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arXiv:2504.14644

The $q$-deformed Calogero's Goldfish Systems

Umpon Pon Jairuk, Thanadon Kongkoom, Sikarin Yoo-Kong

Published Apr 20, 2025• 0 citations
arXivHTMLPDF

Abstract

Searching for integrable models is a central theme in theoretical and mathematical physics, as such systems offer valuable insights into the underlying structure and symmetries of complex physical phenomena. In this work, we contribute to this pursuit by proposing a new class of one-dimensional many-body integrable systems, which we refer to as the $q$-deformed Calogero's Goldfish system. Our construction employs $q$-deformation of logarithmic and exponential functions inspired by Tsallis' formalism in non-extensive statistical mechanics. Notably, the model satisfies the double-zero condition on its solutions, underscoring its integrable nature and offering a novel perspective on deformation techniques within exactly solvable systems.

DOI
10.48550/arxiv.2504.14644

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The qqq-deformed Calogero's Goldfish Systems
Searching for integrable models is a central theme in theoretical and mathematical physics, as such systems offer valuable insights into the underlying structure and symmetries of complex physical phenomena. In this work, we contribute to this pursuit by proposing a new class of one-dimensional many-body integrable systems, which we refer to as the qqq-deformed Calogero's Goldfish system. Our construction employs qqq-deformation of logarithmic and exponential functions inspired by Tsallis' formalism in non-extensive statistical mechanics. Notably, the model satisfies the double-zero condition on its solutions, underscoring its integrable nature and offering a novel perspective on deformation techniques within exactly solvable systems.
We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and nonlinear integrable field equations of Korteweg-de Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models, we provide three alternative deformations: a complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real-valued field equations of nonlinear integrable systems; and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of Korteweg-de Vries type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.
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
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.
Abstract Starting on the basis of Fibonacci calculus and Fibonacci oscillator algebra, we introduce the main properties to develop a new formalism for the two-parameter (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed quantum mechanics, where q1{q}_{1}q1​ q 1 and q2{q}_{2}q2​ q 2 are real positive independent deformation parameters. As applications of such a two-parameter deformed formalism, we investigate the behavior of a quantum particle in some different physical phenomena covering the free particle and the inverse-harmonic potential case. The effect of two deformation parameters on the wave functions for these applications is studied. Another application is carried out onto the quantum statistics of lattice oscillations through a model of the (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed phonon gas, and it is shown that the high- and low-temperature behavior of the model specific heat differs notably from the classical theories for the interval 0<(q1,q2)<∞0<({q}_{1},{q}_{2})<\infty 0<(q1​,q2​)<∞ 0 < ( q 1 , q 2 ) < ∞ . We also construct a two-parameter deformed non-extensive entropy based on some elements of the Fibonacci calculus and discuss its possible connection with the Tsallis entropy in non-extensive statistical mechanics. Finally, other possible application areas of the present two-parameter (q1,q2)({q}_{1},{q}_{2})(q1​,q2​) ( q 1 , q 2 ) -deformed construction on quantum mechanics are discussed.
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.
In classical mechanics, an integrable system is a model that admits a complete set of integrals of motion that are in involution. This notion can be extended to include classical integrable field theories. A natural question is how to lift these notions to the quantum setting. Despite the extensive literature on the subject, these models are still not very well understood. As an example, in the first section of this work, we present a new class of integrable quantum field theories which develop an unusual behaviour that may be interpreted as a Hagedorn transition. The principal reason for the enigmatic nature of these models is the absence of a universal mathematical framework to describe them. One potential solution may be provided by quantum affine Gaudin models, which we examine in greater detail in the second part of this work. In particular, we introduce the first non-trivial Hamiltonian of quartic order for the affine sl2 Gaudin model, as well as the next-to-leading order expression for all higher Hamiltonians. Furthermore, we provide new insights into the double-loop version of the Feigin-Frenkel homomorphism, which is expected to be a crucial component in the construction of the Bethe ansatz for these models.