arXiv subjects

Ryu Sasaki

Publications and source records attributed to Ryu Sasaki.

At least 19 recordsLinked to original sources

arXiv2025

Quantum vs classical Markov chains; Exactly solvable examples

A coinless quantisation procedure of general reversible Markov chains on graphs is presented. A quantum Hamiltonian H is obtained by a similarity transformation of the fundamental transition probability matrix K in terms of the square root of the reversible distribution. The evolution of the classical and quantum Markov chains is described by the solutions of the eigenvalue problem of the quantum Hamiltonian H. About twenty plus exactly solvable Markov chains based on the hypergeometric orthogonal polynomials of Askey scheme, derived by Odake-Sasaki, would provide a good window for scrutinising the quantum/classical contrast of Markov chains. Among them five explicit examples, related to the Krawtchouk, Hahn, q-Hahn, Charlier and Meixner, are demonstrated to illustrate the actual calculations.

quant-ph↗

arXiv2025

Exactly solvable multicomponent spinless fermions

By generalising the one to one correspondence between exactly solvable hermitian matrices $\mathcal{H}=\mathcal{H}^\dagger$ and exactly solvable spinless fermion systems $\mathcal{H}_f=\sum_{x,y}c_x^\dagger\mathcal{H}(x,y)c_y$, four types of exactly solvable multicomponent fermion systems are constructed explicitly. They are related to the multivariate Krawtcouk, Meixner and two types of Rahman like polynomials, constructed recently by myself. The Krawtchouk and Meixner polynomials are the eigenvectors of certain real symmetric matrices $\mathcal{H}$ which are related to the difference equations governing them. The corresponding fermions have nearest neighbour interactions. The Rahman like polynomials are eigenvectors of certain reversible Markov chain matrices $\mathcal{K}$, from which real symmetric matrices $\mathcal{H}$ are uniquely defined by the similarity transformation in terms of the square root of the stationary distribution. The fermions have wide range interactions.

hep-th↗

arXiv2025

Multivariate Hahn polynomials and difference equations

The multivariate Hahn polynomials are constructed explicitly as the common eigenvectors of a family of second order difference operators. They are orthogonal with respect to the hypergeometric multinomial distribution. The main difference operator is adopted from the work of Karlin-McGregor in 1975. The minor ones are the subsets of the main one containing less and less variables. These operators commute with each other. In contrast to the multivariate Krawtchouk and Rahman like polynomials derived recently, the entire multivariate Hahn polynomials are rational functions of the system parameters. Complete sets of multivariate Krawtchouk and Meixner polynomials are derived by limiting procedures.

math.CA↗

arXiv2024

Lattice fermions with solvable wide range interactions

Exactly solvable (spinless) lattice fermions with wide range interactions are constructed explicitly based on {\em exactly solvable stationary and reversible Markov chains} $\mathcal{K}^R$ reported a few years earlier by Odake and myself. The reversibility of $\mathcal{K}^R$ with the stationary distribution $π$ leads to a positive classical Hamiltonian $\mathcal{H}^R$. The exact solvability of $\mathcal{H}^R$ warrants that of a spinless lattice fermion $c_x$, $c_x^\dagger$, $\mathcal{H}^R_f=\sum_{x,y\in\mathcal{X}}c_x^\dagger\mathcal{H}^R(x,y) c_y$ based on the principle advocated recently by myself. The reversible Markov chains $\mathcal{K}^R$ are constructed by convolutions of the orthogonality measures of the discrete orthogonal polynomials of Askey scheme. Several explicit examples of the fermion systems with wide range interactions are presented.

quant-ph↗

arXiv2024

Exactly solvable inhomogeneous fermion systems

15 exactly solvable inhomogeneous (spinless) fermion systems on one-dimensional lattices are constructed explicitly based on the discrete orthogonal polynomials of Askey scheme, e.g. the Krawtchouk, Hahn, Racah, Meixner, $q$-Racah polynomials. The Schrödinger and Heisenberg equations are solved explicitly, as the entire set of the eigenvalues and eigenstates are known explicitly. The ground state two point correlation functions are derived explicitly. The multi point correlation functions are obtained by Wick's Theorem. Corresponding 15 exactly solvable XX spin systems are also displayed. They all have nearest neighbour interactions. The exact solvability of Schrödinger equation means that of the corresponding Fokker-Planck equation. This leads to 15 exactly solvable Birth and Death fermions and 15 Birth and Death spin models. These provide plenty of materials for calculating interesting quantities, e.g. entanglement entropy, etc.

