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The properties of the distribution of Gaussian packets on a spatial network

V. L. Chernyshev, A. A. Tolchennikov

Abstract

The article deals with the description of the statistical behavior of Gaussian packets on a metric graph. Semiclassical asymptotics of solutions of the Cauchy problem for the Schrödinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges ("edge travel times") are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for a finite compact graph packets almost always are distributed evenly. A formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The article also discusses a situation where the times of passage over the edges are not linearly independent over the rationals.

The properties of the distribution of Gaussian packets on a spatial network

Abstract

The article deals with the description of the statistical behavior of Gaussian packets on a metric graph. Semiclassical asymptotics of solutions of the Cauchy problem for the Schrödinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges ("edge travel times") are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for a finite compact graph packets almost always are distributed evenly. A formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The article also discusses a situation where the times of passage over the edges are not linearly independent over the rationals.

Paper Structure

This paper contains 6 sections, 6 theorems, 21 equations.

Table of Contents

  1. Introduction.
  2. Acknowledgment.
  3. Terms and Definitions.
  4. The case of linearly independent over $\mathbb{Q}$ edge travel times.
  5. Main results.
  6. The case of linearly dependent travel times.

Key Result

Theorem 1.1

(See mian) Solution of the Cauchy problem shrodik_n -- nach_dannye for $t \in [0, T]$ ($T$ does not depend on $h$), is given by the following formula where the functions $\varphi_j(t), S_j(t)$ are explicitly expressed in terms of the solutions of two hamiltonian systems.

Theorems & Definitions (10)

  • Theorem 1.1
  • Lemma 1.1
  • Theorem 2.1: About uniformity of distribution
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2: About the leading coefficient of the number of packets
  • Lemma 2.1
  • Remark 2.3
  • Remark 3.1
  • Lemma 3.1