Spectral Residues of Second-order Differential Equations: A New Method for Summation Identities and Inversion Formulas

• Published in: Studies in applied mathematics (Cambridge) Vol. 107; no. 4; pp. 337 - 366
• Main Author: Milson, R.
• Format: Journal Article
• Language: English
• Published: Boston, USA and Oxford, UK Blackwell Publishers Inc 01.11.2001; Blackwell
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Operator theory; Ordinary differential equations; Sciences and techniques of general use; Inversion formula; Second order equation; Summation identity; Spectral method; Frobenius method; Differential equation; Schrödinger operator; Summation; Darboux transformation
• ISSN: 0022-2526; 1467-9590
• DOI: 10.1111/1467-9590.1074190

This article deals with differential equations with spectral parameter from the point of view of formal power series.The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The article focuses on second‐order, self‐adjoint problems. In such a setting, every potential function determines a sequence of spectral residues. This correspondence is invertible and gives rise to a combinatorial inversion formula. Other interesting combinatorial consequences are obtained by considering spectral residues of exactly solvable potentials of one‐dimensional quantum mechanics. It is also shown that the Darboux transformation of one‐dimensional potentials corresponds to a simple negation of the corresponding spectral residues. This fact leads to another combinatorial inversion formula. Finally, there is a brief discussion of applications. The topics considered are enumeration problems and integrable systems.