Boris Gutkin

Can one hear the shape of a graph

We show that the spectrum of the Schroperator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above.

Can billiard eigenstates be approximated by superpositions of plane waves

The plane wave decomposition method (PWDM) is one of the most popular strategies for numerical solution of the quantum billiard problem. The method is based on the assumption that each eigenstate in a billiard can be approximated by a superposition of plane waves at a given energy. From the classical results on the theory of differential operators this can indeed be justified for billiards in convex domains. In contrast, in the present work we demonstrate that eigenstates of non-convex billiards, in general, cannot be approximated by any solution of the Helmholtz equation regular everywhere in R2 (in particular, by linear combinations of a finite number of plane waves having the same energy). From this we infer that PWDM cannot be applied to billiards in non-convex domains. Furthermore, it follows from our results that unlike the properties of integrable billiards, where each eigenstate can be extended into the billiard exterior as a regular solution of the Helmholtz equation, the eigenstates of non-convex billiards, in general, do not admit such an extension.

Collective vs. single-particle motion in quantum many-body systems: Spreading and its semiclassical interpretation in the perturbative regime

We study the interplay between collective and incoherent single-particle motion in a model of two chains of particles whose interaction comprises a non-integrable part. In the perturbative regime, but for a general form of the interaction, we calculate the spectral density for collective excitations. We obtain the remarkable result that it always has a unique semiclassical interpretation. We show this by a proper renormalization procedure which allows us to map our system to a Caldeira-Leggett–type model in which the bath is part of the system.

Particle-time duality in the kicked Ising spin chain

Previously, we demonstrated that the dynamics of kicked spin chains possess a remarkable duality property. The trace of the unitary evolution operator for

Collective versus single-particle motion in quantum many-body systems from the perspective of an integrable model

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Semiclassical prediction of large spectral fluctuations in interacting kicked spin chains

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Hyperbolic Billiards on Surfaces of Constant Curvature

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Hyperbolic Magnetic Billiards on Surfaces¶of Constant Curvature

is related to one of a non-unitary evolution operator for

Entropic Bounds on Semiclassical Measures for Quantized One-Dimensional Maps

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Spectral properties and dynamical tunneling in constant-width billiards

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