f ant (l) = 0, f sym(l) = g sym(0), g ant (0) = 0. Jiří Lipovský Quantum graphs and hedgehog manifolds 20/42

21 Size reduction for general coupling conditions Theorem Let Γ be the graph below obtained from a flower-like graph with N internal and M external edges. Let the coupling be given by arbitrary U (1) and U (2). Let V be an arbitrary unitary 2N 2N matrix, W an arbitrary unitary M M matrix, V (1) := diag (V, W ) and V (2) := diag (I (q 2N) (q 2N), V ) be (2N + M) (2N + M) and q q block diagonal matrices, respectively. Then H on Γ is unitarily equivalent to the Hamiltonian H V,W on topologically the same graph with the coupling given by the matrices [V (1) ] 1 U (1) V (1) and [V (2) ] 1 U (2) V (2). Γ 0 U (2) U (1) l 0 X 1 Jiří Lipovský Quantum graphs and hedgehog manifolds 21/42

22 Size reduction idea of the proof let u be an element of domain of H, u 1,..., u 2N its restrictions to the internal edges between two internal vertices, f 1,..., f M restrictions to the halflines, and u 0 restriction to the rest of the graph Γ 0 then the map (u 1,..., u 2N ) T (v 1,..., v 2N ) T = V 1 (u 1,..., u 2N ) T (f 1,..., f M ) T (g 1,..., g M ) T = W 1 (f 1,..., f M ) T u 0 (x) v 0 (x) = u 0 (x) is a bijection of the domain of H onto the domain of H V,W ; the coupling condition transform into the one with matrices [V (1) ] 1 U (1) V (1) and [V (2) ] 1 U (2) V (2). Jiří Lipovský Quantum graphs and hedgehog manifolds 22/42

23 Effective size of a graph example regular polygon with two halflines attached at each vertex l l l l l criterion whether the graph is Weyl or non-weyl is local, while the effective size of a non-weyl graph depends on topology of the whole graph Jiří Lipovský Quantum graphs and hedgehog manifolds 23/42

24 T unitary rotation operator in L 2 (Γ n ) takes each vertex to the next vertex in the clockwise direction Bloch-Floquet decomposition with respect to the cyclic rotational group L 2 (Γ n ) = n H ω, where H ω = {f L 2 (Γ n ) : Tf = ωf } with ω satisfying ω n = 1 f 2 f 3 f 4 = ωf 1 f 1 0 l 0 l ansatz for functions in H ω f 1 (x) = αe ikx + βe ikx, 0 x l, f 2 (x) = (α + β)e ikx, 0 x <, f 3 (x) = (α + β)e ikx, 0 x <, f 4 (x) = ωf 1 (x), 0 x l, Jiří Lipovský Quantum graphs and hedgehog manifolds 24/42

25 applying coupling conditions leads to ωαe ikl + ωβe ikl = α + β, ωαe ikl ωβe ikl = 2(α + β) + α β. final resonance condition 2(ω 2 + 1) + 4ωe ikl = 0. hence the middle term is zero iff ω = ±i { nl/2 if n 0 mod 4, W n = (n 2)l/2 if n = 0 mod 4. Jiří Lipovský Quantum graphs and hedgehog manifolds 25/42

26 Quantum graphs in magnetic field Hamiltonian acting as (d/dx + ia jn (x)) 2 on jn-th internal edge and as d 2 /dx 2 on the infinite leads coupling conditions: (U I )Ψ + i(u + I )(Ψ + AΨ) = 0 can be after a gauge transformation written as (U A I )Ψ + i(u A + I )Ψ = 0 with U A given by a unitary transformation U A = F 1 UF with with A = diag (A 1 (0), A 1 (l 1 ),..., A N (0), A N (l N ), 0,..., 0) two main results on the asymptotics of magnetic graphs: one cannot change a non-weyl graph into a Weyl one and vice versa for a non-weyl graph it is possible to change the effective size Jiří Lipovský Quantum graphs and hedgehog manifolds 26/42

27 Theorem (Exner, J.L.) Let Γ be a quantum graph with N internal and M external edges with coupling given by a unitary matrix U. Let V be a (2N + ( M) (2N ) + M) block unitary matrix consisting of blocks V1 0 V =. Let Γ 0 V V be a graph with the same topology as Γ 2 and with the coupling given by U V = V 1 UV. Then Γ V is non-weyl iff Γ is. idea of the proof: the graph has non-weyl asymptotics iff at least one of the eigenvalues of Ũ(k) is 1±k 1 k Ũ V (k) = V1 1 U 1 V 1 (1 k)v1 1 U 2 V 2 [(1 k)v2 1 U 4 V 2 (k + 1)I ] 1 V2 1 U 3 V 1 = V1 1 Ũ(k)V 1. eigenvalues of Ũ(k) cannot be changed by this unitary transformation Jiří Lipovský Quantum graphs and hedgehog manifolds 27/42

