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Nikolay Filonov

Publications and source records attributed to Nikolay Filonov.

12 recordsLinked to original sources
arXiv2026

P\'{o}lya's conjecture for higher-dimensional Neumann balls

We prove P\'olya's conjecture for the Neumann eigenvalues of the Laplacian on Euclidean balls in dimensions three and higher. The proof further develops the approach introduced in our earlier work on the two-dimensional case and on Dirichlet eigenvalues in arbitrary dimensions. The main difficulty in the higher dimensional Neumann case is that one has to estimate zeros of the derivatives of ultraspherical Bessel functions, rather than of the usual Bessel functions. For low-lying eigenvalues, we use variational estimates involving dimension-dependent test functions, which is a novel ingredient allowing us to control a larger dimension-scaled frequency range. Other components of the proof include phase-function bounds, lattice-point counting techniques, and computer-assisted arguments.

math.SP↗
arXiv2025

P\'{o}lya's conjecture for Dirichlet eigenvalues of annuli

We prove P\'olya's conjecture for the eigenvalues of the Dirichlet Laplacian on annular domains. Our approach builds upon and extends the methods we previously developed for disks and balls. It combines variational bounds, estimates of Bessel phase functions, refined lattice point counting techniques, and a rigorous computer-assisted analysis. As a by-product, we also derive a two-term upper bound for the Dirichlet eigenvalue counting function of the disk, improving upon P\'olya's original estimate.

math.SP↗
arXiv2024

Uniform enclosures for the phase and zeros of Bessel functions and their derivatives

We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.

math.CA↗
arXiv2022

On spectral bands of discrete periodic operators

We consider discrete periodic operator on $\mathbb Z^d$ with respect to lattices $\Gamma\subset\mathbb Z^d$ of full rank. We describe the class of lattices $\Gamma$ for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed $d-2$.

math.SP↗
arXiv2022

P\'olya's conjecture for Euclidean balls

The celebrated P\'{o}lya's conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl's asymptotics. P\'{o}lya's conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove P\'{o}lya's conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm P\'{o}lya's conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.

math.SP↗
arXiv2012

Polynomials of almost-normal arguments in $C^*$-algebras

The functional calculus for normal elements in $C^*$-algebras is an important tool of analysis. We consider polynomials $p(a,a^*)$ for elements $a$ with small self-commutator norm $\|[a,a^*]\| \le δ$ and show that many properties of the functional calculus are retained modulo an error of order $δ$.

math.OA↗
arXiv2010

A Hilbert-Schmidt analog of Huaxin Lin's Theorem

The paper is devoted to the following question: consider two self-adjoint $n\times n$-matrices $H_1,H_2$, $\|H_1\|\le 1$, $\|H_2\|\le 1$, such that their commutator $[H_1,H_2]$ is small in some sence. Do there exist such self-adjoint commuting matrices $A_1,A_2$, such that $A_i$ is close to $H_i$, $i=1,2$? The answer to this question is positive if the smallness is considered with respect to the operator norm. The following result was established by Huaxin Lin: if $\|[H_1,H_2]\|=δ$, then we can choose $A_i$ such that $\|H_i-A_i\|\le C(δ)$, $i=1,2$, where $C(δ)\to 0$ as $δ\to 0$. Notice that $C(δ)$ does not depend on $n$. The proof was simplified by Friis and Rørdam. A quantitative version of the result with $C(δ)=E(1/δ)δ^{1/5}$, where $E(x)$ grows slower than any power of $x$, was recently established by Hastings. We are interested in the same question, but with respect to the normalized Hilbert-Schmidt norm. An analog of Lin's theorem for this norm was established by Hadwin and independently by Filonov and Safarov. A quantitative version with $C(δ)=12δ^{1/6}$, where $δ=\|[H_1,H_2]\|_{\tr}$, was recently obtained by Glebsky. In the present paper, we use the same ideas to prove a similar result with $C(δ)=2δ^{1/4}$. We also refine Glebsky's theorem concerning the case of $n$ operators.

math.SP↗