arXiv subjects

A. A. Shkalikov

Publications and source records attributed to A. A. Shkalikov.

At least 19 recordsLinked to original sources

arXiv2025

Asymptotics of the spectra of the Dirichlet and Dirichlet-Neumann problems for the Sturm-Liouville equation with integral perturbation

The article studies the Dirichlet and Dirichlet-Neumann problems for the Sturm-Liouville equation perturbed by an integral operator with a convolution kernel. Sharp asymptotic formulas for the eigenvalues of these problems are found. The formulas contain information about the Fourier coefficients of the potential and the kernel, and estimates are obtained for the remainder terms of the asymptotics, which take into account both the rate of decrease as the eigenvvalues tend to infinity and the rate of decrease as the norms of the potential and kernel tend to zero. The formulas are also new in the case of the Sturm-Liouville operator, when the convolution kernel is zero.

math.SP↗

arXiv2022

Spectral asymptotics for solutions of $2\times 2$ system of ordinary differential equations of the first order

The aim of the paper is to find representation for solutions of $2\times 2$ system of ordinary differential equations $$ \mathbf{y^\prime} - B(x)\mathbf{y} = λA(x)\mathbf{y}, \quad \ x \in [0, 1], $$ where $A(x) = diag\{a_1(x), a_2(x)\}$, $B(x) = (b_{ij}(x))$, $a_1(x) > 0, \ a_2(x) < 0$ and all the functions $a_{i}, b_{ij}$ belong to the Sobolev spaces $W^n_1[0,1]$ for given integer $n\geqslant 0$. We prove that there exists a fundamental matrix of solutions for the above system, which have representation $$ Y(x, λ) = M(x)\left(I + \frac{R^1(x)}λ + \dots + \frac{R^n(x)}{λ^n} + o(1)λ^{-n}\right)E(x, λ), $$ where $o(1) \to 0$ uniformly for $x\in [0,1]$ as the spectral parameter $λ\to \infty$ in the half plane $\Re\,λ>-κ$ or $\Re\,λ<κ$, where $κ$ is any fixed real number. The main novelty is that we give explicit formulae for all matrices $M,E$ and $R^m$ in this representation.

math.FA↗

arXiv2021

The Complex Airy Operator as Explicitly Solvable PT-symmetrical Model

We study the Sturm--Liouville operator $$ T(\varepsilon)y=-\frac{1}{\varepsilon}y''+ p(x)y, $$ with concrete $\mathcal{PT}$-- symmetric potential $p(x) = ix$ and Dirichlet boundary conditions on the segment $[-1,1]$. Here $\varepsilon \in (0, \infty)$ is a physical parameter. We explicitly describe a beautiful phenomenon of the eigenvalue behavior when $\varepsilon$ changes from $0$ to $\infty$. All the critical values of $\varepsilon$ which determine the eigenvalue dynamics, are found in terms of the special Airy functions.

math.SP↗

arXiv2019

Odinary differential operators of odd order with distribution coefficients

We work with differential expressions of the form \begin{align} τ_{2n+1} y &=(-1)^ni \{(q_{0}y^{(n+1)})^{(n)}+(q_{0}y^{(n)})^{(n+1)}\}+ \sum\limits_{k=0}^{n}(-1)^{n+k}(p^{(k)}_ky^{(n-k)})^{(n-k)} \\ &\qquad+i\sum\limits_{k=1}^{n}(-1)^{n+k+1}\{(q^{(k)}_{k}y^{(n+1-k)})^{(n-k)}+ (q^{(k)}_{k}y^{(n-k)})^{(n+1-k)}\}, \end{align} where the complex valued coefficients $p_j$ and $q_j$ are subject the following conditions: $ q_0(x) \in AC_{loc}(a,b)$, $Re \,q_0>0$, while all the other functions $$q_1(x),q_2(x),\ldots,q_{n}(x), p_0(x),p_1(x),\ldots,p_n(x)$$ belong to the space $L^1_{loc}(a,b)$. This implies that the coefficients $p^{(k)}_{k}$ and $q^{(k)}_{k}$ in the expression $τ_{2n+1}$ are distributions of singularity order $k$. The main objective of the paper is to represent the differential expression $τ_{2n+1}$ in the other (regularized) form which allows to define the minimal and maximal operators associated with this differential expression.

