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Boris Mityagin

Publications and source records attributed to Boris Mityagin.

At least 19 recordsLinked to original sources
arXiv2026

Eigensystems of Dirac operators with singular potentials

We perturb one-dimensional Dirac operators on a bounded interval subject to Dirichlet boundary conditions by potentials with Fourier coefficients exhibiting power-decay. As a consequence of Paley-Zygmund theorem, this broad family of potentials comprises distributions that are neither integrable functions nor measures. We localize the spectrum of the perturbed operator and, as the main result, show that its eigensystem generates a Riesz basis.

math.SP↗
arXiv2026

Estimates of $\Gamma$-functions and their $\log{}$-derivatives

We provide a lower bound for a weighted ratio of $\Gamma-$functions and its $\log{}-$derivative. Then we apply this result to estimate the norm of the operator associated with the Ces\'{a}ro operator in the Hardy space $H^p({\mathbb C}_+)$, $1<p<\infty$

math.CV↗
arXiv2026

Riesz property in the case of multiple eigenvalues

We analyze spectra and the Riesz property of spectral projections of non-symmetric perturbations of self-adjoint operators with eigenvalues having arbitrary multiplicities, including infinite ones. In particular, we establish the Riesz property for perturbations of the multi-dimensional harmonic oscillator, Landau Hamiltonian and Laplace-Beltrami operator on a sphere by complex-valued $L^r$-potentials if $d/2 < r < \infty$.

math.SP↗
arXiv2026

Spectral projections of an anharmonic oscillator with complex polynomial potential

For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x^b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections $\{P_n\}_n$ of an anharmonic operator $L = - (\mathrm{d}/ \mathrm{d}x)^2 + V(x)$ does not generate a (Riesz) basis in $L^2(\mathbb R)$ if $a - 1 < b < 2a$. Moreover, for $\sigma = [b - (a - 1)]/(1 + a)$ and $\gamma > 0$ small enough, $\limsup_n \|P_n\|/ \exp(\gamma n^\sigma) = \infty$. Proofs are based on two groups of results which are of great interest on their own: (a) relationship between behavior (growth) of the norms of projections $\|P_n\|$ and of the resolvent $\|(z - L)^{-1}\|$ outside of the spectrum $\sigma(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity.

math.SP↗
arXiv2024

Local form subordination without a power decay and a criterion of Riesz basesness

We revisit the local form subordination condition on the perturbation of a self-adjoint operators with compact resolvent, which is used to show the Riesz basis property of the eigensystem of the perturbed operator. Our new assumptions and new proof allow for establishing the Riesz basis property also in the case of slow and non-monotone decay in this subordination condition.

math.SP↗
arXiv2020

The shifted harmonic oscillator and the hypoelliptic Laplacian on the circle

We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half-plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This relies on a novel integral formula for the spectral projections which also gives asymptotics for Laguerre polynomials in a large-parameter regime.

math.SP↗
arXiv2020

Concentration of eigenfunctions of Schroedinger operators

We consider the limit measures induced by the rescaled eigenfunctions of single-well Schrödinger operators. We show that the limit measure is supported on $[-1,1]$ and with the density proportional to $(1-|x|^β)^{-1/2}$ when the non-perturbed potential resembles $|x|^β$, $β>0$, for large $x$, and with the uniform density for super-polynomially growing potentials. We compare these results to analogous results in orthogonal polynomials and semiclassical defect measures.

math.SP↗
arXiv2016

Systems of Dilated Functions: completeness, minimality, basisness

We discuss completeness, minimality, and basisness, in $L^2[0, π]$ and $L^p[0, π]$, $p \neq 2$, of dilated systems $u_n(x) = S(nx)$, $n \in \mathbb{N}$, where $S$ is a trigonometric polynomial $S(x) = \sum_{k = 0}^m a_k \sin(kx), \quad a_0 a_m \neq 0.$ We will present some results and mention a few unsolved questions.

