<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathfrak {sl}(2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">sl</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-algebra in Fock space. In the complex representation of the 3-dimensional Heisenberg algebra, proposed in [<CitationRef CitationID="CR7">7</CitationRef>], this construction is reduced to the linear differential operators in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\frac{\partial }{\partial \overline{z}},\,\frac{\partial }{\partial z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mfrac> <mi>∂</mi> <mrow> <mi>∂</mi> <mover> <mi>z</mi> <mo>¯</mo> </mover> </mrow> </mfrac> <mo>,</mo> <mspace width="0.166667em" /> <mfrac> <mi>∂</mi> <mrow> <mi>∂</mi> <mi>z</mi> </mrow> </mfrac> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> acting on the space of poly-analytic functions in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(z,\overline{z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mover> <mi>z</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

On the spectral theory in the Fock space with polynomial eigenfunctions

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2025年9月15日

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Alexander V. Turbiner,
Nikolai L. Vasilevski

摘要

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 $\mathfrak {sl}(2)$ $sl (2)$ -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 $(\frac{\partial }{\partial \overline{z}},\,\frac{\partial }{\partial z})$ $(\frac{\partial}{\partial \bar{z}}, \frac{\partial}{\partial z})$ acting on the space of poly-analytic functions in $(z,\overline{z})$ $(z, \bar{z})$ . 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.