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Choon-Lin Ho

Publications and source records attributed to Choon-Lin Ho.

At least 19 recordsLinked to original sources
arXiv2025

Effective Free Energy Landscapes and Black Hole Thermodynamic Phase Transitions

A recent interesting development in the dynamics of black hole phase transitions has been the so-called Gibbs free energy landscape approach. In this formalism, it is assumed that there exists a canonical ensemble of a series of black hole spacetimes with arbitrary horizon radius at a given ensemble temperature. An off-shell Gibbs free energy is defined for every spacetime state in the ensemble, with the horizon radius treated as the order parameter. The minima (maxima) of this function correspond to the various stable (unstable) black hole states. This off-shell Gibbs free energy is then treated as a classical effective drift potential of an associated Fokker-Planck equation used to study the stochastic dynamics of black hole phase transition under thermal fluctuations. Additive noise, which is independent of the black hole size, is assumed in obtaining the Fokker-Planck equation. In this work we extend the previous treatment by considering the effects of multiplicative noise, namely, noise that could scale with black hole size. This leads to an effective free energy function that can be used to study the modification of the thermodynamic phase transition of a black hole system. It is realized that it is generally difficult to form black holes under a multiplicative noise, unless the effective and the original free energy become extremal at the same horizon radius. For this latter situation some theoretical noise profiles which are monotonically increasing/deceasing functions of the horizon radius are considered. It is found that stronger noise disfavors the formation of black hole

gr-qc↗
arXiv2025

Multi-parameter isospectral Fokker-Planck equations

From a given Fokker-Planck equation, a multi-parameter deformed partner Fokker-Planck equation is constructed. This is done by first deleting a set of eigenstates of the original FPE by the multi-step Darboux-Crum transformation, and then reinstating the eigen-energy levels by the reverse Darboux-Crum transformation. Extension to fractional Fokker-Planck equation is briefly discussed. A recent study of the one-parameter isospectral FPE applied to black hole in the thermal potential approach is commented.

math-ph↗
arXiv2025

Dirac's variational approach to semiclassical Kramers problem in Smoluchowski limit

Kramers escape from a metastable state in the presence of both thermal and quantum fluctuations under strong damping is treated as a thermally activated process in a quantum modified semiclassical potential. Dirac's time-dependent variational method together with the Jackiw-Kerman function is employed to derive the semiclassical potential. Quantum correction is incorporated in the drift potential, and is determined by quasi-stationary conditions and minimal uncertainty relation. The semiclassical rate obtained here is consistent in form with those from the quantum Smoluchowski equations deduced heuristically by modifying the diffusion coefficient using the path-integral method. Unlike approaches using the path-integral, which involves continuation into imaginary time, the approach here is simpler and more easily understood in terms of classical picture.

quant-ph↗
arXiv2024

Interpolating supersymmetric pair of Fokker-Planck equations

We consider Fokker-Planck equations that interpolate a pair of supersymmetrically related Fokker-Planck equations with constant coefficients. Based on the interesting property of shape-invariance, various one-parameter interpolations of the solutions of the supersymmetric pair of Fokker-Planck systems can be directly constructed.

math-ph↗
arXiv2023

Prepotential Approach: a unified approach to exactly, quasi-exactly, and rationally extended solvable quantal systems

We give a brief overview of a simple and unified way, called the prepotential approach, to treat both exact and quasi-exact solvabilities of the one-dimensional Schr\"odinger equation. It is based on the prepotential together with Bethe ansatz equations. Unlike the the supersymmetric method for the exactly-solvable systems and the Lie-algebraic approach for the quasi-exactly solvable problems, this approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing. In this approach the system is completely defined by the choice of two polynomials and a set of Bethe ansatz equations. The potential, the change of variables as well as the eigenfunctions and eigenvalues are determined in the same process. We illustrate the approach by several paradigmatic examples of Hermitian and non-Hermitian Hamiltonians with real energies. Hermitian systems with complex energies, called the quasinormal modes, are also presented. Extension of the approach to the newly discovered rationally extended models is briefly discussed.

quant-ph↗
arXiv2022

Supersymmetry and Convection-Diffusion-Reaction equations

In this work we are concerned with generating solutions of a class of Convection-Diffusion-Reaction equation from the solutions of another CDR equation through the Darboux transformations. The method is elucidated by cases with certain types of the reaction coefficients. We have also discussed briefly supersymmetric pairs of Convection-Diffusion-Reaction equations connected through similarity solutions.

math-ph↗
arXiv2022

Zero energy states of Dirac equation in $(2+1)$-dimensional curved spacetime

We consider Dirac equation in $(2+1)$ dimensional curved spacetime in the presence of a scalar potential. It is then shown that the zero energy states are degenerate and they can be obtained when the momentum $k_y$ in the $y$ direction satisfies certain constraints involving the mass parameter and the scalar potential coupling.

quant-ph↗
arXiv2021

Time-dependent Darboux transformation and supersymmetric hierarchy of Fokker-Planck equations

