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Article

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Title

On eigenproblem for inverted harmonic oscillators

Authors

[ 1 ] Instytut Matematyki, Wydział Automatyki, Robotyki i Elektrotechniki, Politechnika Poznańska | [ P ] employee

Scientific discipline (Law 2.0)

[7.7] Physical sciences

Year of publication

2021

Published in

BANACH CENTER PUBLICATIONS

Journal year: 2021 | Journal volume: vol. 124

Article type

scientific article

Publication language

english

Keywords
EN
  • inverted harmonic oscillator
  • rigged Hilbert space
  • generalized eigenvalue problem
  • differential operator
Abstract

EN We consider an eigenvalue problem for an inverted one-dimensional harmonic oscillator. We find a complete description for the eigenproblem in C(R). The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is C. The spectrum of the differential operator −d/dx2−ω2x2 is continuous and has physical significance only for the states which are in L2(R) and correspond to real eigenvalues. To identify them we orthonormalize in Dirac sense the states corresponding to real eigenvalues. This leads to the doubly degenerated real line as the spectrum of the Hamiltonian (in L2(R)). We also use two other approaches. First we define a unitary operator between L2(R) and L2 for two copies of R. This operator has the property that the spectrum of the image of the inverted harmonic oscillator corresponds to the spectrum of the operator −i(d/dx). This shows again that the (generalized) spectrum of the inverted harmonic operator is a doubly degenerated real line. The second approach uses rigged Hilbert spaces.

Pages (from - to)

61 - 73

DOI

10.4064/bc124-6

URL

https://www.impan.pl/en/publishing-house/banach-center-publications/all/124/0/114339/on-eigenproblem-for-inverted-harmonic-oscillators

Presented on

Arithmetic Methods in Mathematical Physics and Biology II, 5-11.08.2018, Będlewo, Poland

Ministry points / journal

20

Ministry points / journal in years 2017-2021

20

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