[Paper Review] Quasi-exact solvability and intertwining relations

Sergey M. Klishevich|ArXiv.org|Oct 6, 2004

Quantum Mechanics and Non-Hermitian Physics6 references3 citations

TL;DR

This paper establishes intertwining relations as a universal method for constructing one-dimensional quasi-exactly solvable (QES) operators, generalizing the approach to multidimensional systems. It demonstrates that QES operators with two invariant monomial subspaces are linked to second-order nonlinear parasupersymmetry, constructs new QES Hamiltonians with non-monomial invariant subspaces via deformation, and proves that higher-order nonlinear supersymmetry (beyond cubic) generally does not yield quasi-exact solvability.

ABSTRACT

We emphasize intertwining relations as a universal tool in constructing one-dimensional quasi-exactly solvable operators and offer their possible generalization to the multidimensional case. Considered examples include all quasi-exactly solvable operators with invariant subspaces of monomials. We show that the simplest case of generalized intertwining relations allows to naturally relate quasi-exactly solvable operators with two invariant monomial subspaces to a nonlinear parasupersymmetry of second order. Quantum-mechanical systems with linear and nonlinear supersymmetry are discussed from the viewpoint of quasi-exact solvability. We construct such a general system with a cubic supersymmetry and argue that quantum-mechanical systems with nonlinear supersymmetry of fourth order and higher are generally not quasi-exactly solvable. Besides, we construct two examples of quasi-exactly solvable operators with invariant subspaces which cannot be reduced to monomial spaces.

Motivation & Objective

• To establish intertwining relations as a unifying framework for constructing one-dimensional quasi-exactly solvable (QES) operators.
• To generalize the intertwining approach to multidimensional quantum systems.
• To explore the connection between QES systems with two invariant monomial subspaces and nonlinear parasupersymmetry of second order.
• To construct new QES Hamiltonians with invariant subspaces that are not reducible to monomial bases via variable transformations.
• To investigate the quasi-exact solvability of quantum systems with linear and nonlinear supersymmetry, particularly of cubic and higher orders.

Proposed method

• Utilizes intertwining relations $ AH_0 = H_1A $ and $ A^ op H_1 = H_0 A^ op $ to relate Hamiltonians $ H_0 $ and $ H_1 $, with $ A $ acting as a differential operator.
• Employs the kernel of $ A $, $ \ker A $, as a finite-dimensional invariant subspace for $ H_0 $, ensuring quasi-exact solvability.
• Applies the method to $ \mathfrak{sl}(2,\mathbb{R}) $-based Hamiltonians and constructs QES operators via annihilators of functional subspaces.
• Introduces one-parametric deformations of monomial subspaces, such as $ \mathcal{F} = \text{span}\{1, z, \dots, z^{n-3}, z^{n-2} + \frac{\alpha}{n-1}z^{n-1}, z^n\} $, to generate non-monomial invariant subspaces.
• Derives explicit forms of QES Hamiltonians through coefficient functions $ P(z), Q(z), R(z) $ satisfying constraints derived from the intertwining structure.
• Analyzes the algebraic structure of annihilator algebras, identifying connections to Virasoro and its quadratic deformation for linear and quadratic supersymmetry cases.

Experimental results

Research questions

• RQ1 Can intertwining relations serve as a universal construction method for one-dimensional quasi-exactly solvable operators?
• RQ2 How can the intertwining approach be generalized to multidimensional quantum systems?
• RQ3 What is the algebraic relationship between QES operators with two invariant monomial subspaces and nonlinear parasupersymmetry of second order?
• RQ4 Are there quasi-exactly solvable Hamiltonians with invariant subspaces that are not equivalent to monomial subspaces under gauge or variable transformations?
• RQ5 Under what conditions is a quantum system with nonlinear supersymmetry of order higher than three quasi-exactly solvable?

Key findings

• The kernel of the intertwining operator $ A $ provides a finite-dimensional invariant subspace for $ H_0 $, enabling algebraic solution of part of the spectral problem.
• A 6-parametric quasi-exact solvable Hamiltonian was constructed from a one-parameter deformed monomial subspace $ \mathcal{F} = \text{span}\{1, z, \dots, z^{n-2} + \frac{\alpha}{n-1}z^{n-1}, z^n\} $, with constraints on parameters $ p_4=0 $, $ q_0 = \frac{n+2}{2}p_1 - \alpha p_0 $, etc.
• A 7-parametric QES Hamiltonian was derived for the non-monomial subspace $ \mathcal{F} = \text{span}\{1, z, z^3, z^4 + 6\alpha z^2\} $, with explicit expressions for $ P(z), Q(z), R(z) $.
• The method successfully generates QES operators with non-monomial invariant subspaces that cannot be reduced to monomial form via gauge or variable transformations.
• Quantum systems with cubic supersymmetry admit a general quasi-exactly solvable construction, while those with nonlinear supersymmetry of order four or higher are generally not quasi-exactly solvable.
• The annihilator algebra of the invariant subspace for linear supersymmetry is isomorphic to the Virasoro algebra without central extension, and for quadratic supersymmetry, it is a quadratic deformation of the Virasoro algebra.