Solving an eigenproblem with analyticity of the generating function
| Autor: | U-Rae Kim, Dohyun Kim, Chaehyun Yu, Dong-Won Jung, Jungil Lee |
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| Rok vydání: | 2021 |
| Předmět: |
010302 applied physics
Power series Entire function Mathematical analysis Hilbert space Physical system General Physics and Astronomy 02 engineering and technology 021001 nanoscience & nanotechnology 01 natural sciences Methods of contour integration symbols.namesake 0103 physical sciences symbols 0210 nano-technology Eigenvalues and eigenvectors Variable (mathematics) Generating function (physics) Mathematics |
| Zdroj: | Journal of the Korean Physical Society. 79:113-124 |
| ISSN: | 1976-8524 0374-4884 |
| DOI: | 10.1007/s40042-021-00201-3 |
| Popis: | We present a generating-function representation of a vector defined in either Euclidean or Hilbert space with arbitrary dimensions. The generating function is constructed as a power series in a complex variable whose coefficients are the components of a vector. As an application, we employ the generating-function formalism to solve the eigenproblem of a vibrating string loaded with identical beads. The corresponding generating function is an entire function. The requirement of the analyticity of the generating function determines the eigenspectrum all at once. Every component of the eigenvector of the normal mode can be easily extracted from the generating function by making use of the Schlafli integral. This is a unique pedagogical example with which students can have a practical contact with the generating function, contour integration, and normal modes of classical mechanics at the same time. Our formalism can be applied to a physical system involving any eigenvalue problem, especially one having many components, including infinite-dimensional eigenstates. |
| Databáze: | OpenAIRE |
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