Convex polyhedra quantum billiards in ℝⁿ

Liboff, Richard L.

doi:10.1090/qam/1878259

Convex polyhedra quantum billiards in $\Bbb R^n$

Author: Richard L. Liboff
Journal: Quart. Appl. Math. 60 (2002), 75-85
MSC: Primary 81Q50
DOI: https://doi.org/10.1090/qam/1878259
MathSciNet review: MR1878259
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Abstract: "S-quantum billiards” are defined in ${\mathbb {R}^n}$. These billiards include the regular convex polyhedra as a subset. The “first-excited-state theorem” states that for such quantum billiards: (a) a first excited state (second eigenstate of the Laplacian) exists whose nodal surface is a plane of bisecting symmetry of the billiard; (b) the degeneracy of this state is equal to the dimension in which the billiard exists. It is shown that for such billiards, its bisecting nodal surface is one of minimum energy. The (a) component of the preceding theorem is proved with the latter proof and a conjecture that is based on recent theorems of Lin, Melas, and Alessandrini and a described smoothing procedure. Components of group theory, as well as an ansatz addressing the higher dimensions, come into play in establishing the (b) component of the theorem. An appendix is included describing properties of nodal intersection with a boundary.

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