math.CA↗

arXiv2019

Operator Pencils and Half-range Problem in Operator Theory

This article can be considered as the first version of a book which the author plans to write about half-range problems in operator theory. It consists of two parts. The first part is based on lectures which the author delivered at University of Calgary and Lomonosov Moscow State University. The main attention in this part is paid to the selection of waves which are involved in the formulation of the Mandelstamm radiation principle (the eigen-pairs, corresponding to the real eigenvalues) and to the factorization problems of self-adjoint and dissipative, quadratic and polynomial operator pencils. There is a dramatic difference between finite dimensional and infinite dimensional cases. It is shown that in the finite dimensional case the factorization problems can be solved completely. In the second part we consider abstract models for concrete problems of mechanics. We demonstrate the methods how concrete problems can be represented in an abstract form. The main results concern the factorization of elliptic operator pencils satisfying the resolvent growth condition in a double sector containing the real axis and the investigation of the semi-group properties of a divisor. Using Pontrjagin space methods we obtain a criterium for the stability in the celebrated Sobolev problem about a rotating top with a cavity filled with a viscous liquid.

math.SP↗

arXiv2019

Stability in the inverse resonance problem for the Schr\" odinger operator

We work with the Schr\" odinger equation \begin{equation*} H_q y = -y'' + q(x)y = z^2y, \ x\in [0,\infty), \end{equation*} where $q\in L_1((0,\infty), xdx)$, and asssume that the corresponding operator $H_q$ is defined by the Dirihlet condition $y(0) = 0$ The function $ψ(z) = y(0,z)$ where $y(x,z)$ is the Jost solution of the above equation is analytic in the whole complex plane, provided that the support of the potential $q$ is finite. The zeros of $ψ$ are called the resonances. It is known that $q$ is uniquely determined by the sequence of resonances. Using only finitely many resonances lying in the disk $|z|\le r$ we can recover the potential $q$ with accuracy $\varepsilon(r)\to 0$ as $r \to \infty$. The main result of the paper is the estimate $\varepsilon(r) \le Cr^{-α}$ with some constants $C$ and $α>0$ which are defined by a priori information about the potential $q$.

math.SP↗

arXiv2018

Multipliers in Bessel potential spaces. The case of different sign smooth indices

The objective of this paper is to describe the space of multipliers acting from a Bessel potential space $H^s_p(\mathbb R^n)$ into another space $H^{-t}_q(\mathbb R^n)$, provided that the smooth indices of these spaces have different signs, i.e. $s, t \geqslant 0$. This space of multipliers consists of distributions $u$, such that for all $φ\in H^s_p(\mathbb R^n)$ the product $φ\cdot u$ is well-defined and belongs to the space $H^{-t}_q(\mathbb R^n)$. We succeed to describe this space explicitly, provided that $p \leqslant q$ and one of the following conditions $$ s \geqslant t \geqslant 0, \ s > n/p \ \ \, \text{or} \ \ \, t \geqslant s \geqslant 0, \ t > n/q' \quad (\: \text{where} \; 1/q +1/q' = 1), $$ holds. In this case one has $$ M[H^s_p(\mathbb{R}^n) \to H^{-t}_{q}(\mathbb{R}^n)] = H^{-t}_{q, \: unif}(\mathbb{R}^n) \cap H^{-s}_{p', \: unif}(\mathbb{R}^n), $$ where $H^γ_{r, \: unif}(\mathbb{R}^n), \: γ\in \mathbb{R}, \: r > 1$ is the scale of uniformly localized Bessel potential spaces. In particular but important case $s = t < n/\max (p,q')$ we prove two-sided continuous embeddings $$ H^{-s}_{r_1, \: unif}(\mathbb{R}^n) \subset M[H^s_p(\mathbb{R}^n) \to H^{-s}_q(\mathbb{R}^n)] \subset H^{-s}_{r_2, \: unif}(\mathbb{R}^n), $$ where $r_2 = \max (p', q), \ r_1 =[s/n-(1/p -1/q)]^{-1}$.

