Solvable Quantum Mechanical Systems Research Articles

Numerical study of the SWKB condition of novel classes of exactly solvable systems

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

Exactly solvable discrete quantum mechanical systems and multi-indexed orthogonal polynomials of the continuous Hahn and Meixner–Pollaczek types

Abstract We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of coordinates is a whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner–Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials is $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$: a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

Hall viscosity in the non-Abelian quantum Hall matrix model

Quantum Hall matrix models are simple, solvable quantum mechanical systems which capture the physics of certain fractional quantum Hall states. Recently, it was shown that the Hall viscosity can be extracted from the matrix model for Laughlin states. Here we extend this calculation to the matrix models for a class of non-Abelian quantum Hall states. These states, which were previously introduced by Blok and Wen, arise from the conformal blocks of Wess-Zumino-Witten conformal field theory models. We show that the Hall viscosity computed from the matrix model coincides with a result of Read, in which the Hall viscosity is determined in terms of the weights of primary operators of an associated conformal field theory.

Spectral intertwining relations in exactly solvable quantum-mechanical systems

In exactly solvable quantum-mechanical systems, ladder and intertwining operators play a central role because, if they are found, the energy spectra can be obtained algebraically. In this paper, we propose the spectral intertwining relation as a unified relation of ladder and intertwining operators in a way that can depend on the energy eigenvalues. It is shown that the spectral intertwining relations can connect eigenfunctions of different energy eigenvalues belonging to two different Hamiltonians, which cannot be obtained by previously known structures such as shape invariance. As an application, we find new spectral intertwining operators for the Hamiltonians of the hydrogen atom and the Rosen--Morse potential.

Recurrence relations of the multi-indexed orthogonal polynomials. IV. Closure relations and creation/annihilation operators

We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. Corresponding to the recurrence relations with constant coefficients for the M-indexed orthogonal polynomials, it is expected that the systems satisfy the generalized closure relations. In fact we can verify this statement for small M examples. The generalized closure relation gives the exact Heisenberg operator solution of a certain operator, from which the creation and annihilation operators of the system are obtained.

2d CFT/Gauge/Bethe correspondence and solvable quantum-mechanical systems

The holomorphic accessory parameters for the Heun (Fuchs with four singularities) and the Lamé equations are obtained by taking the classical limit of some null vector decoupling equations appearing in two-dimensional conformal field theory. In addition, a link between the aforementioned accessory parameters and the solution of some Bethe-like equations which occur in the supersymmetric gauge theories, is established.

Solvable discrete quantum mechanics: q-orthogonal polynomials with q=1 and quantum dilogarithm

Several kinds of q-orthogonal polynomials with q=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions, which are a natural extension of the Euler gamma functions and the q-gamma functions (q-shifted factorials). The dimensions of the orthogonal spaces are finite. These q-orthogonal polynomials are expressed in terms of the Askey-Wilson polynomials and their certain limit forms.

The theory of contractions of 2D 2nd order quantum superintegrable systems and its relation to the Askey scheme for hypergeometric orthogonal polynomials

We describe a contraction theory for 2nd order superintegrable systems, showing that all such systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. Analogously, all of the quadratic symmetry algebras of these systems can be obtained by a sequence of contractions starting from S9. By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.This relates the scheme directly to explicitly solvable quantum mechanical systems. Amazingly, all of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2, C). The present paper concentrates on describing this intimate link between Lie algebra and superintegrable system contractions, with the detailed calculations presented elsewhere. Joint work with E. Kalnins, S. Post, E. Subag and R. Heinonen.

Casoratian identities for the Wilson and Askey–Wilson polynomials

Infinitely many Casoratian identities are derived for the Wilson and Askey–Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical systems. Similar identities hold for various reduced form polynomials of the Wilson and Askey–Wilson polynomials, e.g. the continuous q-Jacobi, continuous (dual) (q-)Hahn, Meixner–Pollaczek, Al-Salam–Chihara, continuous (big) q-Hermite, etc.

