Understanding Quantum Graphs P

Vol. 136 (2019) ACTA PHYSICA POLONICA A No. 5

Proceedings of the 9th Workshop on Quantum Chaos and Localisation Phenomena, May 24–26, 2019, Warsaw, Poland Understanding Quantum Graphs P. Kurasov Dept. of Mathematics, Stockholm Univ., 106 91 Stockholm, Sweden Current understanding of spectral asymptotics for quantum graphs is described. It is discussed how these results can be applied to inverse problems. In particular it is analysed whether the inverse spectral problem has a unique solution in the spirit of classical Ambartsumian theorem: a complete overview is provided. DOI: 10.12693/APhysPolA.136.797 PACS/topics: quantum graphs, spectral asymptotics, isospectrality

1. Introduction are asymptotically close and it appears that they always belong to the same spectral cluster. Quantum graphs — ordinary differential equations on Studies of inverse spectral problems for quantum metric graphs — have attracted a lot of attention in graphs is at its starting point, hence we focus here on recent years [1–3]. These models were used by physi- different generalisations of the celebrated Ambartsum- cists due to their simplicity and rich spectral prop- ian theorem (for the Schrödinger operator on a single erties, in particular as an explicit model for transi- interval), which was a source of inspiration for decades tion from regular to chaotic behaviour (see, e.g., [4–8]). when the inverse spectral theory in one dimension One has even performed experiments using microwave was developed. networks to simulate quantum graphs [9, 10]. Mathe- 2. Preliminaries maticians were interested in spectral properties of quantum graphs [11–18], in particular, understanding struc- 2.1. Definitions ture of nodal domains [5, 19, 20] allowing, for example, A quantum graph is a triple consisting of a metric to characterise trees [21]. Most attractive approaches to graph Γ , a differential expression acting on the edges spectral theory of graph Laplacians are based on scat- and vertex conditions imposed at the vertices. A met- tering ideas [7, 22, 23] and on studies of the determinant ric graph is a finite collection of compact intervals manifold [24, 25]. Using the first approach one may prove En = [x2n−1, x2n], n = 1, 2,...,N, with some of the end trace formula [7, 22, 23, 26–30] connecting the spectrum points identified forming vertices Vm (see Fig. 1). to the set of periodic orbits on the metric graph. In this picture every vertex is an equivalence class of 2N The goal of this note is to describe recent develop- end points from V = {xj}j=1. With Γ we associate ments in the spectral theory of quantum graphs [1, 2] the Hilbert space connected with spectral asymptotics and inverse spectral N M problems. Under the inverse spectral problem we un- L2(Γ ) = L2(En). derstand here reconstruction of a quantum graph from n=1 its spectrum. Our focus will be on results that remind Note that L2(Γ ) does not reflect how the edges are con- Ambartsumian theorem from 1929 which states that the nected. We shall reduce our studies to the Schrödinger spectrum uniquely determines the operator in the case 2 − d + q(x) of Neumann Laplacian on a compact interval [31]. differential expressions dx2 with real summable The Schrödinger operators on metric graphs, known as potential q ∈ L1(Γ ). To make such differential expression quantum graphs, possess properties of ordinary and par- self-adjoint in L2(Γ ) one needs to select vertex condi- tial differential equations, but their spectral asymptotics tions connecting the limit values of the functions um and are unusual. For example, for non-trivial metric graphs their directed derivatives ∂um at each vertex Vm: m m (not a single interval, not a loop), it is impossible to im- i(S − I)um = (S + I)∂um, (1) m prove Weyl’s asymptotics in the sense that no lower term where S is any irreducible unitary dm × dm matrix. can be added to Weyl’s term. Its dimension is determined by the degree dm of the ver- It appears that spectra of quantum graphs can be clas- tex coinciding with the number of end points in the equiv- sified by gathering together isospectral Laplacians and alence class [32]. corresponding Schrödinger operators. The spectra of We denote the corresponding self-adjoint operator S the Laplace and the Schrödinger operators (on the same by Lq (Γ ), where S is the collection of all unitary metric graph and with the same vertex conditions) matrices Sm. In the special case of standard vertex conditions   u is continuous at Vm, P ∂u(xj) = 0 (Kirchhoff’s condition), (2) corresponding author; e-mail: [email protected]  xj ∈Vm

