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arXiv:2609.12428

Quasi-exactly solvable deformations of quantum systems associated with exceptional orthogonal polynomials

Siyu Li, Ian Marquette, Sarah Post, Yao-Zhong Zhang

math-phquant-phSubmitted Sep 11, 2026• Announced Sep 14, 2026
v1 · Sep 11, 2026v2 · Sep 28, 2026
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Abstract

Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.

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Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.
We give a systematic construction of new quasi-exactly solvable systems via Bethe ansatz and supersymmetric quantum mechanics (SUSYQM). Methods based on the intertwining of supercharges have been extensively used in the literature for exactly solvable systems. We generalize the state-deleting (Krein-Adler) supersymmetric transformations to quasi-exactly exactly solvable (QES) systems building on the Bethe ansatz method and related Bethe roots. This enables us to construct superpartners for a wide class of known QES systems classified previously through a hidden sl(2)sl(2)sl(2) algebra. We present our constructions of factorizations and intertwining relations related to 1st-order SUSYQM and the n=1n=1n=1 state for 10 nonequivalent types, denoted I,...,X. In order to have a unified treatment we rely on their ODE standard form as this is also the appropriate setting to obtain the Bethe ansatz equations which constrain the polynomial solutions. This setting also allows one to deal with systems with nnn states in a unified manner, using analysis based on the Bethe ansatz equations to build the supersymmetric transformations in terms of the Bethe ansatz roots. We derive the Schrödinger potentials for the n=1n=1n=1 superpartners of the 10 QES cases and give closed-form solutions for the spectra and wavefunctions of the corresponding QES SUSYQM systems. Furthermore, we present numerical results for higher excited states up to the n=10n=10n=10 level. The results obtained may have wider applicability as our framework is built on ODEs with polynomial coefficients.
This thesis concerns two types of quantum many-body systems in one dimension exactly solved by special functions: firstly, systems with interactions localised at points and solved by the (coordinate) Bethe ansatz; secondly, systems of Calogero-Sutherland type, as well as certain recently introduced deformations thereof, with eigenfunctions given by natural many-variable generalisations of classical (orthogonal) polynomials. The thesis is divided into two parts. The first provides background and a few complementary results, while the second presents the main results of this thesis in five appended scientific papers. In the first paper we consider two complementary quantum many-body systems with local interactions related to the root systems CN, one with delta-interactions, and the other with certain momentum dependent interactions commonly known as delta-prime interactions. We prove, by construction, that the former is exactly solvable by the Bethe ansatz in the general case of distinguishable particles, and that the latter is similarly solvable only in the case of bosons or fermions. We also establish a simple strong/weak coupling duality between the two models and elaborate on their physical interpretations. In the second paper we consider a well-known four-parameter family of local interactions in one dimension. In particular, we determine all such interactions leading to a quantum many-body system of distinguishable particles exactly solvable by the Bethe ansatz. We find that there are two families of such systems: the first is described by a one-parameter deformation of the delta-interaction model, while the second features a particular one-parameter combination of the delta and the delta-prime interactions. In papers 3-5 we construct and study particular series representations for the eigenfunctions of a family of Calogero-Sutherland models naturally associated with the classical (orthogonal) polynomials. In our construction, the eigenfunctions are given by linear combinations of certain symmetric polynomials generalising the so-called Schur polynomials, with explicit and rather simple coefficients. In paper 5 we also generalise certain of these results to the so-called deformed Calogero-Sutherland operators.
In this body of work, we will examine two classes of Lie algebra deformations and their appli- cation to the exact solutions of various quantum integrable systems. The first class of Lie algebra deformations are polynomial deformations of the A1 Lie algebra. We construct unitary representations of these algebras in terms of bosonic and spin operators as well as obtain their single-variable differential operator realizations. Using these results, we obtain the exact Bethe ansatz solutions of a variety of quantum integrable systems such as the Bose-Hubbard and Tavis-Cummings model. The second class of Lie algebra deformations are the so-called quasi-Gaudin algebra. This algebra was introduced to construct integrable systems which are only quasi-exactly solvable. Using a suitable representation this algebra, we obtain partial solutions for a class of bosonic models which do not preserve U(1) symmetry
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and find an unexpected and remarkable applications in quantum inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter algebras the standard procedure of QISM one obtains new wide classes of quantum models which, being integrable (i.e. having enough number of commuting integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic Bethe ansatz solution for arbitrarily large but limited parts of the spectrum). These quasi-exactly solvable models naturally arise as deformations of known exactly solvable ones. A general theory of such deformations is proposed. The correspondence ``Yangian --- quasi-Yangian'' and ``XXXXXXXXX spin models --- quasi-XXXXXXXXX spin models'' is discussed in detail. We also construct the classical conterparts of quasi-Yang-Baxter algebras and show that they naturally lead to new classes of classical integrable models. We conjecture that these models are quasi-exactly solvable in the sense of classical inverse scattering method, i.e. admit only partial construction of action-angle variables.
It is a well established fact that there are deeper connections between the theory of orthogonal polynomials on the one hand and properties of Schrodinger operators on the other hand. Those operators are assumed to act in conventionally used Hilbert spaces like for example L 2 (R n ). A pro- minent example for these connections are the classical continuous Hermite polynomials which correspond to Schrodinger operators with a quadratic po- tential. In the one dimensional case, the support of these polynomials is the real line. When dealing with a discretization of this support, one meets the next ingredient which enriches the investigation of orthogonal polynomials: It is the aspect of deformation. The idea of deforming polynomials plays a crucial role in the context of special functions. In the case of a q deformation one sees that the deformation itself can be associated with discretizing the support for orthogonal polynomials. This allows to switch from polynomials dened in the continuum to polynomials dened on a geometric progression. Behind these observations seems to lie a more general concept: Deformati- ons can be related to discretizations or quantizations and vice versa. This concept has also turned out to be an important guideline in structures of non-commutative geometry, thus also in areas quite far away from the theory of orthogonal polynomials. Our purpose is to address to generalized Hermite polynomials on the lattice- like support f+q n ; q n jn 2 Zg. These polynomials will turn out to be one- parameter generalizations of the q discrete Hermite polynomials of type II. Our basic starting point is the introduction of dierence operators which transform these polynomials into each other. These operators contain two parameters, the deformation parameter q which is related to the discrete support f+q n ; q n jn 2 Zg and a parameter which couples one-dimensional Dunkl like operators Mq ;M + q to the dierence operators under consideration. The parameter can be regarded as a further deformation parameter. Having derived the q generalized Hermite polynomials in the course of the second chapter, we classify them. It turns out that they are related to the discrete monic q Laguerre polynomials L () n (x; q). We investigate the related dierence operators in more detail and establish a link to the moment problem for the q generalized Hermite polynomials.
We introduce exactly solvable quantum parametric oscillators, which are generalizations of the quantum problems related with the classical orthogonal polynomials of Hermite, Laguerre, and Jacobi type, introduced in the work of Büyükaşık et al. [J. Math. Phys. 50, 072102 (2009)]. Quantization of these models with specific damping, frequency, and external forces is obtained using the Wei-Norman Lie algebraic approach. This determines the evolution operator exactly in terms of two linearly independent homogeneous solutions and a particular solution of the corresponding classical equation of motion. Then, time-evolution of wave functions and coherent states are found explicitly. Probability densities, expectation values, and uncertainty relations are evaluated and their properties are investigated under the influence of the external terms.