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 $$\mathfrak {sl}(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>sl</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -algebra in Fock space. In the complex representation of the 3-dimensional Heisenberg algebra, proposed in [7], this construction is reduced to the linear differential operators in $$(\frac{\partial }{\partial \overline{z}},\,\frac{\partial }{\partial z})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mi>∂</mml:mi> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mfrac> <mml:mi>∂</mml:mi> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>z</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> acting on the space of poly-analytic functions in $$(z,\overline{z})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>z</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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.