(797) 798 P. Kurasov

3. Inverse problems and exceptional parameters

To solve the inverse problem one has to reconstruct all three members of the quantum graph triple • the metric graph Γ ;

• the real potential q(x) ∈ L1(Γ ); • the vertex conditions, i.e., the matrices Sm. Even in the case of single interval one spectrum in general is not enough to reconstruct the potential (with the ex- ception of the Ambartsumian theorem). Hence we shall need to consider exceptional parameters in the triple: Fig. 1. Connected metric graph. • Single interval [0, `] as a metric graph — this graph st has the smallest Laplacian spectral gap among all the operator will be denoted by Lq (Γ ). The correspond- st graphs of the same total length. ing Laplacian L0 (Γ ) is uniquely determined by the metric graph Γ . • Zero potential q(x) ≡ 0 — the only potential S It is well-known that the spectrum of Lq (Γ ) is always that can be determined a priori without knowing discrete satisfying Weyl’s asymptotics the metric graph. 2 2 π 2 λn = k ∼ n , (3) n L • Standard vertex conditions — these conditions ap- PN pear if one requires just continuity of the functions where L = `n, `n = x2n − x2n−1 is the total length n=1 from the quadratic form domain. of the metric graph Γ . As we already mentioned no further asymptotic expansion of the form In what follows we shall try to understand whether π 1 it is possible to prove an analog of the Ambartsumian kn = n + c0 + c−1 + ... L n theorem in the case where some of the parameters in is possible unless Γ is an interval or a loop. the triple determining quantum graph are assumed to coincide with the exceptional ones and the other param- 2.2. Two examples eters are allowed to vary.

2.2.1. Equilateral graphs 4. Two parameters are fixed Consider the standard Laplacian on any equilateral 4.1. Single interval with Neumann conditions metric graph, then the spectrum is periodic [29] (in k), — Ambartsumian theorem i.e., there exists n1 ∈ N, such that π n The following theorem was proven in 1929 by then k = n + k n n . n 1 { n } 1 ∗ L n1 1 21 year-old Victor Ambartsumian Theorem 1. | π {z } Let the spectrum of the one-dimensional 2 = L n+O(1) d Schrödinger operator − dx2 + q(x), q(x) ∈ R on the in- The spectrum is uniformly discrete but multiple eigen- terval [0, `] with Neumann conditions at the end points values may occur. coincide with the spectrum of the Neumann Laplacian 2.2.2. Lasso graph on the same interval, then the potential is equal to zero The spectrum of the standard Laplacian on the lasso (almost everywhere), or symbolically for n = 1, 2,... st st graph formed by two edges of rationally independent λn(Lq ([0, `])) = λn(L0 ([0, `])) ⇔ q(x) ≡ 0, lengths 2`1 and `2 is given by two series of eigenvalues π 2 π st 2 • n, n = 1, 2,... λn(Lq ([0, `])) = (n − 1) ). (4) `1 ` • solutions to the equation Proof. Modern proof of this theorem [33] is based on 3 sin k(`1 + `2) + sin k(`1 − `2) = 0. first deriving the spectral asymptotics using transforma- tion operator The spectrum is discrete but not uniformly discrete since arbitrarily close eigenvalues occur. We would like to understand how two such different types of spectral ∗It surprised me that motivation for these studies is pure quan- behaviour can be unified. In physics language we may tum mechanical despite it was just 1929. Try to read this old paper speak about regular and chaotic systems. — it is worth doing! Understanding Quantum Graphs 799

R ` 2 ! st π 0 q(x)dx 1 ` 1 Proof. The secular equation for the Robin Laplacian kn(Lq ([0, `])) = n+ +o(1/n) , ` ` 2 π n h0h1 k − sin (k`) = (h + h ) cos (k`) k 0 1 which implies that ` depends on the sum h0 + h1 and product h0h1 of the pa- Z rameters, hence the parameters are determined up to q(x)dx = 0, the exchange of the end points. 0 st π These calculations show that the parameters are de- since kn(Lq ([0, `])) = ` n by the assumption of the theorem. Then plugging in u(x) ≡ 1 as a trial function into termined by the spectrum in this special case, not only the quadratic form if the boundary conditions are Neumann. ` ` Z Z |ψ0(x)|dx + q(x)|ψ(x)|2 dx = 0, 5. One parameter is ﬁxed