math.FA↗

arXiv2017

Eigenvalue Dynamics of a PT-symmetric Sturm-Liouville Operator. Criteria of the Similarity to a Self-adjoint or Normal Operator

The goal of the paper is to investigate the dynamics of the eigenvalues of the Sturm-Liouville operator with summable PT-symmetric potential on the finite interval. It turns out that the case of a complex Airy operator presents an exactly solvable model which allows us to trace the dynamics of the movement of the eigenvalues in all details and to find explicitly the critical parameter values, in particular, to specify precisely the number $\varepsilon_1$ such that for $0<\varepsilon<\varepsilon_1$ the operator has a real spectrum and is similar to a self-adjoint operator.

math.SP↗

arXiv2017

Asymptotics of the Solutions of the Sturm--Liouville Equation with Singular Coefficients

We obtain asymptotic representations as $λ\to \infty$ in the upper and lower half-planes for the solutions of the Sturm--Liouville equation $$ -y"+p(x)y'+q(x)y= λ^2 ρ(x)y, \qquad x\in [a,b] \subset \mathbb{R}, $$ under the condition that $q$ is a distribution of the first-order singularity, $ρ$ is a positive absolutely continuous function, and $p$ belongs to the space $L_2[a,b]$. In supplementary part, the results are generilized on equation of the following type $$-(r^2y')'+py'+qy=λ^2 ρ^2 y, \qquad x\in [a,b] \subset \mathbb{R},$$ where $λ^2$ is the large parameter, $r$ and $ρ$ are positive functions, while $p$ and $q$ are complex valued ones. It is assumed that $p\in L_1[a,b],\quad q\in W_2^{-1}[a,b], \quad ρ,r \in AC[a,b] =W_1^1[a,b],$ moreover, \begin{equation*} ρ'u, r'u, pu \in L_1[a,b], \quad \text{where }\, u=\int q \, dx, \end{equation*} and the antiderivative is understood in the sense of distributions.

math.SP↗

arXiv2017

Unconditional bases of subspaces related to non-self-adjoint perturbations of self-adjoint operators

Assume that $T$ is a self-adjoint operator on a Hilbert space $\mathcal{H}$ and that the spectrum of $T$ is confined in the union $\bigcup_{j\in J}Δ_j$, $J\subseteq\mathbb{Z}$, of segments $Δ_j=[α_j, β_j]\subset\mathbb{R}$ such that $α_{j+1}>β_j$ and $$ \inf_{j} \left(α_{j+1}-β_j\right) = d > 0. $$ If $B$ is a bounded (in general non-self-adjoint) perturbation of $T$ with $\|B\|=:b<d/2$ then the spectrum of the perturbed operator $A=T+B$ lies in the union $\bigcup_{j\in J} U_{b}(Δ_j)$ of the mutually disjoint closed $b$-neighborhoods $U_{b}(Δ_j)$ of the segments $Δ_j$ in $\mathbb{C}$. Let $Q_j$ be the Riesz projection onto the invariant subspace of $A$ corresponding to the part of the spectrum of $A$ lying in $U_{b}\left(Δ_j\right)$, $j\in J$. Our main result is as follows: The subspaces $\mathcal{L}_j=Q_j(\mathcal H)$, $j\in J$, form an unconditional basis in the whole space $\mathcal H$.

