LMS/EPSRC Short Course Analysis on Graphs and its Applications Gregynog Hall, University of Wales, 10-15 January 2007 Quantum Graphs Part I. Basic Structures Peter Kuchment kuchment@math.tamu.edu Department of Mathematics Texas A& M University College Station, TX, USA 1. An outline of the three lectures • What is a quantum graph? interesting? Why are quantum graphs • Self-adjoint vertex conditions. • Spectral theory. Usual and unusual spectral features. • Special problems: justification of the model, gaps, slowing down waves, etc. • Inverse problems. • Generalizations: “leaky graphs”, “quantum buildings”,etc. • The outer space. 2. Books, surveys and collections of articles (MANY more references at the end of the file) Books and collections of articles (a) G. Berkolaiko et al (Editors), Quantum Graphs and Their Applications, Contemp. Math., v. 415, Amer. Math. Soc. 2006. (b) V. A. Kozlov, V. G. Mazýa, and A. B. Movchan, Asymptotic Analysis of Fields in Multi-Structures, Oxford Sci. Publ. 1999. (c) P. Kuchment (Editor), Waves in Periodic and Random Media, Contemp. Math. v. 339, Amer. Math. Soc. 2004. (d) P. Kuchment (Editor), Quantum graphs and their applications, a special issue of Waves in Random media, 14(2004), no.1. (e) F. Mehmeti et al (Editors), Partial Differential Equations on Multistructures, CRC 2001 (f) Yu.V. Pokornyi et al, Differential equations on geometrical graphs. (in Russian) Moscow: Fizmatlit, 2004. Surveys and introductory articles (a) P. Exner and P. Seba, Electrons in semiconductor microstructures: a challenge to operator theorists, in Proc. Workshop “Schrödinger Operators, Standard and Nonstandard” (Dubna 1988), World Sci., Singapore 1989; pp. 79–100. (b) V. Kostrykin and R. Schrader, Kirchhoff’s rule for quantum wires, J. Phys. A 32(1999), 595-630. (c) T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Phys. 274 (1999), 76–124. (d) P. Kuchment, The Mathematics of Photonic Crystals, Ch. 7 in Mathematical Modeling in Optical Science, Bao, G., Cowsar, L. and Masters, W.(Editors), 207–272, Philadelphia: SIAM, 2001. (e) P. Kuchment, Graph models of wave propagation in thin structures, Waves in Random Media 12(2002), no. 4, R1-R24. (f) P. Kuchment, Differential and pseudo-differential operators on graphs as models of mesoscopic systems, in “Analysis and Applications”, H. Begehr et al (Editors), Kluwer 2003, 7–30. (g) P. Kuchment, Quantum graphs I. Some basic structures, Waves in Random media, 14(2004), S107–S128. (h) P. Kuchment, On some spectral problems of mathematical physics, in Partial Diff. Equat. and Inverse Problems, C. Conca et al(Editors), Contemp. Math. v. 362, AMS 2004 (i) Yu. V. Pokornyi and V. L. Pryadiev, Some problems of the qualitative SturmLiouville theory on a spatial network, Russian Math. Surveys 59 (2004), no. 3, 515–552. Books on spectral graph theory (a) F. Chung, Spectral Graph Theory, Amer. Math. Soc., Providence R.I., 1997. (b) Y. Colin de Verdière, Spectres De Graphes, Societe Mathematique De France, 1998 (c) D. Cvetkovic, M. Doob, and H. Sachs, Spectra of Graphs, Acad. Press., NY 1979. (d) D. Cvetkovic, M. Doob, I. Gutman, A. Targasev, Recent Results in the Theory of Graph Spectra, Ann. Disc. Math. 36, North Holland, 1988. 3. What is a metric graph? Locally finite graph Γ = (V, E). Graphs will be assumed to be connected. Metric graph - graph equipped with a length 0 < le ≤ ∞ assigned to each edge e. Example: Γ embedded into Rn, le - the “true” length of e. We consider here Γ as a singular manifold. Metric graphs are also sometimes (in combinatorial and geometric group theory) called R-graphs. 