Abstract
The paper addresses the nodal count (i.e., the number of nodal domains) for eigenfunctions of Schrödinger operators with Dirichlet boundary conditions in bounded domains. The classical Sturm theorem states that in dimension one, the nodal and eigenfunction counts coincide: the nth eigenfunction partitions the interval into n nodal domains. The Courant Nodal Theorem claims that in any dimension, the number of nodal domains ν n of the nth eigenfunction cannot exceed n. However, it follows from an asymptotically stronger upper bound by Pleijel that in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, in most cases a “nodal deficiency” d n = n−ν n arises. One can say that the nature of the nodal deficiency has not been understood. It was suggested in recent years that, rather than starting with eigenfunctions, one can look at partitions of the domain into ν sub-domains, asking which partitions can correspond to eigenfunctions, and what would be the corresponding deficiency. To this end one defines an “energy” of a partition, for example, the maximum of the ground state energies of the sub-domains. One notices that if a partition does correspond to an eigenfunction, then the ground state energies of all the nodal domains are the same, i.e., it is an equipartition. It was shown in a recent paper by Helffer, Hoffmann-Ostenhof and Terracini that (under some natural conditions) partitions minimizing the energy functional correspond to the “Courant sharp” eigenfunctions, i.e. to those with zero nodal deficiency. In this paper it is shown that it is beneficial to restrict the domain of the functional to the equipartition, where it becomes smooth. Then, under some genericity conditions, the nodal partitions correspond exactly to the critical points of the functional. Moreover, the nodal deficiency turns out to be equal to the Morse index at the corresponding critical point. This explains, in particular, why the minimal partitions must be Courant sharp.
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Berkolaiko, G., Kuchment, P. & Smilansky, U. Critical Partitions and Nodal Deficiency of Billiard Eigenfunctions. Geom. Funct. Anal. 22, 1517–1540 (2012). https://doi.org/10.1007/s00039-012-0199-y
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DOI: https://doi.org/10.1007/s00039-012-0199-y