Fluctuation statistics for quantum star graphs

Contemporary Mathematics Fluctuation statistics for quantum star graphs J.P. Keating Abstract. Star graphs are examples of quantum graphs in which the spectral and eigenfunction statistics can be determined exactly and explicitly. They show behaviour intermediate between that expected for quantum chaotic systems and that expected for integrable systems. I review here the main results relating to star graphs in the context of our general understanding of the statistical properties of quantum systems in the semiclassical limit. In particular, I focus on recent results concerning quantum ergodicity and scarring in the eigenfunctions.. Introduction.. Quantum Chaos. Within classical mechanics there is a broad spectrum of qualitatively different types of dynamics. At one end there is integrability and at the other chaos. The subject of Quantum Chaos is concerned with the question of how this centrally important fact manifests itself in quantum mechanics, in the semiclassical limit as. For example, how does it influence the distribution of quantum eigenvalues and the morphology of eigenfunctions? As an example, consider quantum billiards in R 2. A billiard is an enclosure with hard walls, so that the classical trajectories are straight line segments with specular reflections (angle of incidence equal to angle of reflection at the boundary. In some billiards the classical trajectories are integrable (e.g. in a rectangle or a circle, in others they are strongly chaotic. The Schrödinger equation for billiards is just the Helmholtz equation (. 2 Ψ = k 2 Ψ with appropriate (e.g. Dirichlet conditions on the boundary. The question then is: how do the eigenfunctions Ψ n and eigenvalues k n reflect the integrable or chaotic nature of the underlying classical dynamics in the limit as k? Given that many of the principal consequences of chaos, such as ergodicity and mixing, are statistical, it is natural to expect its influence on quantum mechanics in the semiclassical limit to be seen most clearly in the statistical properties of the eigenfunctions and eigenvalues. 2 Mathematics Subject Classification. Primary 8Q5, 8U5; Secondary 8Q. Key words and phrases. quantum graphs, quantum chaos. The author is supported by an EPSRC Senior Research Fellowship. c (copyright holder

2 J.P. KEATING Let me start with eigenvalue statistics. There are two very important conjectures relating to energy level correlations on the scale of the mean level spacing in the semiclassical limit: in 977, Berry and Tabor [BT] suggested that in generic integrable systems these should be the same as those arising from a Poisson process (i.e. the same as those of uncorrelated random numbers; and in 984 Bohigas, Giannoni and Schmit [BGS] proposed that in generic chaotic systems they are the same as those relating to the eigenvalues of random matrices in the limit as the matrix size becomes infinite. In the case of chaotic systems, the appropriate ensemble of random matrices depends on the symmetries (e.g. time-reversibility of the classical dynamics. In two-dimensional billiards, Weyl s law gives that (.2 #{n: k n k} ck 2 when k, where the constant c is proportional to the area of the billiard. Let us define (.3 X n = ck 2 n. In this case the Berry-Tabor conjecture asserts that if the classical dynamics of a billiard is integrable (and sufficiently generic then the X n have the same local statistical properties as independent random variables from a Poisson process. This means that (.4 N(T, L := #{n: X n [T, T + L]}, the number of X n s in a randomly shifted interval [T, T + L] of fixed length L, is distributed according to the Poisson law Lk k! e L. By contrast, if the classical dynamics is chaotic (and sufficiently generic then the X n have the same local statistical properties as the eigenvalues of certain ensembles of random matrices in the limit as the matrix-size tends to infinity. Now let me turn to eigenfunction statistics. One of the fundamental results in the field of Quantum Chaos concerns quantum ergodicity. Originally put forward by Schnirelman in 974, the quantum ergodicity theorem asserts, in its simplest form, that in systems in which the classical trajectories are ergodic, quantum eigenfunctions (specifically Ψ n 2 become uniformly distributed (with respect to Liouville measure as one approaches the semiclassical limit through subsequences of eigenstates that have density one with respect to all subsequences [S, CdV, Z]. For example, in billiards in which the classical trajectories are ergodic, the integral of Ψ n 2 over an interior region γ tends, as n through almost all subsequences of eigenstates, to the ratio of the area of γ to that of the whole billiard. This prompts the important question as to whether the eigenfunctions are ergodic with respect to all subsequences that is whether they exhibit quantum unique ergodicity [RS] or whether exceptional subsequences exist. This is a rather delicate issue: it has been proved that the eigenfunctions of the Laplacian on certain arithmetic surfaces of constant negative curvature exhibit quantum unique ergodicity [L], but that the eigenvectors of the quantum cat maps do not [FNdB]. Related to this is the long-standing and important issue of scarring: in some chaotic systems one finds eigenfunctions with an enhanced modulus near to short classical periodic orbits [H]. These eigenfunctions are said to be scarred. Are there subsequences of eigenstates for which this persists in the semiclassical limit? Obviously quantum ergodicity means that such subsequences, if they exist, must be