4. What is a quantum graph? Introduce coordinate xe (or just x) on e with values in [0, le]. Measure dx on Γ, integration, and differentiation along edges are defined, as well as some functional spaces. L 2 2 L (e). E.g., space L (Γ) = e∈E One can define Sobolev space H 1(Γ) as follows: include functions u that satisfy: u ∈ H 1(e) for each edge e ∈ E, P kuk2 1 (e) < ∞ (needed for infinite graphs only), H e∈E u is continuous at each v ∈ V, and thus on the whole Γ. Higher order Sobolev spaces are not uniquely defined on Γ. One can define differential operators on Γ. Main operator (Hamiltonian) H acts on each edge on a function u as d2u(x) + q(x)u(x), Hu = − dx2 ∞ (Γ). Schrödinger operator on (Γ) ∩ L real potential q ∈ L2 loc the graph Γ. Domain D(H) consists of u s.t. u ∈ H 2(e) for each edge e ∈ E, P kuk2 2 (e) < ∞ (needed for infinite graphs only), H e∈E u satisfies “appropriate” boundary conditions at each v ∈ V. We will describe all vertex conditions that give rise to selfadjoint operators H. The simplest: Neumann conditions u is continuous at each vertex, and thus on the whole Γ, P du dxe (v) = 0 for each v ∈ V e∈Ev (derivatives taken in outgoing directions from v). Definition. A quantum graph Γ is a metric graph equipped with a self-adjoint operator H. More general operators are also considered in applications. E.g., magnetic Schrödinger operators 2 d + A(x) + q(x)u(x), −i dx operators of higher orders, pseudo-differential operators (see later on in these lectures and also in Exner’s lectures), etc. 5. Why are quantum graphs interesting? ♣ Modeling waves in thin branching “graph-like ”structures: narrow waveguides, quantum wires, photonic crystals, blood vessels, lungs. Applications in nanotechnology, optics, chemistry, medicine. ♣ Modeling electron spectra of aromatic molecules Naphthalene and Carbon nano-structures Graphene Nanotube ♣ Averaging in dynamical systems. ♣ Toy models in quantum mechanics and other areas 6. Justification of “fattened graph” models “Fattened graph” domain Ω around Γ. Laplace operator −∆N, (or more general Schrödinger) with Neumann b.c. in Ω. Does σ(−∆N,) converge when → 0 to the spectrum of a quantum graph Γ? Yes, it does (Freidlin&Wentzel, Rubinstein&Schatzmann, P.K.&Zeng, Saito, Exner&Post, Post). When neighborhoods of vertices decay slowly, additional spectrum attached to the vertices arises, or the graph completely decouples (Freidlin&Wentzel, P.K.&Zeng, Exner&Post). Technique used: Constructing operators of extension E : H 1(Γ) 7→ H 1(Ωε) and averaging P : H 1(Ωε) 7→ H 1(Γ) such that they can only slightly increase the quadratic forms and only slightly decrease L2-norms. Then the Rayleigh-Ritz representation of eigenvalues is used: (Hu, u) . inf sup λn = dimW =n u∈W,u6=0 (u, u) Similar question about the Dirichlet b.c.. Yes in smooth case (i.e., no vertices allowed). (Exner et al.) Graph case: the answer is in general a No. Partial results show different spectral behavior below the lowest energy of transversal modes (around π 2/2) and above it. Below: additional spectrum arises “attached to vertices” (Exner et al.) Above: a graph model might be possible (Post, Molchanov&Vainberg). Electromagnetic waves Spectral problems for the operators: Maxwell 1 ~ = λU ~,∇ · U ~ =~ ∇U 0. ∇× ε(x) Divergence type −∇ · 1 ∇H = λH. ε(x) Helmholtz type −∆E = λε(x)E. ε(x) - periodic electric permittivity. For the Helmholtz case, some modes converge to the ones of the problem −∆E = µδΓ(x)E + cE, δΓ(x) – the delta-function supported on the graph Γ.(Figotin & P.K., P.K.&Kunyansky) When c = 0, this reduces to the spectrum of the (pseudodifferential) Dirichlet-to-Neumann operator on the graph Γ: ∂E ]Γ = µE. [ ∂ν Similar spectral problem arises also for the “leaky wires” (see Exner’s lectures). Relations to quantum graphs with differential Hamiltonians (sometimes of higher order) exist (P.K.&Kunaynsky). 