Kernel inspired factorizations of partial differential operators
type: Item
instance of: scholarly article
title: Kernel inspired factorizations of partial differential operators
zbMATH Open document ID: 0934.35027
author: A. Kasman
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-04-02T00:00:00Z
review text: As a practical tool to factor some constant coefficient linear partial differential operators, the following result is proved: Let \(\varphi(x_1,\dots,x_n)\) be a nonzero differentiable function, let \(q(z_1,\dots,z_n)\) be an irreducible polynomial. Let \(K^\varphi_q\) denote the operator \(\varphi(x) \circ q(\partial) \circ 1/\varphi(x)\). Then there exists a set \(F^\varphi_q \subset \ker K^\varphi_q\) such that a differential operator \(L\) factors through \(K^\varphi_q\) if and only if \(F^\varphi_q \subset \ker L\). The set \(F^\varphi_q\) constructed in the proof is parametrized by the algebraic variety \(q^{-1}(0)\). Notes on applications to Darboux transformations, quantum integrability, and bispectrality close the paper.
Mathematics Subject Classification ID: 35E20
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 13N10
zbMATH DE Number: 1334728
zbMATH Keywords: rational-exponential function
zbMATH Keywords: Nullstellensatz
zbMATH Keywords: Darboux transformation
zbMATH Keywords: bispectrality
zbMATH Keywords: quantum integrability
reviewed by: Michal Marvan
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6383
OpenAlex ID: W1965771308
cites work: Q4938184
cites work: \(\mathcal{D}\)-modules and Darboux transformations
cites work: Quantum integrable systems and differential Galois theory
cites work: Commutative rings of partial differential operators and Lie algebras
cites work: Integrability in the theory of Schrödinger operator and harmonic analysis
cites work: The duality of the generalized Calogero and Ruijsenaars problems
cites work: Differential equations in the spectral parameter
cites work: Q4387367
cites work: Darboux transformations of bispectral quantum integrable systems
cites work: Formal solutions and factorization of differential operators with power series coefficients
cites work: Factorization of differential operators with rational functions coefficients
cites work: Bispectral Darboux transformations: The generalized Airy case
cites work: Darboux transformations from \(n\)-KdV to KP
cites work: Q4387370
cites work: Spectral difference equations satisfied by KP soliton wavefunctions
cites work: Q4002235
cites work: Q4387374
cites work: New situation in quantum mechanics (wonderful potentials from the inverse problem)
DOI: 10.1006/JMAA.1999.6383
description: scientific article; zbMATH DE number 1334728