Leading off-diagonal correction to the form factor of large graphs

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, K(tau), of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution -2tau(2) to the form factor which agrees with random-matrix theory.     

Eigenspaces of symmetric graphs are not typically irreducible

We construct rich families of Schrödinger operators on symmetric graphs, both quantum and combinatorial, whose spectral degeneracies are persistently larger than the maximal dimension of an irreducible representation of the symmetry group.     

Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schr?dinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph??s eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.     

Nodal deficiency,spectral flow,and the Dirichlet-to-Neumann map

Letters in Mathematical Physics - It has been recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the...     

Quantum Ergodicity for Graphs Related to Interval Maps

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakónski et al (J. Phys. A, 34, 9303-9317 (2001)). As observables we take the L ² functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.     

Critical Partitions and Nodal Deficiency of Billiard Eigenfunctions

The paper addresses the nodal count (i.e., the number of nodal domains) for eigenfunctions of Schr?dinger operators with Dirichlet boundary conditions in bounded domains. The classical Sturm theorem states that in dimension one, the nodal and eigenfunction counts coincide: the nth eigenfunction partitions the interval into n nodal domains. The Courant Nodal Theorem claims that in any dimension, the number of nodal domains ?? _n of the nth eigenfunction cannot exceed n. However, it follows from an asymptotically stronger upper bound by Pleijel that in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, in most cases a ??nodal deficiency?? d _n ?=?n??? _n arises. One can say that the nature of the nodal deficiency has not been understood. It was suggested in recent years that, rather than starting with eigenfunctions, one can look at partitions of the domain into ?? sub-domains, asking which partitions can correspond to eigenfunctions, and what would be the corresponding deficiency. To this end one defines an ??energy?? of a partition, for example, the maximum of the ground state energies of the sub-domains. One notices that if a partition does correspond to an eigenfunction, then the ground state energies of all the nodal domains are the same, i.e., it is an equipartition. It was shown in a recent paper by Helffer, Hoffmann-Ostenhof and Terracini that (under some natural conditions) partitions minimizing the energy functional correspond to the ??Courant sharp?? eigenfunctions, i.e. to those with zero nodal deficiency. In this paper it is shown that it is beneficial to restrict the domain of the functional to the equipartition, where it becomes smooth. Then, under some genericity conditions, the nodal partitions correspond exactly to the critical points of the functional. Moreover, the nodal deficiency turns out to be equal to the Morse index at the corresponding critical point. This explains, in particular, why the minimal partitions must be Courant sharp.