Refined spectral properties of Dirichlet and Neumann problems for the Laplace operator in a rectangular domain

V. I. Voytitsky ; A. S. Prudkii

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 1, pp. 20-32

Résumé

In one-dimensional boundary value spectral problems the dimensions of eigen-subspaces are not greater than some known number (as a rule 1 or 2). In multidimensional self-adjoint problems with a discrete spectrum the sequence of multiplicities can be unbound despite the finite dimensions of all eigen-subspaces. It is realized even for classical boundary value problems solved by the method of separation of variables. In the case of Dirichlet or Neumann problems for the Laplace operator given in a rectangular domain $\Omega=(0;a)\times(0;b)$ the formula $\lambda_{km} = \big(\frac{\pi k}{a}\big)^2 + \big(\frac{\pi m}{b}\big)^2$ for eigenvalues is well known (indexes $k, m$ are correspondingly positive or nonnegative integers for Dirichlet or Neumann problem). The problem of multiplicities reduces to counting the number of ordered pairs $(k, m)$ which determine the same number $\lambda_{km}$. Using classical and new results of number theory and the theory of diophantine approximations we study problems of relative arrangement, multiplicities and asymptotic behavior of eigenvalues $\lambda_{km}$ depending on parameters $a$ and $b$. In the case of square domain ($a=b$) we formulate explicit algorithm for counting the multiplicities of eigenvalues based on decomposition of a natural number into prime factors and counting devisors of the form $4k+1$. For a rectangular domains we establish relationship between the distribution of multiplicities and rationality of numbers $f:=a/b$ and $f^2$. For the case $f, f^2 \not\in \mathbb{Q}$ we prove that all eigenvalues are simple but infinitely many pairs of them are located at an arbitrarily close distance. Using the refined estimation of the remainder in the Gauss circle problem we establish Weyl's asymptotic formula with the first two members and qualified assessment of residual member.