Abstract
We consider the linear, second-order elliptic, Schrödinger-type differential operator L:=−12∇2+r22. Because of its rotational invariance, that is it does not change under SO(3) transformations, the eigenvalue problem −12∇2+r22f(x,y,z)=λf(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.
Highlights
In this paper, we intend to treat an elliptic PDE
We focus on accidental degeneracy and on its relationship with ladder operators
Since the spectrum of the operator O has no degeneracy, it follows that the two sets of eigenfunctions {yi(1)} and {y(j2)} are the same set, but this conclusion is absurd because the operators U1, U2 do not commute with each other, and there cannot exist a basis of the Hilbert space formed by all simultaneous eigenfunctions of the non-commuting operators U1, U2
Summary
We intend to treat an elliptic PDE The main notions to tackle the typical mathematical physics problems can be found in [2], for example.) with a special focus on the property of the degeneracy of its spectrum. If λi are independent of l, accidental degeneracy occurs. We focus on accidental degeneracy and on its relationship with ladder operators (a similar procedure applied to spherical hydrogen atom eigenfuctions can be found in [3]). The paper is organized as follows: In Section 2 the main notions and a selection of useful results on invariance and degeneracy are presented.
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