Multiplicities of the eigenvalues of the discrete Schrödinger equation in any dimension

Burghelea, Dan; Kappeler, Thomas

doi:10.1090/S0002-9939-1988-0920982-3

Multiplicities of the eigenvalues of the discrete Schrödinger equation in any dimension
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by Dan Burghelea and Thomas Kappeler

Proc. Amer. Math. Soc. 102 (1988), 255-260

DOI: https://doi.org/10.1090/S0002-9939-1988-0920982-3

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Abstract:

The following von Neumann-Wigner type result is proved: The set of potentials $a:\;\Gamma \to {\mathbf {R}}(\Gamma \subseteq {{\mathbf {Z}}^N})$, with the property that the corresponding discrete Schrödinger equation ${\Delta _d} + a$ has multiple eigenvalues when considered with certain boundary conditions, is an algebraic set of ${\text {codimension}} \geq {\text {2}}$ within ${{\mathbf {R}}^\Gamma }$.

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