A procedure is presented for solving the Fokker-Planck equation with constant diffusion but non-stationary drift. It is based on the correspondence between the Fokker-Planck equation and the non-stationary Schr\"odinger equation. The formalism of supersymmetric quantum mechanics is extended by applying the Darboux transformation directly to the non-stationary Schr\"odinger equation. From a solution of a Fokker-Planck equation a solution of the Darboux partner is obtained. For drift coefficients satisfying the condition of shape invariance, a supersymmetric hierarchy of Fokker-Planck equations with solutions related by the Darboux transformation is obtained.

math-ph↗
arXiv2019

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↗
arXiv2017

A spectral analysis of discrete-time quantum walks related to the birth and death chains

In this paper, we consider a spectral analysis of discrete time quantum walks on the path. For isospectral coin cases, we show that the time averaged distribution and stationary distributions of the quantum walks are described by the pair of eigenvalues of the coins as well as the eigenvalues and eigenvectors of the corresponding random walks which are usually referred as the birth and death chains. As an example of the results, we derive the time averaged distribution of so-called Szegedy's walk which is related to the Ehrenfest model. It is represented by Krawtchouk polynomials which is the eigenvectors of the model and includes the arcsine law.

quant-ph↗
arXiv2016

Periodicity for space-inhomogeneous quantum walks on the cycle

In this paper, we consider periodicity for space-inhomogeneous quantum walks on the cycle. For isospectral coin cases, we propose a spectral analysis. Based on the analysis, we extend the result for periodicity for Hadamard walk to some isospectral coin cases. For non-isospectral coin cases, we consider the the system that uses only one general coin at the origin and the identity coin at the other sites. In this case, we show that the periodicity of the general coin at the origin determines the periodicity for the whole system.

quant-ph↗
arXiv2014

Multi-indexed Extensions of Soliton Potential and Extended Integer Solitons of KdV Equation

We calculate infinite set of initial profiles of higher integer KdV solitons, which are both exactly solvable for the Schrodinger equation and for the Gel'fand-Levitan-Marchenko equation in the inverse scattering transform method of KdV equation. The calculation of these higher integer soliton solutions is based on the recently developed multi-indexed extensions of the reflectionless soliton potential.

quant-ph↗
arXiv2011

Similarity solutions of Fokker-Planck equation with time-dependent coefficients

In this work, we consider the solvability of the Fokker-Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker-Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulted ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker-Planck equations are presented, and their properties analyzed.

math-ph↗
arXiv2011

Similarity Solutions of a Class of Perturbative Fokker-Planck Equation

In a previous work, a perturbative approach to a class of Fokker-Planck equations, which have constant diffusion coefficients and small time-dependent drift coefficients, was developed by exploiting the close connection between the Fokker-Planck equations and the Schrodinger equations. In this work, we further explore the possibility of similarity solutions of such a class of Fokker-Planck equations. These solutions possess definite scaling behaviors, and are obtained by means of the so-called similarity method.

math-ph↗
arXiv2010

Prepotential approach to quasinormal modes

In this paper we demonstrate how the recently reported exactly and quasi-exactly solvable models admitting quasinormal modes can be constructed and classified very simply and directly by the newly proposed prepotential approach. These new models were previously obtained within the Lie-algebraic approach. Unlike the Lie-algebraic approach, the prepotential approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present three new models with quasinormal modes: a new exactly solvable Morse-like model, and two new quasi-exactly solvable models of the Scarf II and generalized P\"oschl-Teller types.

math-ph↗
arXiv2009

Properties of the Exceptional ($X_{\ell}$) Laguerre and Jacobi Polynomials

We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 684} (2010), 173-176]. These $X_{\ell}$ polynomials are global solutions of second order Fuchsian differential equations with $\ell+3$ regular singularities and their confluent limits. We derive equivalent but much simpler looking forms of the $X_{\ell}$ polynomials. The other subjects discussed in detail are: factorisation of the Fuchsian differential operators, shape invariance, the forward and backward shift operations, invariant polynomial subspaces under the Fuchsian differential operators, the Gram-Schmidt orthonormalisation procedure, three term recurrence relations and the generating functions for the $X_{\ell}$ polynomials.

math-ph↗
arXiv2008

Fermion pair production in planar Coulomb and Aharonov--Bohm potentials

Exact analytic solutions are found for the Dirac equation in 2+1 dimensions for a spin-one-half particle in a combination of the Lorentz 3-vector and scalar Coulomb as well as Aharonov--Bohm potentials. We employ the two-component Dirac equation which contains a new parameter introduced by Hagen to describe the spin of the spin-1/2 particle. We derive a transcendental equations that implicitly determine the energy spectrum of an electron near the negative-energy continuum boundary and the critical charges for some electron states. Fermion pair production from a vacuum by a strong Coulomb field in the presence of the magnetic flux tube of zero radius is considered. It is shown that the presence of the Ahanorov--Bohm flux tends to stabilize the system.

hep-th↗