7. Self-adjoint vertex conditions ♣ Local (vertex) conditions involve values at a single vertex at a time. Av F (v) + Bv F 0(v) = 0 with d × d matrices Av , Bv . d - degree of v. Theorem 1. (Kostrykin&Schrader ’99) Let Γ - finite. H is self-adjoint iff matrix (Av Bv ) has the maximal rank and the matrix Av Bv∗ is self-adjoint. (1) Parametrization (A, B) is redundant. ♣ Non-local conditions - the same, matrices involve all vertex values. Theorem still holds. Conclusions: (i) topology of a quantum graph is encoded in boundary conditions. (ii) Allowing arbitrary conditions, every quantum graph is a rose. 8. Alternative descriptions of self-adjoint vertex conditions ♣ (M. Harmer) Let U = Uv - unitary d × d. i(Uv − I)F (v) + (Uv + I)F 0(v) = 0. Alternatively, Pv - ortho-projector onto eigenspace of Uv corresponding to −1 (Pv could be zero), Qv = I − Pv . Then P F (v) = 0 (the “Dirichlet” part of conditions), v Qv F 0(v) = Sv QF (v), Uv −I is s.-a. in Q C|B| . Quadratic form: where Sv = −i U v v +I X Z X du 2 | | dx − hSv U (v), U (v)i h[u, u] = e∈E e dx v∈V Domain: u ∈ H 1(e) ∀e ∈ E(G), P Sv F (v) = 0 ∀v ∈ V (G). So called vertex and bond scattering matrices, as well as trace formulas will not be covered in these lectures. 9. “Popular” types of vertex conditions δ-type conditions: f (x) is continuous on Γ P df at each vertex v , dxe (v) = αv f (v), αv ∈ e∈Ev R . (2) Quadratic form defined on functions from H 1(Γ): X Z X df 2 | | dx + αv |f (v)|2. e∈E e dx v∈V (3) When αv = 0, we get the most common case of Neumann (Kirchhoff) conditions: f (x) is continuous on Γ P df . at each vertex v , dxe (v) = 0 e∈Ev (4) Conditions of δ 0-type: df Derivative dxe (v) is the same for all edges e s.t. v ∈ e e P . df f (v) = α (v) e v dx e∈Ev (5) The quadratic form X Z X X 1 df 2 | | dx + | fe(v)|2. α e∈E e dx {v∈V | αv = 6 0} v {e∈Ev } is defined on functions in L e H 1(e) that have at each vertex v where αv = 0 the sum of the vertex values along all entering edges equal to zero. Vertex Dirichlet conditions: at each vertex the boundary values of the function are equal to zero. Operator decouples into direct sum of Dirichlet Laplacians on the edges. Quadratic form X Z df | |2dx e∈E e dx is defined on f ∈ H 1(Γ) s.t. f (v) = 0 at any vertex v. Vertex Neumann conditions: derivatives fe0 (v) at all vertices are equal to zero. Operator decouples into the sum of Neumann problems on each edge. Quadratic form X Z df | |2dx e∈E e dx on L e∈E(Γ) H 1(e). 10. Dirichlet-to-Neumann map A graph G with a subset B ⊂ V (G) designated as a boundary: Graph G Blue vertices - the “boundary” Hu = λu on G “Dirichlet problem” u| = 0 B Σ = {λn}. has discrete spectrum Let λ ∈ / Σ, φ : B 7→ C. Dirichlet-to-Neumann map Λ(λ) is: ∂u |B , Λ(λ)φ := ∂ν where Hu = λu on G u| = φ. B For v ∈ B, ∂u ∂ν (v) is the sum of ingoing derivatives of u at v. Λ(λ) is a meromorphic |B| × |B| - matrix function. 11. Scattering graphs Changing the view of the graph, the problem can be recast as a vector-valued on R+: − d2 U = λU, t > 0 dt2 U 0(0) = Λ(λ)U (0) I.e., spectral problems for scattering graphs reduce to star graphs & eigenvalue-dependent vertex conditions Related to the Schur complement in the matrix theory. 12. Some “graphic” peculiarities :-) ♣ Unlike for ODEs or second order elliptic PDEs, there is no uniqueness of continuation. One can, for instance, have compactly supported eigenfunctions (bound states) for quantum graph operators: This influences many aspects of quantum graph research. ♣ No monodromy. Consider a graph like this: Whatever graph Easy to check: there are values of λ such that given initial data left from the obstacle, solution of Hu = λu cannot be extended to the right. 