Speaker : Ryu Sasaki (Shinshu University ; retired from Kyoto University)
Title : Simplest Quantum Mechanics; Novel Introduction to Orthogonal Polynomials of a Discrete Variable in the Askey Scheme
Time : 2017-10-20 (Fri) 10:30 - 11:30
Place : Seminar Room 617, Institute of Mathematics (NTU Campus)
Abstract: By pursuing the similarity and parallelism between the ordinary Quantum Mechanics (QM) and the eigenvalue problem of hermitian matrices, I present simplest forms of exactly solvable QM.
Their Hamiltonians ($Schr\ddot{o}dinger$ operators) are a special class of tri-diagonal real symmetric (Jacobi) matrices.
The super (sub) diagonal elements of the Jacobi matrices are interpreted as the positive (negative) shift operators acting on vectors which are expressed as functions defined on a finite (infinite) integer lattice, x=0,1,2, ...,. Now the Jacobi matrices are second order difference operators which are Hamiltonians ($Schr\ddot{o}dinger$ operators) of discrete Quantum Mechanics with real shifts (rdQM) in one dimension.
Their eigenfunctions (vectors) are the well known classical orthogonal polynomials of a discrete variable belonging to the Askey scheme of hypergeometric orthogonal polynomials, e.g.the Charlier, Meixner, (dual) Hahn, Racah etc and their q-versions.
Here I present the simplest QM counterparts of the prominent results of exactly solvable QM; Crum's theorem, shape invariance, Heisenberg operator solutions together with the duality and dual polynomials which are the special features of these polynomials. Birth and Death problems are the important applications.