Exactly Solvable Quantum Mechanics

A comprehensive review of exactly solvable quantum mechanics is presented with the emphasis of the recently discovered multi-indexed orthogonal polynomials. The main subjects to be discussed are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modifications), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, coherent states, various deformation schemes (multiple Darboux transformations) and the infinite families of multi-indexed orthogonal polynomials, the exceptional orthogonal polynomials, and deformed exactly solvable scattering problems.Comment: LaTeX 48 pages, 5 figures. arXiv admin note: text overlap with arXiv:1104.047

Exact Heisenberg operator solutions for multi-particle quantum mechanics

Exact Heisenberg operator solutions for independent `sinusoidal coordinates' as many as the degree of freedom are derived for typical exactly solvable multi-particle quantum mechanical systems, the Calogero systems 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 multi-particle systems. At the same time they can be interpreted as multi-variable generalisation of the three term recursion relations for multi-variable orthogonal polynomials constituting the eigenfunctions.Comment: 17 pages, no figure

Extensions of solvable potentials with finitely many discrete eigenstates

We address the problem of rational extensions of six examples of shape-invariant potentials having finitely many discrete eigenstates. The overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions, which are an essential ingredient for multi-indexed and iso-spectral extensions of these potentials. They have exactly the same form as the eigenfunctions but their degrees are much higher than n_{max} so that their energies are lower than the groundstate energy.Comment: 22 pages, 3 figures. Typo corrected, comments and references added. To appear in J.Phys.A. arXiv admin note: text overlap with arXiv:1212.659

Boundary Effects in Integrable Field Theory on a Half Line

Abstarct: Boundary effects caused by the boundary interactions in various integrable field theories on a half line are discussed at the classical as well as the quantum level. Only the so-called ``integrable" boundary interactions are discussed. They are obtained by the requirement that certain combinations of the lower members of the infinite set of conserved quantities should be preserved. Contrary to the naive expectations, some ``integrable" boundary interactions can drastically change the character of the theory. In some cases, for example, the sinh-Gordon theory, the theory becomes ill-defined because of the instability introduced by ``integrable" boundary interactions for a certain range of the parameter.Comment: 12 page

Krein-Adler transformations for shape-invariant potentials and pseudo virtual states

For eleven examples of one-dimensional quantum mechanics with shape-invariant potentials, the Darboux-Crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to Krein-Adler transformations deleting multiple eigenstates with shifted parameters. These are based upon infinitely many polynomial Wronskian identities of classical orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials, which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape-invariant potentials.Comment: 33 pages, 1 figure. Typo corrected, comments and references added. To appear in J.Phys.

Orthogonal Polynomials from Hermitian Matrices II

This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended $\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schr\"odinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the $q$-Meixner ($q$-Charlier) polynomials do not form a complete basis of the $\ell^2$ Hilbert space, based on the fact that the dual $q$-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the $q$-Meixner polynomials is obtained by constructing the duals of the dual $q$-Meixner polynomials which require the two component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big $q$-Jacobi family and their duals and $q$-Meixner ($q$-Charlier) polynomials.Comment: 65 pages. Comments, references and table of contents are added. To appear in J.Math.Phy

Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states

Various examples of exactly solvable `discrete' quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schr\"odinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.Comment: 46 pages, 2 figure

Reflectionless Potentials for Difference Schr\"odinger Equations

As a part of the program `discrete quantum mechanics,' we present general reflectionless potentials for difference Schr\"odinger equations with pure imaginary shifts. By combining contiguous integer wave number reflectionless potentials, we construct the discrete analogues of the $h(h+1)/\cosh^2x$ potential with the integer $h$, which belong to the recently constructed families of solvable dynamics having the $q$-ultraspherical polynomials with $|q|=1$ as the main part of the eigenfunctions. For the general ($h\in\mathbb{R}_{>0}$) scattering theory for these potentials, we need the connection formulas for the basic hypergeometric function ${}_2\phi_1(\genfrac{}{}{0pt}{}{a,b}{c}|q;z)$ with $|q|=1$, which is not known. The connection formulas are expected to contain the quantum dilogarithm functions as the $|q|=1$ counterparts of the $q$-gamma functions. We propose a conjecture of the connection formula of the ${}_2\phi_1$ function with $|q|=1$. Based on the conjecture, we derive the transmission and reflection amplitudes, which have all the desirable properties. They provide a strong support to the conjectured connection formula.Comment: 24 pages. Comments and references added. To appear in J. Phys.

Discrete Quantum Mechanics

A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics. The main subjects to be covered are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modification), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, the unified theory of exact and quasi-exact solvability based on the sinusoidal coordinates, the infinite families of new orthogonal (the exceptional) polynomials. Two new infinite families of orthogonal polynomials, the X_\ell Meixner-Pollaczek and the X_\ell Meixner polynomials are reported.Comment: 61 pages, 1 figure. Comments and references adde