FLUCTUATION STATISTICS FOR QUANTUM STAR GRAPHS 3 of density zero. And in systems where quantum unique ergodicity can be established there are no such sequences. Prior to the work reviewed here, the quantum cat maps were the only systems for which it had been proved rigorously that scarred sequences do indeed exist [FNdB]. Quantum ergodicity provides information about the semiclassical limit of eigenfunctions on fixed (i.e. -independent scales. What do these eigenfunctions look like on the scale of the de Broglie wavelength (which is of the order of? It was conjectured by Berry in 977 that on this scale they may be modelled statistically by random superpositions of plane waves and so exhibit Gaussian random fluctuations [B]. There is considerable experimental and numerical evidence in support for this, but proving it remains an important open problem..2. Intermediate systems. Recently a number of systems have been studied in which the eigenvalue statistics exhibit behaviour intermediate between Poisson and random-matrix, that is, intermediate between integrable and chaotic [BGS2, BGS3]. Examples include polygonal billiards, so-called barrier billiards, and certain quantum maps (see, for example, [GMO, G]. One class of intermediate systems that have been studied in considerable detail are integrable billiards perturbed by a point scatterer, for example a δ-function scatterer in a rectangular billiard. In the context of quantum chaos (i.e. in relation to the statistical properties of quantum energy levels and wavefunctions such systems were introduced by Šeba, and are known as Šeba billiards [Še]. While the spectral statistics of the Šeba billiards are relatively well understood [BGS3], their eigenfunction statistics remain largely undetermined: do the eigenfunctions exhibit quantum ergodicity, scarring, and/or Gaussian random fluctuations? These questions provide the main motivations for the work reviewed here..3. Quantum graphs. Graphs are some of the simplest systems that exhibit quantum chaotic behaviour [KS]. Here by graphs I mean specifically systems of bonds connected at vertices, with the one-dimensional Laplacian acting on each bond and with matching conditions for the solutions at the vertices. The classical dynamics then corresponds to a Markov random walk on the graph, with transition probabilities determined by a transition matrix M. The matrix elements M ij coincide with the squared moduli of the corresponding elements of the quantum S-matrix, which describes the scattering of waves at the vertices of the graph. Let b denote the number of bonds. It was conjectured by Tanner that, in a sequence of graphs in which b, if the spectral gap of M (the gap between the largest eigenvalue and the rest of the spectrum decreases more slowly than /b then the eigenvalues of the Laplacian will generically have a limiting statistical distribution coinciding with one of the ensembles of random matrices (depending on the presence or not of time-reversal symmetry [T]. The eigenfunctions are also expected in this case to behave like those of quantum chaotic systems. My purpose here is to review some recent results relating to a particular family of graphs in which the spectral gap of M closes at precisely the critical rate /b as b. The graphs in question are known as star graphs. The results I shall describe were obtained in collaboration with, variously, Gregory Berkolaiko, Eugene Bogomolny, Jens Marklof and Brian Winn in [BK, BBK, KMW, BKW, BKW2].

