On the spectral theory in the Fock space with polynomial eigenfunctions

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.