0 0 5.1. Standard Schrödinger operators we see that it is a minimiser and hence is the ground state In this section we assume only that the vertex con- 00 −ψ (x) + q(x)ψ(x) = 0 ⇒ q(x) ≡ 0. ditions are standard, but the graph and potential are arbitrary. Theorem 4. 4.2. Standard Laplacian on arbitrary metric graph If the spectrum of the standard Γ — geometric version of Ambartsumian theorem Schrödinger operator on a metric graph coincides with the spectrum of the Neumann Laplacian on an interval, Theorem 2. Assume that the spectrum of the stan- then the graph (essentially) coincides with the interval dard Laplacian on a metric graph Γ coincides with and the potential is zero, or symbolically ( the spectrum of the Neumann Laplacian on the inter- st st Γ = [0, `], λn(L (Γ )) = λn(L ([0, `]) ⇔ (7) val, then Γ (essentially) coincides with the interval, q 0 q(x) ≡ 0. or symbolically st st λn(L0 (Γ )) = λn(L0 ([0, `])) ⇔ Γ = [0, `]. (5) Note that this theorem is not a simple combination of Proof. From Weyl’s law we see that the metric graph the classical Ambartsumian Theorem 1 and its geometric and the interval should have the same total length. version (Theorem 2). Moreover, the spectral gaps for the two Laplacians co- Proof. The theorem can be proven in four steps incide. Nicaise [34], Friedlander [18], and Naboko [35] (see [36]):

(see also [15]) proved by diﬀerent methods that inter- st val has the smallest spectral gap among all graphs of 1. The spectra of the Schrödinger Lq (Γ ) and Laplace st the same length. Hence, Γ is the interval since it has L0 (Γ ) operators on the same graph with the same the smallest possible spectral gap. One should only be vertex conditions are asymptotically close: careful and remove all dummy vertices of degree two, st st kn(L (Γ )) − kn(L (Γ )) = o(1). (8) since standard conditions at such vertices are equivalent q 0 to continuity of the function and its ﬁrst derivative. Here it is important to take into account that the eigenvalues satisfy Weyl’s asymptotic. The proof is 4.3. Laplacian on the interval elementary if q ∈ L∞(Γ ), but is extended for q ∈ L1(Γ ) in [36]. We add this case for the sake of completeness. Let h L0 ([0, `]) denote the Laplacian on the interval [0, `] 2. The spectrum of the Laplacian is given by with the Robin conditions at the end points a trigonometric polynomial and therefore is close ( to integers, provided ` = π, if and only if it coin- 2 ∂u(0) = h0u(0), h = (h0, h1) ∈ R ⇒ cides with the integers. It follows that the spectrum ∂u(`) = h1u(`). of the Laplacian on Γ coincides with the spectrum of the Neumann Laplacian on the interval. Theorem 3. Two Laplacians on an interval [0, `] with the Robin conditions at the end points are isospec- 3. Geometric version of the Ambartsumian theorem tral if and only if the Robin parameters are equal up to (Theorem 2) implies that the graph Γ is essentially permutation the interval. h λn(L0 ([0, `])) = 4. Classical Ambartsumian Theorem 1 implies that " 0 0 0 h0 = h and h1 = h , q(x) ≡ 0. λ (Lh ([0, `)) ⇔ 0 1 n 0 0 0 (6) h0 = h1 and h1 = h0. 800 P. Kurasov