4 J.P. KEATING 2. Star graphs A star graph is a metric graph with b outlying vertices all connected only to one central vertex. Thus there are b + vertices and b bonds (figure. Let L R b denote the vector of bond lengths L j. Figure. A star graph with 5 bonds The corresponding quantum star graph may be defined in the following way. Let H denote the real Hilbert space (2. H := L 2 ([, L ] L 2 ([, L b ] with inner product (2.2 f g := b j= Lj f j (xg j (xdx. Elements of H are denoted f = (f,..., f b. Let F H be the subset of functions f which are twice-differentiable in each component and satisfy the conditions (2.3 (2.4 (2.5 f j ( = f i ( =: f, j, i =,..., b b f j( = λ f j= f j(l j =, j =,..., b. The parameter λ may be varied to give different boundary conditions at the central vertex of the graph. Henceforth I shall concentrate on the case /λ =, the socalled Neumann condition. The Laplace operator on F is defined by (2.6 f := ( d 2 f dx 2,..., d2 f b dx 2 defined on F is self-adjoint. Since the space on which the functions in F are defined is compact, the operator has a discrete spectrum of eigenvalues; that is, the equation (2.7 ψ = k 2 ψ.

FLUCTUATION STATISTICS FOR QUANTUM STAR GRAPHS 5 has non-trivial solutions for k = k, k 2,.... Let ψ (n := (ψ (n i (x b i= denote the eigenfunction corresponding to k = k n. Solving (2.7 with boundary conditions (2.3 (2.5, it may be seen straightforwardly that the component of the n th normalized eigenfunction of the Laplace operator on the i th bond of a star graph is (2.8 ψ (n i (x = A (n i cos k n (x L i where the amplitude is given by ( (2.9 A (n i = and k n is the n th positive solution to (2. Z(k, L := 2.. Spectral statistics. Let /2 2 sec 2 k n L i b j= L j sec 2 k n L j b tan kl j =. j= (2. w n := k n 2π b L j. Then the real numbers w n have mean density, in the sense that j= (2.2 lim W W #{w n [, W ]} =. It was shown in [BK] that, assuming the bond lengths L j are not rationally related, the two-point correlation function R 2 (x, defined by (2.3 lim satisfies (2.4 where (2.5 C M = ( 2 M lim b N N n,m N f(w n w m = R 2 (x exp(2πixτdx = δ(τ + exp( 4τ + k +...+k j+n +...+n j=m f(xr 2 (xdx, (K + j!(n + j! (M + j! j=2 M= j i= 4 j j! C M τ M+j+, ( ni+k i n i (n i +!(k i +! with K = j i= k i, N = j i= n i, and the sum being performed over the 2j variables k i and n i. It was later shown in [BBK] that, again assuming the bond lengths L j are not rationally related, (2.6 R 2 (x = + e πxm(u+2i(u+u2 [ J 2 (2 u u 2 + J 2 (2 u u 2 ] du. D

6 J.P. KEATING Here the domain of integration D includes the first and third quadrants of the u u 2 -plane and (iu r (iu 2 s (r + s 2! (2.7 M(u = u + u 2 2i sign(u. r!s!(r!(s! r,s= Formulae (2.4 and (2.6 were shown in [BBK] to be equivalent: (2.4 may be obtained from (2.6 by Fourier transforming and expanding the integral in (2.6. The original derivations of these two expressions were, however, different from each other: (2.4 was first obtained using the trace formula to express the pair correlation function in terms of pairs of closed cycles on the graph, and then by a combinatorial evaluation of the contributions from these cycles (the diagonal contribution, i.e. that coming from pairs of identical cycles, gives τ + exp( 4τ( τ 4τ 2 ; the difference between this expression and the right hand side of (2.4 thus represents the off-diagonal contribution; (2.6, on the other hand, was derived directly from (2., without recourse to the trace formula. I now come to the main point, which is that the expression (2.6 for the two-point spectral correlation function for the star graphs coincides exactly with the corresponding expression for Šeba billiards [BBK]. The star graphs are thus examples of intermediate systems. Specifically, the Fourier transform of the stargraph two-point correlation function (cf. (2.4 takes the value when τ = and then, as τ increases, dips before rising again to when τ ; by contrast, the Fourier transform of the Poisson two-point correlation function is equal to for all τ, and the Fourier transforms of the random-matrix two-point correlation functions take the value when τ = and then increase to when τ. It is worth remarking again in this context that the spectral gap of the Markov transition matrix for the star graphs closes at precisely the critical value /b as b, so this does not contradict Tanner s criterion outlined in the previous subsection. 2.2. Eigenfunction statistics. My goal now is to describe some recent results relating to the eigenfunction statistics for star graphs. One motivation for doing this is the expectation that these results may extend, appropriately modified, to other intermediate systems. 2.2.. Amplitude distribution. The first result concerns the existence of a limit distribution for the amplitudes A (n i semiclassical limit [KMW]. = A (n i (L; b for a fixed star graph in the Theorem 2.. Let the components of L be linearly independent over Q. Then there exists a probability density Q b (η such that lim N { N # n {,..., N} : b 2 A (n i } (L; b < R = R Q b (ηdη where the density Q b (η is independent of the choice of bond i but depends on L. The second result concerns the limit of the probability density Q b (η for a sequence of star graphs with b [KMW]. Theorem 2.2. For each b let the bond lengths L j, j =,..., b lie in the range [ L, L + L] and be linearly independent over Q. If b L as b then for

