Dynamical Invariants for Variable Quadratic Hamiltonians

TL;DR

This paper explores the properties of dynamical invariants in variable quadratic Hamiltonians, establishing links with the Schrödinger equation and introducing a superposition principle for generalized Ermakov systems.

Contribution

It introduces a comprehensive analysis of linear and quadratic invariants, connecting eigenvalue problems with initial value solutions and deriving a nonlinear superposition principle.

Findings

Eigenfunction expansion for the Schrödinger equation solutions

Relations between invariants and initial value problems

Superposition principle for generalized Ermakov systems

Abstract

We consider linear and quadratic integrals of motion for general variable quadratic Hamiltonians. Fundamental relations between the eigenvalue problem for linear dynamical invariants and solutions of the corresponding Cauchy initial value problems for the time-dependent Schroedinger equation are emphasized. An eigenfunction expansionof the solution of the initial value problem is also found. A nonlinear superposition principle for the generalized Ermakov systems is established as a result of decomposition of the general quadratic invariant in terms of the linear ones.