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Search Results: 1 - 10 of 2860 matches for " Ryu Sasaki "
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Exactly and quasi-exactly solvable `discrete' quantum mechanics
Ryu Sasaki
Physics , 2010, DOI: 10.1098/rsta.2010.0262
Abstract: Brief introduction to the discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators, dynamical symmetry algebras including the $q$-oscillator algebra and the Askey-Wilson algebra. A simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics is presented. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and quasi-exactly solvable ones are constructed. The sinusoidal coordinate plays an essential role.
New Quasi Exactly Solvable Difference Equation
Ryu Sasaki
Physics , 2007, DOI: 10.2991/jnmp.2008.15.s3.36
Abstract: Exact solvability of two typical examples of the discrete quantum mechanics, i.e. the dynamics of the Meixner-Pollaczek and the continuous Hahn polynomials with full parameters, is newly demonstrated both at the Schroedinger and Heisenberg picture levels. A new quasi exactly solvable difference equation is constructed by crossing these two dynamics, that is, the quadratic potential function of the continuous Hahn polynomial is multiplied by the constant phase factor of the Meixner-Pollaczek type. Its ordinary quantum mechanical counterpart, if exists, does not seem to be known.
Quasi Exactly Solvable Difference Equations
Ryu Sasaki
Physics , 2007, DOI: 10.1063/1.2818560
Abstract: Several explicit examples of quasi exactly solvable `discrete' quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogues of the well-known quasi exactly solvable systems, the harmonic oscillator (with/without the centrifugal potential) deformed by a sextic potential and the 1/sin^2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.
Exactly Solvable Birth and Death Processes
Ryu Sasaki
Physics , 2009, DOI: 10.1063/1.3215983
Abstract: Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable `matrix' quantum mechanics, which is recently proposed by Odake and the author. The ($q$-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of $q^x$ ($x$ being the population) corresponding to the $q$-Racah polynomial.
Exactly Solvable Quantum Mechanics
Ryu Sasaki
Physics , 2014,
Abstract: 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.
Perturbations around the zeros of classical orthogonal polynomials
Ryu Sasaki
Physics , 2014, DOI: 10.1063/1.4918707
Abstract: Starting from degree N solutions of a time dependent Schroedinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,...,N-1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
Ryu Sasaki
Mathematics , 2014, DOI: 10.1063/1.4880200
Abstract: We address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics are exactly solvable. With an additional time dependence these potentials are identified as the soliton solutions of the KdV hierarchy. An $N$-soliton potential has the time $t$ and $2N$ positive parameters, $k_1<...
Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity
Ryu Sasaki,Kouichi Takemura
Symmetry, Integrability and Geometry : Methods and Applications , 2012,
Abstract: Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent 2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.
Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
Satoru Odake,Ryu Sasaki
Physics , 2009, DOI: 10.1063/1.3371248
Abstract: We provide analytic proofs for the shape invariance of the recently discovered (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417) two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux-P\"oschl-Teller potential by a degree \ell (\ell=1,2,...) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3\ell involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.
Another set of infinitely many exceptional (X_{\ell}) Laguerre polynomials
Satoru Odake,Ryu Sasaki
Physics , 2009, DOI: 10.1016/j.physletb.2009.12.062
Abstract: We present a new set of infinitely many shape invariant potentials and the corresponding exceptional (X_{\ell}) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one dimensional quantum mechanics and the corresponding X_{\ell} Laguerre and Jacobi polynomials (Odake and Sasaki, Phys. Lett. B679 (2009) 414-417). The new X_{\ell} Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X_{\ell} Jacobi polynomials and the potentials, whereas the known X_{\ell} Laguerre polynomials and the potentials are obtained in the same manner from the mirror image of the known X_{\ell} Jacobi polynomials and the potentials.
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