Spectral $$\varvec{\zeta }$$-functions and $$\varvec{\zeta }$$-regularized functional determinants for regular Sturm–Liouville operators
| Autor: | Fritz Gesztesy, Klaus Kirsten, Jonathan Stanfill, Guglielmo Fucci |
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| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Applied Mathematics Analytic continuation Zero (complex analysis) Sturm–Liouville theory Function (mathematics) Theoretical Computer Science Computational Mathematics Mathematics (miscellaneous) Piecewise Boundary value problem Functional determinant Constant (mathematics) Mathematics |
| Zdroj: | Research in the Mathematical Sciences. 8 |
| ISSN: | 2197-9847 2522-0144 |
| DOI: | 10.1007/s40687-021-00289-w |
| Popis: | The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and $$\zeta $$ -functions to efficiently compute values of spectral $$\zeta $$ -functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm–Liouville differential expressions $$\tau $$ . Depending on the underlying boundary conditions, we express the $$\zeta $$ -function values in terms of a fundamental system of solutions of $$\tau y = z y$$ and their expansions about the spectral point $$z=0$$ . Furthermore, we give the full analytic continuation of the $$\zeta $$ -function through a Liouville transformation and provide an explicit expression for the $$\zeta $$ -regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrodinger operators with zero, piecewise constant, and a linear potential on a compact interval. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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