Accidental Degeneracy of an Elliptic Differential Operator: A Clarification in Terms of Ladder Operators

Roberto De Marchis, Arsen Palestini and Stefano Patrì
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Roberto De Marchis: MEMOTEF, Faculty of Economics, Sapienza University of Rome, Via del Castro Laurenziano 9, 00161 Rome, Italy
Arsen Palestini: MEMOTEF, Faculty of Economics, Sapienza University of Rome, Via del Castro Laurenziano 9, 00161 Rome, Italy
Stefano Patrì: MEMOTEF, Faculty of Economics, Sapienza University of Rome, Via del Castro Laurenziano 9, 00161 Rome, Italy

Mathematics, 2021, vol. 9, issue 23, 1-14

Abstract: We consider the linear, second-order elliptic, Schrödinger-type differential operator L : = − 1 2 ∇ 2 + r 2 2 . Because of its rotational invariance, that is it does not change under S O ( 3 ) transformations, the eigenvalue problem − 1 2 ∇ 2 + r 2 2 f ( x , y , z ) = λ f ( x , y , z ) can be studied more conveniently in spherical polar coordinates. It is already known that the eigenfunctions of the problem depend on three parameters. The so-called accidental degeneracy of L occurs when the eigenvalues of the problem depend on one of such parameters only. We exploited ladder operators to reformulate accidental degeneracy, so as to provide a new way to describe degeneracy in elliptic PDE problems.

Keywords: degeneracy; elliptic PDE; ladder operator; commuting operator; eigenvalues (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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