FLUCTUATION STATISTICS FOR QUANTUM STAR GRAPHS 7 each R R + R Q b (ηdη R Q(ηdη. as b. The limiting density is given by the function (2.8 Q(η = 2π 3/2 η Im exp ( ξ2 4 Lηm(ξ 2 ( Lηm(ξ erfc 8 2i dξ 2 which is continuous on (,. Here Furthermore, (2.9 Q(η = as η. m(ξ := 2 π e ξ2 /4 + ξ erf(ξ/2. 2 Lπ2 η 3/2 e ξ2 /4 m(ξ dξ + O(η 5/2 Crucially, the expression for the limiting density Q(η is markedly different from that which one would expect on the basis of a random wave model. Statistical fluctuations of the eigenfunctions of quantum star graphs on the smallest scale (a single bond thus do not conform to typical quantum chaotic behaviour. 2.2.2. Quantum ergodicity. In order to probe issues relating to quantum ergodicity we need to determine how the eigenfunctions behave on a fixed portion of the graph as b. We thus need to consider an observable that picks out a positive proportion of the graph. To this end, let us take b = αv, where α, v N, and define B = (B i (x αv i= by (2.2 B i := { for i =,..., v for i = v +,..., αv. B may be thought of as the indicator function of the first v bonds. The classical average of B is approximately /α. With the inner product defined as in (2.2, the main results relating to the quantum expectation values of B can be stated as follows. The first result concerns the existence of a limit distribution for the expectation values in the basis of eigenstates ψ (n for a fixed star graph in the semiclassical limit [BKW, BKW2]. Theorem 2.3. For each v let the components of L be linearly independent over Q. Then there exists a probability density p v (η such that for any continuous function h, (2.2 lim N N N h( ψ (n B ψ (n = n= The density p v (η is supported on the interval [, ]. h(ηp v (ηdη. The second result concerns the limit of the probability density p v (η when v [BKW, BKW2].

8 J.P. KEATING Theorem 2.4. For each v let the bond lengths L j, j =,..., αv lie in the range [ L, L + L] and be linearly independent over Q. If v L as v then there exists a probability distribution function F (R such that for any R (,, (2.22 lim v where (2.23 F (R = 2 πα Re and R P η (ξ = ( iπ exp πη 4 + iξ2 4η τ η (ξ = 2 ( iπ η exp π 4 + iξ2 4η + exp ( 2(α π p v (ηdη = F (R P η (ξ (arg(τ η (ξ i log τ η (ξ dξ, η=/r ( (α iπ + exp π + ξ erf iπ4 iξ2 4 ( e iπ/4 ξ 2 η 4 iξ2 4 + ξ(α erf, ( e iπ/4 ξ The key point to make now is that if star graphs satisfied quantum ergodicity, then F (R would be the step-function {, for R > /α (2.24 F erg (R =, for R < /α. The fact that F (R F erg (R means that in the limit as v the star graphs are not quantum ergodic. This is an initially somewhat surprising conclusion, because the classical dynamics on any given star graph is ergodic. The point is, however, that as v the sum of the bond lengths increases, and so in the sequence of systems we are considering the set of points that is classically accessible increases. It seems likely that the main feature of the star graphs that renders them non-quantum-ergodic in the limit is the fact that the spectral gap in the Markov transition matrix decreases at precisely the critical rate /b. 2.2.3. Scars. I now turn to the question of whether the eigenfunctions on quantum star graphs exhibit scarring in the strong sense of the term; that is, in the sense that sequences of eigenfunctions exist whose limit is supported by (i.e. localized on one or more periodic orbits. The following theorem shows that indeed the eigenfunctions of a given (fixed star graph do exhibit strong scarring with respect to two-bond periodic orbits in the semiclassical limit [BKW, BKW2]. Theorem 2.5. Let the elements of L be linearly independent over Q. Given any two distinct bonds, indexed by i and i 2, of a b-bond star graph, there exists a subsequence (k nr (k n such that for any f = (f i b i= smooth in each component, ( (2.25 lim r ψ(nr f ψ (nr Li Li2 = f i (xdx + f i2 (xdx. L i + L i2 It is worth mentioning the main ideas behind the proof of this theorem. The eigenvalues k n are the solutions of (2.. Each term in the sum in (2. contributes a sequence of poles. These poles correspond to the eigenvalues of the Laplacian on the individual bonds of the graph; that is, if the bonds are disconnected at the central vertex. It is not difficult to see that the poles and zeros of Z(k, L interlace. 2.

