Resonance asymptotics on quantum graphs and hedgehog manifolds

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

0.0 0.5 1.0 1.5 2.0 2.5 3.0 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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) + 1 4 tan kl π 4. Jiří Lipovský Quantum graphs and hedgehog manifolds 38/42

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

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

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. 729-756. arxiv:1003.0051. E.B. Davies, P. Exner, J. Lipovský: Non-Weyl asymptotics for quantum graphs with general coupling conditions J. Phys. A43 (2010), 474013. arxiv: 1004.0856. P. Exner, J. Lipovský: Non-Weyl resonance asymptotics for quantum graphs in a magnetic field, Phys. Lett. A 375 (2011), 805 807. arxiv: 1011.5761. P. Exner, J. Lipovský: Resonances in hedgehog manifolds, Acta Polytechnica 53 (2013), no. 5, 416 426. arxiv: 1302.5269. Jiří Lipovský Quantum graphs and hedgehog manifolds 40/42

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. 269 279. Berkolaiko, G., Bogomolny, E., and Keating, J., Star graphs and Šeba billiards, J. Phys. A (2001), vol. 34, pp. 335 350. J. Bolte, J. Harrison: Spectral statistics for the Dirac operator on graphs, J. Phys. A: Math. Gen. (2003), vol. 36, 2747 2769. 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. 703 716. Jiří Lipovský Quantum graphs and hedgehog manifolds 41/42

Kottos, T. and Smilansky, U., Quantum chaos on graphs, Phys. Rev. Lett. (1997), vol. 79, pp. 4794 4797. Pauling, L.: The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys. (1936), vol. 4, pp. 673 677. Ruedenberg, K. and Scherr, C.: Free-electron network model for conjugated systems, I. Theory, J. Chem. Phys. (1953), vol. 21, pp. 1565 1581. Jiří Lipovský Quantum graphs and hedgehog manifolds 42/42