math.CA↗
arXiv2016

Local form-subordination condition and Riesz basisness of root systems

We exploit the so called form-local subordination in the analysis of non-symmetric perturbations of unbounded self-adjoint operators with isolated simple positive eigenvalues. If the proper condition relating the size of gaps between the unperturbed eigenvalues and the strength of perturbation, measured by the form-local subordination, is satisfied, the root system of the perturbed operator contains a Riesz basis and usual asymptotic formulas for perturbed eigenvalues and eigenvectors hold. The power of the abstract perturbation results is demonstrated particularly on Schrödinger operators with possibly unbounded or singular complex potential perturbations.

math.SP↗
arXiv2015

Root system of singular perturbations of the harmonic oscillator type operators

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space H, i.e. of the self-adjoint operator with simple positive eigenvalues $μ_k$ satisfying $μ_{k+1}-μ_k \geq Δ>0$. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system forms a Riesz basis.

math.SP↗
arXiv2014

The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions

We consider the operator $ L = - (d/dx)^2 + x^2 y + w(x) y , y \in L^2(\mathbb{R}) $, where $ w(x) = s [ δ(x - b) - δ(x + b)], b \neq 0,$ real, $s \in \mathbb{C}$. This operator has a discrete spectrum: eventually the eigenvalues are simple and $λ_n = (2n + 1) + s^2 (κ(n) / n) + ρ(n)$, where $ κ(n) = \frac{1}{2π} [(-1)^{n + 1} \sin ( 2 b \sqrt{2n} ) - \frac{1}{2} \sin ( 4 b \sqrt{2n} ) ]$ and $ |ρ(n) | \leq C (\log n) / (n^{3/2})$ If $s = i γ$, $γ$ real, the number $T(γ)$ of non-real eigenvalues is finite, and $T(γ) \leq [ C (1 + | γ|) \log (e + | γ|)]^2.$ The analogue of the above equations is given in the case of any two-point interaction perturbation $w(x) = c_+ δ(x - b) + c_- δ(x + b), c_+, c_- \in \mathbb{C}. $

math.SP↗
arXiv2014

Riesz basis property of Hill operators with potentials in weighted spaces

Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,π]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We describe classes of complex potentials $v(x)= \sum_{2\mathbb{Z}} V(k) e^{ikx}$ in weighted spaces (defined in terms of the Fourier coefficients of $v$) such that the periodic (or antiperiodic) root function system of $L(v) $ contains a Riesz basis if and only if $$V(-2n) \asymp V(2n) \quad \text{as} \;\; n \in 2\mathbb{N}\;\; (\text{or} \; n \in 1+ 2\mathbb{N}), \;\; n \to \infty.$$ For such potentials we prove that $λ_n^+ - λ_n^- \sim \pm 2\sqrt{V(-2n)V(2n)} $ and $$μ_n - \frac{1}{2}(λ_n^+ + λ_n^-) \sim -\frac{1}{2} (V(-2n) + V(2n)).$$

math.SP↗
arXiv2013

Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators

Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,π].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \; n\in \mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $γ_n = λ_n^+ - λ_n^-$ and deviations $ δ_n =μ_n - λ_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, for special potentials that are trigonometric polynomials we provide precise asymptotics of $γ_n$ and $δ_n.$

math.SP↗
arXiv2012

Divergence of spectral decompositions of Hill operators with two exponential term potentials

We consider the Hill operator $$ Ly = - y^{\prime \prime} + v(x)y, \quad 0 \leq x \leq π, $$ subject to periodic or antiperiodic boundary conditions ($bc$) with potentials of the form $$ v(x) = a e^{-2irx} + b e^{2isx}, \quad a, b \neq 0, r,s \in \mathbb{N}, r\neq s. $$ It is shown that the system of root functions does not contain a basis in $L^2 ([0,π], \mathbb{C})$ if $bc$ are periodic or if $bc$ are antiperiodic and $r, s$ are odd or $r=1$ and $s \geq 3. $

math.SP↗