5.2. Schrödinger operators with Robin conditions on the interval 6. Further extensions In this section the graph is fixed to coincide with Counterexamples presented in the last two subsec- the interval. We should analyse whether the spectrum of tions imply that no general Ambartsumian-type theo- the Schrödinger operator with the Robin conditions de- rem, where all three parameters may vary is possible. termine the operator uniquely. It is well-known that to Therefore, we discuss here two more extensions close to reconstruct potential one needs to know two spectra cor- the Ambartsumian theorem in their spirit. responding to different boundary conditions [33, 37–39]. We present here an explicit counterexample using 6.1. A theorem by Brian Davies Crum’s article [40], where Darboux transform was ap- Theorem [41] Let arbitrary finite compact metric plied to add eigenvalues to the Schrödinger operator graph Γ be fixed, then the standard Schrödinger opera- on the interval. 2 tor is isospectral to the standard Laplacian if and only if λ = πn Start with the Dirichlet Laplacian: n ` , the potential is zero, or symbolically n = 1, 2,.... Its spectrum differs from the Neumann λ (Lst(Γ )) = λ (Lst(Γ )) ⇒ q(x) ≡ 0. Laplacian by just one eigenvalue λ = 0. We add this n q n 0 (10) eigenvalue by Crum’s method. If we scale the interval Proof. To prove the theorem it is enough to show that Z to [0, 1], then we get the potential and Robin parameters q(x)dx = 0, (11) equal to Γ −1 1 then the proof of classical Ambartsumian Theorem 1 q(x) = , h = −1, h = . x + 1 0 1 2 can just be repeated step by step, since we know that λ (Lst(Γ )) = λ (Lst(Γ )) = 0. Moreover, the corresponding eigenfunctions are explicit the ground state is zero 1 q 1 0 H Lst(Γ ) 1 1 sin (nx) Let Γ be the heat kernel for 0 : √ ψ1(x) = , ψn+1 = − 2 2 n cos (nx) − . 1 M x + 1 π n x + 1 lim tHΓ (t, x, x) = √ , x ∈ Γ \ (∪m=1Vm). t→0 4π (9) We constructed a family of Robin Schrödinger op- Perturbation formula for traces of the heat semigroups erators isospectral to the Neumann Laplacian, hence implies that −Lst(Γ)t −Lst(Γ)t no Ambartsumian-type theorem holds in this case. Tr(e q ) − Tr(e 0 ) = 5.3. Laplace operators on arbitrary graphs Z with arbitrary vertex conditions −t HΓ (t, x, x)q(x)dx + ρ(t), Here we again present an explicit counterexample im- Γ plying that no Ambartsumian-type theorem is possi- where ρ(t) = O(t3/2). The left hand side is identically ble. Let Γ be the graph formed by the two intervals of zero, hence taking the limit t → 0 we get the desired length 1/2 connected at one vertex. We assume standard equality (11). (= Neumann) conditions at the outer vertices. Condi- tions at the central vertex are given as The theorem is proven for q ∈ L∞(Γ ) in [41], but it is u(x ) ∂u(x ) indicated that it should hold also for q ∈ L1(Γ ). i(S − I) 1 = (S + I) 1 u(x2) ∂u(x2) 6.2. A theorem by P.K. and Rune Suhr with the 2 × 2 matrix S unitary and Hermitian Using that the spectra of standard Laplacians are de- S−1 = S∗ = S. scribed by trigonometric polynomials, which are almost periodic holomorphic functions one may prove the follow- Then the following holds: ing theorem [42, 43]. S,st st λn(L0 (Γ )) = λn(L0 (I)). Theorem 6. Assume that the Laplacians on Γ1 and 2 Γ2 are asymptotically isospectral The proof is essentially based on the fact that S = I st st and explicit calculation of the ground state. Every such kn(L0 (Γ1)) − kn(L0 (Γ2)) → 0 (12) 2 × 2 matrix possesses the representation then the operators are isospectral. √ a 1 − a2 e i θ It is important to bear in mind that there exists S(a, θ) = √ . 2 − i θ isospectral Laplacian in the case of rationally dependent 1 − a e −a edge lengths [22]. In fact, this theorem can be generalised This family interpolates between the case of single inter- to include arbitrary almost periodic functions, and not val (standard vertex conditions at the central vertex) and only trigonometric polynomials. It implies that zeroes two intervals with the Dirichlet and Neumann conditions of almost periodic functions have certain rigidity — they 0 1 1 0 cannot be perturbed a little. In applications all measure- ↔ . ments lead to eigenvalues known with certain precision. 1 0 0 −1 Using rigidity of the zeroes one may improve the mea- | {z } | {z } a=0 a=1 sured values. Understanding Quantum Graphs 801

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