On the spectral theory in the Fock space with polynomial eigenfunctions

TL;DR

This paper explores the spectral theory of polynomial eigenfunctions in Fock space, classifying eigenvalue problems via $rak{sl}(2)$ representations and analyzing specific solvable operators like Hermite, Laguerre, and Heun.

Contribution

It introduces a classification of eigenvalue problems in Fock space using $rak{sl}(2)$ representations and studies (quasi)-exactly solvable operators with concrete examples.

Findings

Eigenvalue problems classified by $rak{sl}(2)$-algebra representations.

Detailed analysis of the number operator and (quasi)-exactly solvable operators.

Examples include Hermite, Laguerre, Heun, Lame, and sextic QES polynomial operators.

Abstract

The notion of the eigenvalue problem in the Fock space with polynomial eigenfunctions is introduced. This problem is classified by using the finite-dimensional representations of the $\mathfrak{sl}(2)$-algebra in Fock space. In the complex representation of the 3-dimensional Heisenberg algebra, proposed by Turbiner-Vasilevski (2021) in Ref.7, this construction is reduced to the linear differential operators in $(\frac{\partial}{\partial \overline{z}}\,,\,\frac{\partial}{\partial z})$ acting on the space of poly-analytic functions in $(z,\overline{z})$. The number operator, equivalently, the Euler-Cartan operator appears as fundamental, it is studied in detail. The notion of (quasi)-exactly solvable operators is introduced. The particular examples of the Hermite and Laguerre operators in Fock space are proposed as well as the Heun, Lame and sextic QES polynomial operators.