28 Example: loop graph in a magnetic field f(x) 0 l g 1(x) g 2(x) one loop and two halflines attached at one point, standard coupling conditions f (0) = f (l) = g 1 (0) = g 2 (0), f (0) f (l) + g 1(0) + g 2(0) = 0. constant magnetic field with constant value of tangent component of the magnetic potential A Jiří Lipovský Quantum graphs and hedgehog manifolds 28/42

29 the ansatz f (x) = e iax (ae ikx + be ikx ) for the wavefunction on the loop leads to ik[a b e ial (ae ikl be ikl + 2e θ/2 g θ (0))] = 0, final resonance condition a + b = e ial (ae ikl + be ikl ) = e θ/2 g θ (0). 2 cos Al + e ikl = 0, for Φ = Al = ±π/2 (mod π) the l-independent term disappears and hence effective size of the graph is zero Jiří Lipovský Quantum graphs and hedgehog manifolds 29/42

30 Effective size of a one-loop magnetic graph Lemma (Exner, J.L.) The effective size of a graph with a single internal edge is zero iff it is non-weyl and its effective coupling matrix Ũ(k) satisfies ũ 12 + ũ 21 = 0. Theorem (Exner, J.L.) For any non-weyl quantum graph with one internal edge and ũ 12 = ũ 21 there is a magnetic field such that the graph under its influence has at most finite number of resonances. idea of the proof: the( gauge transformation ) changes the effective coupling to ũ11 e iφ ũ 12 e iφ ũ 21 ũ 22 we adjust the phase shift in order to satisfy the condition ũ 12 + ũ 21 = 0 conclusion: it is possible to change the size of a non-weyl graph Jiří Lipovský Quantum graphs and hedgehog manifolds 30/42

31 Hedgehog manifolds compact Riemannian manifold Ω R N with N {1, 2, 3} without boundary or with smooth boundary endowed with metric g rs we denote by Γ a manifold Ω with n j halflines attached at points x j, j {1,..., n j } distinct from the boundary the Hilbert space we consider consists of direct sum of square integrable functions on every component of Γ H = L 2 (Ω, g dx) M i=1 L 2 (R (i) + ) where M = j n j is the total number of halflines and g = det (g rs ) Jiří Lipovský Quantum graphs and hedgehog manifolds 31/42

32 asymptotical expansion of f (x) near x j is where f (x) = c j (f )F 0 (x, x j ) + d j (f ) + O(r(x, x j )) F 0 (x, x j ) = { q 2(x,x j ) 2π ln r(x, x j ) N = 2 q 3 (x,x j ) 4π (r(x, x j )) 1 N = 3 q 2, q 3 are continuous functions of x with q i (x j, x j ) = 1 and r(x, x j ) denotes distance between x and x j function F 0 is the leading (not energy-dependent) term of asymptotics of the Green s function near x j. G(x, x j ; k) = F 0 (x, x j ) + F 1 (x, x j ; k) + R(x, x j ; k) with R(x, x j ; k) = O(r(x, x j )). Jiří Lipovský Quantum graphs and hedgehog manifolds 32/42

33 Asymptotics of resonances for hedgehog manifolds the Hamiltonian acting as Laplace-Beltrami operator on Ω and as d2 on the infinite leads dx 2 the family of corresponding Hamiltonians can be constructed as selfadjoint extensions of H 0 the restriction of the free Laplace-Beltrami operator on Ω functions in W 2,2 (Ω) satisfying f (x j ) = 0, j coupling conditions binding the coefficients by the leading two terms of expansion of f has the standard form (U I )Ψ + i(u + I )Ψ = 0, where Ψ = (d 1 (f ),..., d n (f ), f 1 (0),..., f M (0)) T and Ψ = (c 1 (f ),..., c n (f ), f 1 (0),..., f M (0))T Jiří Lipovský Quantum graphs and hedgehog manifolds 33/42

34 an effective condition for a manifold without halflines (Ũj(k) I )d j (f ) + i(ũj(k) + I )c j (f ) = 0 condition of solvability of the system leads to [(Ũ(k) ] det I )Q 0(k) + i(ũ(k) + I ) = 0 where { G(xi, x Q 0 (k) = j ; k) i j F 1 (x i, x i ; k) i = j in particular, if there is only one point where the halfline is attached we get F (k) = F 1 (k) + i(ũ(k) I ) 1 (Ũ(k) + I ) = 0, where Ũ(k) is energy dependent effective coupling matrix (in this case 1 1) Jiří Lipovský Quantum graphs and hedgehog manifolds 34/42

