arXiv subjects

Chun-Kong Law

Publications and source records attributed to Chun-Kong Law.

4 recordsLinked to original sources

arXiv2023

A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation

We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation which determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.

math.CA↗

arXiv2019

Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV

We show that there is a full correspondence between the parameters space of the degenerate biconfluent Heun connection (BHC) and that of Painlevé IV that admits special solutions. The BHC degenerates when either the Stokes' data for the irregular singularity at $\infty$ degenerates or the regular singular point at the origin becomes an apparent singularity. We show that if the BHC is written as isomonodromy family of biconfluent Heun equations (BHE), then the BHE degenerates precisely when it admits eigen-solutions of the biconfluent Heun operators, after choosing appropriate accessory parameter, of specially constructed invariant subspaces of finite dimensional solution spaces spanned by parabolic cylinder functions. We have found all eigen-solutions over this parameter space apart from three exceptional cases after choosing the right accessory parameters. These eigen-solutions are expressed as certain finite sum of parabolic cylinder functions. We extend the above sum to new convergent series expansion in terms of parabolic cylinder functions to the BHE. The infinite sum solutions of the BHE terminates precisely when the parameters of the BHE assumes the same values as those of the degenerate biconfluent Heun connection except at three instances after choosing the right accessory parameter.

math.CA↗

arXiv2019

Periodic spectrum of $n$-cubic quantun graphs

We study the spectrum of some periodic differential operators, in particular the periodic Schrödinger operator acting on infinite $n$-cubic graphs. Using Floquet-Bloch theory, we derive and analyze on the dispersion relations of the periodic quantum graph generated by 2-dimensional rectangles, and also $n$-cubes. Our proof is analytic. These dispersion relations define the spectra of the associated periodic operator, thus facilitating further analysis of the spectra.

math.SP↗

arXiv2019

Dispersion relations of periodic quantum graphs associated with Archimedean tiling (I)

There are totally 11 kinds of Archimedean tiling for the plane. Applying the Floquet-Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tiling, namely the triangular tiling {$(3^6)$}, the elongated triangular tiling {$(3^3,4^2)$}, the trihexagonal tiling {$(3,6,3,6)$} and the truncated square tiling {$(4,8^2)$}. The derivation makes use of characteristic functions, with the help of the symbolic software Mathematica. The resulting dispersion relations are surprisingly simple and symmetric. They show that in each case the spectrum is composed of point spectrum and an absolutely continuous spectrum. We further analyzed on the structure of the absolutely continuous spectra. Our work is motivated by the studies on the periodic quantum graphs associated with hexagonal tiling in \cite{KP} and \cite{KL}.

math.SP↗