[GS_C_QI] On exact and quasi-exact solvability in quantum mechanics
ABSTRACT
In context of quantum mechanics, solvable systems were used in different contexts to obtain insights into different phenomena. From a mathematical perspective solvable systems may be seen as rare, and a much larger class of models consist of quasi-exactly solvable systems. I will introduce different definitions of those quantum solvable and quasi-exactly solvable systems. The quasi-exactly solvable systems are characterized by the fact that only a finite number of states can be obtained using algebraic or analytical approaches. There are different methods which build on the Heun equation and its generalizations, Bethe Ansatz methods and underlying algebraic structures. We will start the talk by presenting a review of the sl(2) hidden symmetry approach and the analytical setting consisting of using Bethe equations. We will discuss different models including the generalized quantum isotonic oscillator, non-polynomially deformed oscillator, and the Schrödinger system from the kink stability analysis of field theory. We will present a unified treatment of their states and sl(2) algebraization.
In second part of the talk, we will discuss how well-known models in condensed matter have underlying hidden algebra which is beyond Lie algebras. We introduce novel polynomial deformations of Lie algebras. We construct their finite-dimensional irreducible representations and the corresponding differential operator realizations. We apply our results to a class of spin models with hidden polynomial algebra symmetry and obtain the closed-form expressions for their energies and wave functions by means of the Bethe ansatz method. The general framework enables us to give a unified algebraic and analytic treatment for three interesting spin models with hidden cubic algebra symmetry: the Lipkin-Meshkov-Glick (LMG) model, the molecular asymmetric rigid rotor, and the two-axis countertwisting squeezing model. We provide analytic and numerical insights into the structures of the roots of the Bethe ansatz equations (i.e. the so-called Bethe roots) of these models. We give descriptions of the roots on the spheres using the inverse stereographic projection. The changes in nature and pattern of the Bethe roots on the spheres indicate the existence of different phases of the models. We also present the fidelity and derivatives of the ground-state energies (with respect to model parameters) of the models. The results indicate the presence of critical points and phase transitions of the models.
Finally, we will come back on the notion of exactly solvable systems and their related orthogonal polynomials. We will point out how large amount of exactly solvable systems do not have wave functions written in terms of well-known classical orthogonal polynomial among the Askey-Wilson Scheme such as Hermite, Laguerre and Jacobi polynomials. We will point they are connected in general to exceptional orthogonal polynomials which admit gaps in the sequence of polynomials while still forming a complete set of orthogonal polynomials. This point how wide range of quantum models are beyond the usual solvable systems exist and have states which can be calculated explicitly and be used in applications.
