Solutions of Helmholtz and Schr\"odinger Equations with Side Condition and Nonregular Separation of Variables

TL;DR

This paper explores nonregular separation of variables for Helmholtz and Schrödinger equations on Riemannian manifolds, linking symmetry operators and generalized Stäckel matrices to new solution methods.

Contribution

It introduces a framework connecting nonregular separation with symmetry operators and generalized Stäckel matrices for Hamiltonian systems.

Findings

Nonregular separation corresponds to $N-1$ commuting symmetry operators.

Solutions differ from those obtained via regular separation.

The theory is supported by explicit examples.

Abstract

Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems, namely Hamilton-Jacobi, Helmholtz and time-independent Schr\"odinger equations with potential on $N$-dimensional Riemannian and pseudo-Riemannian manifolds, but with a linear side condition, where more structure is available. We show that the requirement of $N-1$ commuting second-order symmetry operators, modulo a second-order linear side condition corresponds to nonregular separation of variables in an orthogonal coordinate system, characterized by a generalized St\"ackel matrix. The coordinates and solutions obtainable through true nonregular separation are distinct from those arising through regular separation of variables. We develop the theory for these systems and provide examples.