13. Spectra of quantum graphs ♣ Standard decomposition of the spectrum: σ(H) = σpp(H) ∪ σac(H) ∪ σsc(H) 1 0.8 0.6 0.4 0.2 Pure point σpp (localized states) -4 -2 0 0 2 4 x 1 0.5 x -20 -10 0 0 10 20 -0.5 Absolutely continuous σac (extended states) -1 Singular continuous σsc (?) Discrete σdis - isolated eigenvalues of finite multiplicity Essential σess = σ − σdisc ♣ A connection between quantum and discrete graph spectra Γ - finite. λ ∈ σ(H) ⇔ ∃u ∈ L2, Hu = λu. Assume Neumann conditions: u is continuous, sum of outgoing derivatives at each v is 0. On edges, resolve Hu = λu, assuming vertex values of u known. This gives sum of vertex derivatives of u at v ∈ V (Γ) as combination (with coefficients depending on λ) of values of u at v and at neighboring vertices. We need to make the sum of the derivatives equal to zero. This produces discrete linear equation T (λ){u|V (Γ)} = 0. Thus, quantum graph spectral problem reduces to a discrete one, at the price of λ entering nonlinearly. Observation: this works when λ is not in the Dirichlet spectrum of H on any edge e. Incorporating “fake” vertices of degree 2 along edges, one can avoid this difficulty. 2 d H = − dx2 , all edge lengths are equal to 1, all Example: valences are d. Solve along edge e = (v, w): ue(x) = sin 1 √ λle √ √ ue(v) sin λ(le − x) + ue(we) sin λx . (6) This works if λ 6= n2π 2 with an integer n 6= 0. Simple calculation shows that T (λ)u = 0 becomes √ ! √ X sin l λ u(w) = α √ + dv cos λ u(v). l λ {w| e=(v,w)∈E } v I.e., λ 6= n2π 2 is in σ(H) iff √ √ α sin√l λ + d cos λ is in σ(∆), l λ ∆ being the discrete Laplace operator on Γ. This reduction to discrete problems is harder to deduce for infinite graphs, due to appearance of continuous spectrum. It is established in several cases: 1. Graphs with equal lengths of edges (Pankrashkin). 2. Periodic graphs (P.K., must be folklore, uses Floquet theory, see later in the lectures). 3. Graphs of sub-exponential growth (P.K., uses Shnol’s Theorem and generalized eigenfunctions, described later in the lectures). 4. Point spectrum (Cattaneo). ♣ Genericity of simple spectrum There are known results by Albert and by Uhlenbeck claiming that generically the spectrum of an elliptic operator on a compact manifold is simple. Is this true for quantum graphs? Theorem 2. (L. Friedlander) d2 and Neumann Let Γ be a finite quantum graph with H = − dx 2 conditions. Consider the set M of points {le}|e∈E(Γ) ∈ R|E(Γ)| such that the spectrum of H is simple. If Γ is not topologically a circle, then M is residual. The statement is incorrect for Γ a circle. ♣ Shnol’s Theorem How can one detect that a particular λ belongs to σ(H)? Theorem 3.(a) λ is an eigenvalue ⇔ there exists an eigenfunction φ ∈ L2(Γ) (suffices for compact graphs). (b) λ ∈ σ(H) ⇔ there exists a sequence φn ∈ L2 of approximate eigenfunctions, i.e. kφnkL2(Γ) = 1 and k(H − λ)φnk →n→∞ 0 (Weyl’s criterion). (c) If for every > 0 there exists a generalized eigenfunction φ (i.e., satisfying Hφ = λφ) such that kφkL2(Br ) < Cer , then λ ∈ σ(H). (Shnol’s Theorem) ♣ Generalized eigenfunctions Shnol’s theorem: having a generalized eigenfunction of a subexponential growth implies being on spectrum. Is converse true: does one have generalized eigenfunctions of a controllable growth? Yes (Gelfand-Kostyuchenko, Berezansky). Let 0 < Φ ∈ L∞(Γ) ∩ L2(Γ). Theorem 4. (Hislop&Post ’06) For almost every (w.r.t. spectral measure) λ ∈ σ(H) there exists a generalized eigenfunction ψ s.t. Φψ ∈ L2(Γ). The system of such functions is complete. In particular, if Φ decays slower than exponentially, i.e. min Φ ≥ Br Ce−r for any positive , then the generalized eigenfunction is of Shnol’s Theorem’s class. ♣ Spectral gaps It is often desirable to have gaps in the spectrum of a graph operator (photonic crystals, expander graphs, etc.) Some known mechanisms of opening gaps: • Periodic media. According to Floquet theory, spectra of operators in periodic media tend (but do not have) to have gaps. Hard to control. • Resonant gap opening by “decorating” graphs (Pavlov, Aizenman&Schenker, P.K.