quant-ph↗

arXiv2024

Towards verifications of Krylov complexity

Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace $\mathcal{K}_M(\mathcal{H},η)$ spanned by the multiple applications of the Liouville operator $\mathcal{L}$ defined by the commutator in terms of a Hamiltonian $\mathcal{H}$, $\mathcal{L}:=[\mathcal{H},\cdot]$ acting on an operator $η$, $\mathcal{K}_M(\mathcal{H},η)=\text{span}\{η,\mathcal{L}η,\ldots,\mathcal{L}^{M-1}η\}$. For a given inner product $(\cdot,\cdot)$ of the operators, the orthonormal basis $\{\mathcal{O}_n\}$ is constructed from $\mathcal{O}_0=η/\sqrt{(η,η)}$ by Lanczos algorithm. The moments $μ_m=(\mathcal{O}_0,\mathcal{L}^m\mathcal{O}_0)$ are closely related to the important data $\{b_n\}$ called Lanczos coefficients. I present the exact and explicit expressions of the moments $\{μ_m\}$ for 16 quantum mechanical systems which are {\em exactly solvable both in the Schrödinger and Heisenberg pictures}. The operator $η$ is the variable of the eigenpolynomials. Among them six systems show a clear sign of `non-complexity' as vanishing higher Lanczos coefficients $b_m=0$, $m\ge3$.

quant-ph↗

arXiv2024

Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as more encompassing characterisation of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree $\ell_{\mathcal D}\ge1$, the three term recurrence relations are broken. The extra $\ell_{\mathcal D}$ `lower degree polynomials', which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree $\ell_{\mathcal D}+\mathcal{N}$. %changed The discrete orthogonality relations are shown to hold for another group of `new' orthogonal polynomials called Krein-Adler polynomials based on the Hermite, Laguerre and Jacobi polynomials.

math.CA↗

arXiv2023

Rahman polynomials

Two very closely related Rahman polynomials are constructed explicitly as the left eigenvectors of certain multi-dimensional discrete time Markov chain operators $K_n^{(i)}({\boldsymbol x},{\boldsymbol y};N)$, $i=1,2$. They are convolutions of an $n+1$-nomial distribution $W_n({\boldsymbol x};N)$ and an $n$-tuple of binomial distributions $\prod_{i}W_1(x_i;N)$. The one for the original Rahman polynomials is $K_n^{(1)}({\boldsymbol x},{\boldsymbol y};N) =\sum_{\boldsymbol z}W_n({\boldsymbol x}-{\boldsymbol z};N-\sum_{i}z_i) \prod_{i}W_1(z_i;y_i)$. The closely related one is \ $K_n^{(2)}({\boldsymbol x},{\boldsymbol y};N) =\sum_{\boldsymbol z}W_n({\boldsymbol x}-{\boldsymbol z};N-\sum_{i}y_i) \prod_{i}W_1(z_i;y_i)$. The original Markov chain was introduced and discussed by Hoare, Rahman and Grünbaum as a multivariable version of the known soluble single variable one. The new one is a generalisation of that of Odake and myself. The anticipated solubility of the model gave Rahman polynomials the prospect of the first multivariate hypergeometric function of Aomoto-Gelfand type connected with solvable dynamics. The promise is now realised. The $n^2$ system parameters $\{u_{i\,j}\}$ of the Rahman polynomials are completely determined. These $u_{i\,j}$'s are irrational functions of the original system parameters, the probabilities of the multinomial and binomial distributions.

math.PR↗

arXiv2023

Multivariate Kawtchouk polynomials as Birth and Death polynomials

Multivariate Krawtchouk polynomials are constructed explicitly as Birth and Death polynomials, which have the nearest neighbour interactions. They form the complete set of eigenpolynomials of a birth and death process with the birth and death rates at population $x=(x_1,\ldots,x_n)$ are $B_j(x)=\bigl(N-\sum_{i=1}^nx_i\bigr)$ and $D_j(x)=p_i^{-1}x_j$, $0<p_j$, $j=1,\ldots,n$. The corresponding stationary distribution is the multinomial distribution with the probabilities $\{η_i\}$, $η_i= p_i/(1+\sum_{j=1}^np_j)$. The polynomials, depending on $n+1$ parameters ($\{p_i\}$ and $N$), satisfy the difference equation with the coefficients $B_j(x)$ and $D_j(x)$ $j=1,\ldots,n$, which is the straightforward generalisation of the difference equation governing the single variable Krawtchouk polynomials. The polynomials are truncated $(n+1,2n+2)$ hypergeometric functions of Aomoto-Gelfand. The divariate Rahman polynomials are identified as the dual polynomials with a special parametrisation.