Equivalences of the multi-indexed orthogonal polynomials

Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion. Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only (or type II indices only) with shifted parameters.

Confluence of apparent singularities in multi-indexed orthogonal polynomials: the Jacobi case

The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the Pöschl–Teller potential (Odake–Sasaki). By fine-tuning the parameter(s) of the Pöschl–Teller potential, we obtain several families of explicit and global solutions of certain second-order Fuchsian differential equations with an apparent singularity of characteristic exponents −2 and −1. They form orthogonal polynomials over x ∈ ( − 1, 1) with weight functions of the form (1 − x)α(1 + x)β/{(ax + b)4q(x)2}, in which q(x) is a polynomial in x.

Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite families of the exceptional Laguerre and Jacobi polynomials of types I and II constructed by the present authors. The totality of the integer indices of the new polynomials are finite and they correspond to the degrees of the ‘virtual state wavefunctions’ which are ‘deleted’ by the generalisation of Crum–Adler theorem. Each polynomial has another integer n which counts the nodes.

Asymptotics of orthogonal polynomial’s entropy

This is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree. This study is motivated primarily by quantum mechanics, where the wave functions and the densities of the states of solvable quantum-mechanical systems are expressed by means of orthogonal polynomials. Moreover, the uncertainty principle, lying in the ground of quantum mechanics, is best formulated by means of position and momentum entropies. In this sense, the behavior for large values of the degree is intimately connected with the information characteristics of high energy states. But the entropy functionals and their behavior have an independent interest for the theory of orthogonal polynomials. We describe some results obtained in the last 15 years, as well as sketch the ideas behind their proofs.

Exact Heisenberg operator solutions for multiparticle quantum mechanics

Exact Heisenberg operator solutions for independent “sinusoidal coordinates” as many as the degree of freedom are derived for typical exactly solvable multiparticle quantum mechanical systems, the Calogero systems [J. Math. Phys. 12, 419 (1971)] based on any root system. These Heisenberg operator solutions also present the explicit forms of the annihilation-creation operators for various quanta in the interacting multiparticle systems. At the same time they can be interpreted as multivariable generalization of the three term recurrence relations for multivariable orthogonal polynomials constituting the eigenfunctions.

Exact solution in the Heisenberg picture and annihilation–creation operators

The annihilation–creation operators of the harmonic oscillator, the basic and most important tools in quantum physics, are generalised to most solvable quantum mechanical systems of single degree of freedom including the so-called ‘discrete’ quantum mechanics. They admit exact Heisenberg operator solution. We present unified definition of the annihilation–creation operators ( a ( ± ) ) as the positive/negative frequency parts of the exact Heisenberg operator solution.

Dynamical Processes in Exactly Solvable Quantum Mechanical Systems: III

Dynamical Processes in Exactly Solvable Quantum Mechanical Systems: III

Dynamical Processes in Exactly Solvable Quantum Mechanical Systems. II

Dynamics and decoherence are studied in a theoretical model in which an incident particle interacts with an array of harmonic oscillators. This system is analogous to the generalized Coleman-Hepp model treated in the previous paper and can also be formulated systematically and solved exactly. In spite of the similarities there exist certain differences between the two. We give detailed numerical calculations and discussions on the decoherence processes.

Dynamical Processes in Exactly Solvable Quantum Mechanical Systems I.

Dynamics and decoherence are studied in a theoretical model in which an incident particle interacts with an array of harmonic oscillators. This system is analogous to the generalized Coleman-Hepp model treated in the previous paper and can also be formulated systematically and solved exactly. In spite of the similarities there exist certain differences between the two. We give detailed numerical calculations and discussions on the decoherence processes.

Exactly solvable quantum-mechanical systems on lattices associated with classical Lie algebras

In this paper we find the eigenvalues of the generating function of higher Hamiltonian for the special class of exactly solvable quantum systems. These systems describe the lattice of interacting nodes with interaction between closest neighbors. The Hamiltonian of such a system is invariant with respect to the action of a classical Lie group. In the conclusion we discuss some properties of the systems considered.