Abstract: 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.
Keywords: discrete spectrum, multiplicities of eigenvalues, prime numbers, diophantine approximations, power asymptotic, Gauss circle problem
For citation: Voytitsky, V. I. and Prudkii, A. S. Refined Spectral Properties of Dirichlet and Neumann Problems for the Laplace Operator in a Rectangular Domain, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp. 20-32 (in Russian). DOI 10.46698/u2067-6110-4876-g
1. Savchuk, À. Ì. and Shkalikov, À. À. On the Eigenvalues of the Sturm-Liouville
Operator with Potentials from Sobolev Spaces, Mathematical Notes, 2006, vol. 80,
no. 6, pp. 814-832. DOI: 10.1007/s11006-006-0204-6.
2. Levitan, B. M. and Sargsyan, I. S. Operatory Shturma-Liuvillya i Diraka
[Sturm-Liouville and Dirac Operators], Moscow, Nauka, 1988, 208 p. (in Russian).
3. Pikulin, V. P. and Pohozhaev, S. I. Prakticheskiy kurs po uravneniyam matematicheskoy fiziki
[Equations in Mathematical Physics: A Practical Course], Moscow, MCCME, 2004, 208 p. (in Russian).
4. Antunes, P. R. S. and Freitas, P. Optimal Spectral Rectangles and Lattice Ellipses,
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,
2013, vol. 469. DOI: 10.1098/rspa.2012.0492.
5. Birman, M. Sh. and Solomyak, M. Z. Asymptotic Properties of the Spectrum
of Differential Equations, Journal of Soviet Mathematics, 1979, vol. 12, no. 3,
pp. 247-283. DOI: 10.1007/BF01098368.
6. Ivrii, V. Sharp Spectral Asymptotics for Operators with Irregular Coefficients.
II. Domains with Boundary and Degeneration, Communications in Partial Differential
Equations, 2003, vol. 28, no. 1-2, pp. 103-128. DOI: 10.1081/PDE-120019376.
7. Safarov, Yu. G. and Filonov, N. D. Asymptotic Estimates of the Difference Between
the Dirichlet and Neumann Counting Functions, Functional Analysis and its Applications,
2010, vol. 44, no. 4, pp. 286-294. DOI: 10.1007/s10688-010-0039-5.
8. Huxley, M. N. Exponential Sums and Lattice Points III,
Proceedings of the London Mathematical Society,
2003, vol. 87, no. 3, pp. 591-609. DOI: 10.1112/S0024611503014485.
9. Jacobi, Ñ. G. J. Fundamenta Nova Theoriae Functionum Ellipticarum,
Sumtibus fratrumBorntraeger, 1829.
10. Bagis, N. D. and Glasser, M. L. On the Number of Representation
of Integers into Quadratic Forms, 2014, arXiv:1406.0466v5 [math.GM].
11. Bukhshtab, A. A. Teoriya chisel [The Number Theory], Moscow,
Prosveshchenie, 1960. (in Russian).
12. Leng, S. Vvedenie v Teoriyu Diofantovykh Priblizheniy
[Introduction to the Theory of Diophantine Approximations],
Moscow, Mir, 1970, 102 p. (in Russian).
13. Shmidt, V. Diofantovy priblizheniya
[Diophantine Approximations], Moscow, Mir, 1983. (in Russian).
14. Nowak, W. G. Primitive Lattice Points Inside an Ellipse,
Czechoslovak Mathematical Journal, 2005, vol. 55, no. 2, pp. 519-530.
DOI: 10.1007/s10587-005-0043-8.
15. Bleher, P. On the Distribution of the Number of Lattice Points Inside a Family of Convex Ovals,
Duke Mathematical Journal, 1992, vol. 67, no. 3, pp. 461-481. DOI: 10.1215/S0012-7094-92-06718-4.
16. Nowak, W. G. On the Mean Lattice Point Discrepancy of a Convex Disc,
Archiv der Mathematik, 2002, vol. 78, no. 3, pp. 241-248. DOI: 10.1007/s00013-002-8242-0.
17. Kratzel, E. Lattice Points in Planar Convex Domains,
Monatshefte fur Mathematik, 2004, vol. 143, no. 2, pp. 145-162.
DOI: 10.1007/s00605-003-0146-y.
18. Hooley, C. Applications of Sieve of Methods to the Theory of Numbers,
London, Cambridge University Press, 1976, 136 p.
19. Hardy, G. H. On the Expression of a Number as the Sum of Two Squares,
Quarterly Journal of Mathematics, 1915, vol. 46, pp. 263-283.
20. Voytitsky, V. I. On Multiplicities and Asymptotic of Eigenvalues for Dirichlet
and Neumann Problems for the Laplace Operator in a Rectangle,
International Conference Dedicated to I. G. Petrovskiy: Book of Abstracts,
Moscow, MSU, 2021, pp. 195-197 (in Russian).
