Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time evolution and the Schrödinger equation on time dependent quantum graphs

Uzy Smilansky, Gilad Sofer

Abstract

The purpose of the present paper is to discuss the time dependent Schrödinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schrödinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schrödinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.

Time evolution and the Schrödinger equation on time dependent quantum graphs

Abstract

The purpose of the present paper is to discuss the time dependent Schrödinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schrödinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schrödinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.
Paper Structure (9 sections, 7 theorems, 58 equations, 3 figures)

This paper contains 9 sections, 7 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quantum graphs with time dependent edge lengths and standard boundary conditions
  3. A time dependent interval
  4. Compact metric graphs with time dependent edge lengths
  5. Example -- equilateral graph with linearly growing edges
  6. Generalizations and applications
  7. Generalization of the results to time dependent vertex conditions
  8. Application -- the geometric phase
  9. Secular equation for the instantaneous graph

Key Result

Theorem 2.1

The Schrödinger equation (grapheq) on the stationary metric graph $\Gamma_{0}$ equipped with the magnetic vertex conditions (graphvc, graphvc2) induces a unitary flow on $L^{2}\left(\Gamma_{0}\right)$. This allows to define a solution to the Schrödinger equation (laplacian) on the time dependent gra

Figures (3)

  • Figure 2.1: A time dependent metric graph.
  • Figure 2.2: An equilateral star graph with three edges of varying lengths $\Gamma_{t}$, along with the associated stationary graph $\Gamma_{0}$ parametrized so that the central vertex corresponds to $\xi=1/2$.
  • Figure 3.1: The first four eigenvalues of the Robin interval as a function of $t/T$. As $t/T\rightarrow1$, two of the eigenvalue curves approach $-\infty$.

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6
  • Corollary 3.7