Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math-ph/0501057

Orthogonal polynomials with discontinuous weights

Yang Chen·Gunnar Pruessner
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w_0 and a standard jump function, then A_n(z) and B_n(z) have apparent simple poles at these jumps. We exemplify the approach by taking w_0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painleve IV.

Explore related subjects

math-phmath.MPnlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yang Chen, Gunnar Pruessner. 2005-01-22. Orthogonal polynomials with discontinuous weights. https://doi.org/10.1088/0305-4470%2F38%2F12%2Fl01

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Functorial Theory of Defects in Abelian Chern-Simons Theory

Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.

math-ph↗
arXiv2026

Gradient nature of Laplacian growth

For a class of growth processes of Laplacian type in the plane, we suggest an interpretation as a ``gradient descent'' in the space of smooth closed curves. More precisely, we show that boundary of a growing domain moves along a gradient of a certain functional in the space of curves. In the simplest cases this functional is $\log (1/r)$, where $r$ is the external conformal radius of the growing domain.

math-ph↗
arXiv2026

Entanglement-Inducing Quantum Markov Processes

We introduce a new model for a system of interacting bosons placed in an array of sites. At its core is a nonlinear, nonlocal evolution equation, which we have dubbed the Schrödinger-Dirichlet equation. The construction is closely related to the Bose-Hubbard model and to a specific type of generalized bosons. In contrast to conventional mean-field closures, the resulting nonlinear dynamics need not preserve product structure and can generate entanglement from initially separable states. The relevant methods of analysis are based on harmonic analysis for the multiplicative group of positive rationals.

math-ph↗