FLUCTUATION STATISTICS FOR QUANTUM STAR GRAPHS 9 Taking the elements of L to be linearly independent over Q means that the poles are Poisson-distributed. One can therefore find an infinite sequence of poles which are arbitrarily close to their nearest neighbouring pole and well-separated from the rest, and where the two poles in question are associated with a given pair of bonds. The zeros between these almost-degenerate poles correspond to the eigenvalues of states whose eigenfunctions inherit the properties of the eigenfunctions of the disconnected star graph corresponding to the near-degenerate poles in question and so localize on the two corresponding bonds. To-date, star graphs and the quantum cat maps are the only systems in which strong scarring has be established rigorously. 3. Conclusions The spectral statistics of star graphs fall within the class of intermediate statistics and, in particular, coincide with those of Šeba billiards. Quantum star graph eigenfunctions exhibit fluctuations on the scale of the individual bonds that do not conform to the usual random wave models which apply to quantum chaotic systems; in the limit as the number of bonds tends to infinity they are not quantum ergodic, despite the fact that the classical dynamics on a given star graph is ergodic; the eigenfunctions on a given star graph in which the bond lengths are not rationally related exhibit strong scarring in the semiclassical limit sequences of states exist which localize onto two-bond periodic orbits. These properties extend, conjecturally, to the eigenfunctions of Šeba billiards [KMW, BKW, BKW2]. Acknowledgements The work I have reviewed here was done in collaboration with, variously, Gregory Berkolaiko, Eugene Bogomolny, Jens Marklof and Brian Winn. References [BBK] G. Berkolaiko, E.B. Bogomolny and J.P. Keating, Star graphs and Šeba billiards, J. Phys. A 34 (2, 335 35. [BK] G. Berkolaiko and J.P. Keating, Two-point spectral correlations for star graphs, J. Phys. A 32 (999, 7827 784. [BKW] G. Berkolaiko, J.P. Keating and B. Winn, Intermediate wave-function statistics, Phys. Rev. Lett 9 (23, art. no. 343. [BKW2] G. Berkolaiko, J.P. Keating and B. Winn, No quantum ergodicity for star graphs, Commun. Math. Phys. 25 (24, 259 285. [B] M.V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A (977, 283 29. [BT] M.V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. Roy. Soc. Lond. A 356 (977, 375 394. [BGS2] E.B. Bogomolny, U. Gerland and C. Schmit, Models of intermediate spectral statistics, Phys. Rev. E 59 (999, R35 R38. [BGS3] E.B. Bogomolny, U. Gerland and C. Schmit, Singular statistics, Phys. Rev. E 63 (2, art. no. 3626. [BGS] O. Bohigas, M.-J. Giannoni and C. Schmit, Characterisation of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52 (984, 4. [CdV] Y. Colin De Verdière, Ergodicité et fonctions propres du Laplacien, Commun. Math. Phys 2 (985, 497 52.

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