35 Hedgehog manifold with one halfline using the results of Avramidi for high mass asymptotics of operator + m 2 one can find Lemma The asymptotics of regularized Green s function is 1 For d = 1: F 1 (k) = 2 For d = 2: F 1 (k) = { 1 { 1 2ik + O( Im k 2 ) Im k < 0, 1 2ik + O( Im k 2 ) Im k > 0. 2π (ln ik ln 2 γ) + O( Im k 1 ) Im k < 0, 1 2π (ln ik ln 2 γ) + O( Im k 1 ) Im k > 0. { ik 3 For d = 3: F 1 (k) = 4π + O( Im k 1 ) Im k < 0, ik 4π + O( Im k 1 ) Im k > 0. where γ stands for the Euler constant. Jiří Lipovský Quantum graphs and hedgehog manifolds 35/42

36 Example 1: abscissa of length 2l with M halflines attached l 0 l the regularized Green s function F 1 is F 1 (x, y; k) = n ψ n (x)ψ n (y) λ n k 2 = 1 l cos (2n 1)πx 2l cos (2n 1)πy 2l ( ) 2 (2n 1)π n=1 2l k 2. F 1 (0, 0; k) = 1 l n=1 ( (2n 1)π 2l 1 ) 2 k 2 = 1 tan kl. 2k Jiří Lipovský Quantum graphs and hedgehog manifolds 36/42

37 the number of phase jumps along the circle in the k plane gives the difference between the number of resonances in the system and the number of eigenvalues of a decoupled operator Figure: Phase of the regularized Green s function for d = 1 Figure: Phase of the regularized Green s function plus i/(2k) for d = 1 Jiří Lipovský Quantum graphs and hedgehog manifolds 37/42

38 Example 2: circle with M halflines attached l Green s function G(x, 0; k) = 1 4 Y 0(kr) + c(k)j 0 (kr), using previous properties one can write F 1 (k) = 1 2π (ln k ln 2 + γ) + Y 0(kl) 4J 0 (kl) 1 2π (ln k ln 2 + γ) tan kl π 4. Jiří Lipovský Quantum graphs and hedgehog manifolds 38/42

39 Open questions How can one determine effective size of a graph? What about hedgehog manifolds with halflines attached at more points? Is there a chance to influence two leading coefficients? Thank you for your attention! Jiří Lipovský Quantum graphs and hedgehog manifolds 39/42

40 Open questions How can one determine effective size of a graph? What about hedgehog manifolds with halflines attached at more points? Is there a chance to influence two leading coefficients? Thank you for your attention! Jiří Lipovský Quantum graphs and hedgehog manifolds 39/42

41 Articles on which the talk was based E.B. Davies, A. Pushnitski: Non-Weyl Resonance Asymptotics for Quantum Graphs, Analysis and PDE 4 (2011), no. 5, pp arxiv: E.B. Davies, P. Exner, J. Lipovský: Non-Weyl asymptotics for quantum graphs with general coupling conditions J. Phys. A43 (2010), arxiv: P. Exner, J. Lipovský: Non-Weyl resonance asymptotics for quantum graphs in a magnetic field, Phys. Lett. A 375 (2011), arxiv: P. Exner, J. Lipovský: Resonances in hedgehog manifolds, Acta Polytechnica 53 (2013), no. 5, arxiv: Jiří Lipovský Quantum graphs and hedgehog manifolds 40/42

42 Other references Aguilar, J. and Combes, J.-M., A class of analytic perturbations for one-body Schrödinger operators, Commun. Math. Phys. (1971), vol. 22, pp Berkolaiko, G., Bogomolny, E., and Keating, J., Star graphs and Šeba billiards, J. Phys. A (2001), vol. 34, pp J. Bolte, J. Harrison: Spectral statistics for the Dirac operator on graphs, J. Phys. A: Math. Gen. (2003), vol. 36, Kostrykin, V. and Schrader, R.: Kirchhoff s rule for quantum wires. II: The inverse problem with possible applications to quantum computers. Fortschritte der Physik (2000) vol. 48, pp Jiří Lipovský Quantum graphs and hedgehog manifolds 41/42

43 Kottos, T. and Smilansky, U., Quantum chaos on graphs, Phys. Rev. Lett. (1997), vol. 79, pp Pauling, L.: The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys. (1936), vol. 4, pp Ruedenberg, K. and Scherr, C.: Free-electron network model for conjugated systems, I. Theory, J. Chem. Phys. (1953), vol. 21, pp Jiří Lipovský Quantum graphs and hedgehog manifolds 42/42