& Ong): or This mandatorily opens spectral gaps at locations that depend on the decoration only. This is how it works: equation of the type ∆u = λu on the decorated graph after replacing the decoration by its Dirichlet-to-Neumann map, reduces to ∆u + Λ(λ)u = λu. The energy dependent “potential” Λ(λ) has poles at certain locations, which makes the equality impossible near these poles, and thus no spectrum arises there. UCLA antenna ground plane. • Gap opening by using appropriate edge potentials (Pankrashkin). ♣ Spectra of periodic graphs. Periodic graph Γ: free co-compact action of Zn: Edge lengths and operators periodic w.r.t. the action. Floquet-Bloch theory: let k ∈ Rn - quasimomentum. Restrict the operator to functions u such that u(x + p) = eik·pu(x), x ∈ Γ, p ∈ Zn. Get the (Bloch Hamiltonian) operator H(k). Theorem 5. (Floquet-Bloch theorem): i) H(k) has discrete spectrum λm(k) that tends to ∞. ii) σ(H) = ∪ σ(H(k)). k E.g., dispersion curves for graphene and carbon nano-tubes can be easily found explicitly (Korotyaev&Lobanov, P.K.&Post). Theorem 6. (L. Thomas) For a periodic elliptic operator (including periodic quantum graph Hamiltonians), σsc(H) = ∅ Theorem 7. (L. Thomas, Birman&Suslina, A. Sobolev, L. Friedlander, P.K.& Levendorskii, ...) For periodic elliptic operators of 2nd order, σpp(H) = ∅, i.e. σ(H) = σac(H). This statement does not hold for quantum graphs: Theorem 8. (P.K.) If λ ∈ σpp(H), then the (∞-dimensional) eigenspace is generated by compactly supported functions. Amenable case due to Veselic. Embedded eigenvalues. Localized perturbation of a periodic quantum graph by changing topology, vertex conditions, or potential in a compact domain can only add isolated eigenvalues either in gaps, or inside the spectral bands (embedded eigenvalues). (Weyl’s theorem) Theorem 9. (1D Rofe-Beketov, 2D, 3D P.K.&Vainberg) There are no embedded eigenvalues due to localized perturbations of a periodic Schrödinger operator. This is not true for periodic quantum graphs. Embedded eigenvalues can be constructed. Theorem 10. (P.K.&Vainberg) Eigenfunctions corresponding to the embedded eigenvalues of a locally perturbed quantum graph are compactly supported near the support of the perturbation. ♣ Slowing down wave packets ψ(x, ω) - Bloch solution with quasimomentum k(ω) of Hψ = ω 2ψ (x ∈ Γ, ω ∈ R). v = ψe−iωt solves non-stationary wave equation vtt = −Hv and “propagates with the velocity” ω/k(ω). Wave packet u= Z |ω−ω0 |< ψ(x, ω)α(ω − ω0)e−iωtdω. Group velocity Vg = (k0(ω0))−1. It can be shown that −i(k(ω0 )x−ω0 t) e u ∼ f (x)α(x/V g − t)e So, flat pieces of dispersion curves lead to low group velocities. Possible/suggested ways to slow down packets: • Near spectral edges. Does not work well. • Near inflection points of the dispersion (Figotin&Vitebsky). Uses layered anisotropic media. • Necklace structures (using poles of the Dirichlet-to-Neumann map). (Molchanov&Vainberg) • CROWS = coupled resonator optical waveguides (uses slow tunnelling between resonators). (Yariv et al, P.K.&Kunyansky) • SCISSORS = side-coupled integrated spaced sequence of resonators(Heebner&Boyd) 14. Inverse problems: can one hear the shape of a quantum graph? One wanders whether the spectrum σ(H) contains sufficient information for recovery of the topology of graph, vertex conditions, and potentials along the edges. Without extra assumptions on the graph, none of these is impossible (von Below, Kurasov&Stenberg). 