Generalized Symmetries of Partial Differential Equations and Quasiexact Solvability

TL;DR

This paper explores the relationship between generalized symmetries of PDEs and quasiexact solvability, providing explicit methods to analyze symmetry coefficients and their dependence on variables.

Contribution

It introduces a novel approach linking symmetry analysis with quasiexactly solvable models, expanding the theoretical framework for PDE solvability.

Findings

Explicit determination of symmetry coefficient dependence on variables

Connection established between symmetry structures and quasiexact solvability

Generalizations of the symmetry analysis approach are discussed

Abstract

Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant variables) on specific finite-dimensional subspaces of the space of generalized symmetries of some system of partial differential equations, we explicitly determine the dependance of coefficients of generalized symmetries from these subspaces on the above-mentioned variables. We establish the connection of our results with the theory of quasiexactly solvable models. Some generalizations of the approach proposed also are discussed.