Spectral Residues of Second-order Differential Equations: towards a New Class of Summation Identities and Inversion Formulas

The article introduces a method of generating sequences by means of a formal eigenvalue problem. This is accomplished by considering the spectral residues of a second-order Schrödinger operator. It needs to be noted that the n spectral residue is a certain homogeneous polynomial of the first n coefficients of the corresponding potential’s power series, and that it serves as a measure of obstruction to the existence of a power series eigenfunction when the eigenvalue parameter is such that the roots of the indicial equation differ by n. The transformation from potential to spectral residue sequence is invertible, and gives rise to a combinatorial inversion formula. Other interesting combinatorial consequences are obtained by considering spectral residues of the exactly-solvable potentials of 1dimensional quantum mechanics. Finally, it is shown that the Darboux transformation of 1-dimensional potentials corresponds to a simple negation of the corresponding spectra residues. This fact leads to another combinatorial inversion formula. 1991 Mathematics Subject Classification. Primary: 05A19. Secondary: 81C05. Email: [email protected]. This research supported by a Dalhousie University grant.