2 d and Neumann conditions. Consider the case of H = − dx 2 It is clear that vertices of degree 2 need to be excluded (a graph without such vertices is sometimes called clean). Theorem 11. (Gutkin&Smilansky) If a clean compact graph has no loops and multiple edges, and if the lengths of its edges are rationally independent, then the topology of the graph is uniquely determined by its spectrum. Later extension to clean graphs with incommensurate lengths of the edges (Kurasov&Nowaczyk). The proofs are based on trace formulas due to Smilansky et al that allow one to recover the lengths of periodic orbits in Γ. See also work by Pivovarchik and by Yurko. 15. Generalizations: “leaky graphs”, higher dimensional “quantum buildings”, multi-structures. There are various generalizations of quantum graphs: ♣ Pseudo-differential Hamiltonians, photonic crystals, and leaky wires. Study of high contrast photonic crystals leads to −∆u = λδΓu + cu (Figotin & P.K.) Parameter c - coupling of two dielectrics. When c = 0 (no coupling), −∆u = λδΓu - spectral problem for D-to-N operator on Γ. Same spectral problem describes “leaky wires”, while there c is the spectral parameter and λ is a coupling constant. ♣ Higher order Hamiltonians. The pseudo-differential problem was studied by P.K. & Kunyansky. Existence of good differential approximations was shown for some geometries (the square of D-to-N operator is “almost” Laplacian). This lead to energy dependent vertex conditions. In some cases, e.g. for Γ = Z2, the resulting differential operator is of fourth order. These results were heuristic and numerical, and there is no rigorous treatment of this problem. An analytic approach was suggested by Ola and Paivarinta. ♣ Quantum buildings and multistructures Some popular photonic crystal structures look like thin surface (rather than graph) structures in 3D. Such is, for instance, the inverse opal structure (what is left after close packing by spheres): (The picture is from work of Ames group headed by Prof. K. Soukoulis.) Thus (and for some other reasons) it is worthwhile to attempt a generalizations of quantum graph theory to higher dimension CW complexes (“quantum buildings”). In various applications there is an interest towards equations on “multistructures,” i.e. complexes combining cells of different dimensions. See, e.g. books by Mehmeti et al and by V. A. Kozlov, V. G. Mazýa, and A. B. Movchan. 16. The outer space We mention here a moduli space of metric graphs, which might be useful in quantum graph studies. It has been studied intensively as a tool for investigating automorphism groups of free groups. The fundamental group of a connected graph is a free group. Indeed, contracting a maximal tree T in Γ into a point produces a rose Rn of n petals and π1(Rn) is isomorphic to the free group Fn with n generators. Consider the space of all connected graphs Γ whose fundamental group is Fn and whose total length is normalized to be equal to 1. We call a homotopy equivalence mapping g : Rn 7→ Γ a marking. A pair (g, Γ) is a marked metric graph. Two such marked graphs (g1, Γ1), (g2, Γ2) are said to be 7 Γ2 such equivalent, if there exists an isometry i : Γ1 → that the diagram is commutative. The set Xn of these equivalence classes is the Outer space. If C is the set of conjugacy classes in Fn, one can naturally embed Xn into the infinite dimensional projective space RP C . This is done by coordinate functions: let c ∈ C, w - a representative of c, and (g, Γ) - marked graph. One can construct the number l(g,Γ)(c) = the length of the shortest loop in Γ in the free homotopy class of g∗(w). It has been shown that Xn consists of simplices, is contractible, and has dimension 3n − 4. Shrinking one edge to a point moves a graph to the boundary of a simplex, where several simplices meet. 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