On the spectral theory in the Fock space with polynomial eigenfunctions

2025/08/08 by Turbiner, A. V., Vasilevski, N. L.
#Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2508.05924

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 \mathfraksl(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∂∂ z , (∂)/(∂ z)) acting on the space of poly-analytic functions in (z,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.

Citations

• Coherent state transforms attached to generalized Bargmann spaces on the complex plane
• Lie Algebras in Fock Space
• Differential calculus on compact matrix pseudogroups (quantum groups)

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• Complex symmetric differential operators on Fock space
• The ∂-complex on the Fock space
• The number of eigenvalues for an Hamiltonian in Fock space
• Different faces of harmonic oscillator
• q-Fock Space of q-Analytic Functions and its realization in L²(ℂ; e^{-z z} dx dy)