math.CA↗

arXiv2023

Multivariate Meixner polynomials as Birth and Death polynomials

Based on the framework of Plamen Iliev, multivariate Meixner polynomials are constructed explicitly as Birth and Death polynomials. They form the complete set of eigenpolynomials of a birth and death process with the birth and death rates at population $x=(x_1,\ldots,x_n)\in\mathbb{N}_0^n$ are $B_j(x)=\bigl(β+\sum_{i=1}^nx_j\bigr)$ and $D_j(x)=c_j^{-1}x_j$, $0<c_j$, $j=1,\ldots,n$, $\sum_{j=1}^nc_j<1$. The corresponding stationary distribution is $(β)_{\sum_{j=1}^nc_j}\prod_{j=1}^n(c_j^{x_j}/x_j!)(1-\sum_{j=1}^nc_j)^β$, the trivial $n$-variable generalisation of the orthogonality weight of the single variable Meixner polynomials. The polynomials, depending on $n+1$ parameters ($\{c_i\}$ and $β$), satisfy the difference equation with the coefficients $B_j(x)$ and $D_j(x)$ $j=1,\ldots,n$, which is the straightforward generalisation of the difference equation governing the single variable Meixner polynomials. The polynomials are truncated $(n+1,2n+2)$ hypergeometric functions of Aomoto-Gelfand. The polynomials and the derivation are very similar to those of the multivariate Krawtchouk polynomials reported recently.

math.CA↗

arXiv2023

Exactly solvable piecewise analytic double well potential $V_{D}(x)=min[(x+d)^2,(x-d)^2]$ and its dual single well potential $V_{S}(x)=max[(x+d)^2,(x-d)^2]$

By putting two harmonic oscillator potential $x^2$ side by side with a separation $2d$, two exactly solvable piecewise analytic quantum systems with a free parameter $d>0$ are obtained. Due to the mirror symmetry, their eigenvalues $E$ for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function ${}_1F_1$ of $d$ and $E$, which are common to $V_{D}$ and $V_{S}$ but in two different branches. The eigenfunctions are the piecewise square integrable combinations of ${}_1F_1$, the so called $U$ functions. By comparing the eigenvalues and eigenfunctions for various values of the separation $d$, vivid pictures unfold showing the tunneling effects between the two wells.

quant-ph↗

arXiv2022

Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples

A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable Markov chains contain many adjustable free parameters which could be helpful for various simulation purposes.

quant-ph↗

arXiv2022

"Diophantine'' and Factorisation Properties of Finite Orthogonal Polynomials in the Askey Scheme

A new interpretation and applications of the ``Diophantine'' and factorisation properties of {\em finite} orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the ($q$-)Racah, (dual, $q$-)Hahn, Krawtchouk and five types of $q$-Krawtchouk. These ($q$-)hypergeometric polynomials, defined only for the degrees of $0,1,\ldots,N$, constitute the main part of the eigenvectors of $N+1$-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The {\em monic} versions of these polynomials all exhibit the ``Diophantine'' and factorisation properties at higher degrees than $N$. This simply means that these higher degree polynomials are zero-norm ``eigenvectors'' of the $N+1$-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

math.CA↗

arXiv2022

Markov Chains Generated by Convolutions of Orthogonality Measures

About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms of convolutions of orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, $q$-Hahn, $q$-Meixner and little $q$-Jacobi polynomials. By construction, the stationary probability distributions, the complete sets of eigenvalues and eigenvectors are provided by the polynomials and the orthogonality measures. An interesting property possessed by these stationary probability distributions, called `convolutional self-similarity,' is demonstrated.

math.PR↗

arXiv2021

Exactly solvable discrete time Birth and Death processes

We present 15 explicit examples of discrete time Birth and Death processes which are exactly solvable. They are related to the hypergeometric orthogonal polynomials of Askey scheme having discrete orthogonality measures. Namely, they are the Krawtchouk, three different kinds of q-Krawtchouk, (dual, q)-Hahn, (q)-Racah, Al-Salam-Carlitz II, q-Meixner, q-Charlier, dual big q-Jacobi and dual big q-Laguerre polynomials. The birth and death rates are determined by the difference equations governing the polynomials. The stationary distributions are the normalised orthogonality measures of the polynomials. The transition probabilities are neatly expressed by the normalised polynomials and the corresponding eigenvalues. This paper is simply the discrete time versions of the known solutions of the continuous time birth and death processes.

math.PR↗

arXiv2018

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 Schrödinger 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.

math.CA↗

arXiv2017

Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.

math.CA↗

arXiv2017

Multi-indexed Meixner and Little $q$-Jacobi (Laguerre) Polynomials

As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little $q$-Jacobi (Laguerre) polynomials in the framework of `discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the ($q$-)Racah systems defined on a finite lattice, in which the `virtual state' vectors satisfy the matrix Schrödinger equation except for one of the two boundary points.

math.CA↗