Solving an eigenproblem with analyticity of the generating function

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 gener...

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Bibliographic Details
Published in:Journal of the Korean Physical Society Vol. 79; no. 2; pp. 113 - 124
Main Authors: Kim, U-Rae, Jung, Dong-Won, Kim, Dohyun, Lee, Jungil, Yu, Chaehyun
Format: Journal Article
Language:English
Published: Seoul The Korean Physical Society 01.07.2021
Springer Nature B.V
한국물리학회
Subjects:
Beads
Bernoulli Hypothesis
Classical mechanics
Complex variables
Eigenvalues
Eigenvectors
Entire functions
Euclidean geometry
Formalism
Hilbert space
Mathematical analysis
Mathematical and Computational Physics
Original Paper
Particle and Nuclear Physics
Physics
Physics and Astronomy
Power series
Theoretical
물리학
Eigenproblem
Generating function
Normal modes
Analyticity
ISSN:0374-4884, 1976-8524
Online Access:Get full text
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Summary: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 Schläfli 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.
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ISSN:0374-4884
1976-8524
DOI:10.1007/s40042-021-00201-3