math.SP↗

arXiv2016

The Limit Spectral Graph in the Semi-Classical Approximation for the Sturm-Liouville Problem With a Complex Polynomial Potential

The limit distribution of the discrete spectrum of the Sturm-Liouville problem with complex-valued polynomial potential on an interval, on a half-axis, and on the entire axis is studied. It is shown that at large parameter values, the eigenvalues are concentrated along the so-called limit spectral graph; the curves forming this graph are classified. Asymptotics of eigenvalues along curves of various types in the graph are calculated.

math.SP↗

arXiv2016

The estimation of the ratio of two entire functions with the same zeros in the ball

The paper studies entire functions of finite order of growth for which a representation of the form $ψ(z) = 1+ O(|z|^{-μ}), μ>0,$ as $z\to \infty$, is valid on a fixed ray of the complex plane. The main result is the following. Assume that the zeros of two functions $ψ_1, ψ_2$ of this class coincide in the circle of radius $R$ with the center in zero. Then given arbitrary small $δ\in (0,1)$ and $\varepsilon >0$ the relation of these functions admits the estimate $|ψ_1(z)/ψ_2(z) -1| \leqslant \varepsilon R^{-μ(1-δ)}$ for all $|z|\leqslant R^{1-δ}$, provided that $R\geqslant R_0$ and $R_0 =R_0(\varepsilon, δ)$ is sufficiently large. This result is of considerable interest in the analysis of the stability in the inverse resonance problem for the Schroedinger equation.

math.CV↗

arXiv2013

On the Interpolation of Analytic Maps

Let (E_0,E_1) and (H_0,H_1) be a pair of Banach spaces with dense and continuous embeddings E_1 into E_0, H_1 into H_0. For $θ\in [0,1]$ denote by $B_θ(0,R)$ the ball of radius R centered at zero in the interpolation spaces E_θ. Assume that an analytic map $Φ$ maps the ball B_0(0,R) into H_0, $Φ$ maps B_1(0,R) into H_1 and for $θ=0,1$ the estimates $$ \|Φ(x)\|_{H_θ} \le C_θ\|x\|_{H_θ}, \forall\ x\in B_θ(0,R), $$ hold. Then for all $θ\in(0, 1)$ and r<R $Φ$ maps the ball $B_θ(0,r)$ into $H_θ$ and the same estimate holds for $x\in B_θ(0,r)$ if the constant $C_θ$ is replaced by $C_0^{1-θ}C_1^θR/(R-r)$.

math.FA↗

arXiv2012

Eigenvalue Asymptotics of Perturbed Self-adjoint Operators

We study perturbations of a self-adjoint positive operator $T$, provided that a perturbation operator $B$ satisfies "local" subordinate condition $\|Bφ_k\|\leqslant bμ_k^β$ with some $β<1$ and $b>0$. Here $\{φ_k\}_{k=1}^\infty$ is an orthonormal system of the eigenvectors of the operator $T$ corresponding to the eigenvalues $\{μ_k\}_{k=1}^\infty$. We introduce the concept of $α$-non-condensing sequence and prove the theorem on the comparison of the eigenvalue-counting functions of the operators $T$ and $T+B$. Namely, it is shown that if $\{μ_k\}$ is $α-$non-condensing then the difference of the eigenvalue-counting functions is subject to relation $$|n(r,\, T)- n(r,\, T+B)| \leqslant C[n(r+ar^γ,\, T) - n(r-ar^γ,\, T)] +C_1 $$ with some constants $C, C_1, a$ and $γ= \max(0, β, 2β+α-1)\in [0,1)$.

math.SP↗

arXiv2011

Spectral and oscillation properties for a linear pencil of fourth-order differential operators

The present paper deals with the spectral and the oscillation properties of a linear pencil $A-λB$. Here $A$ and $B$ are linear operators generated by the differential expressions $(py")"$ and $-y"+ cry$, respectively. In particular, it is shown that the negative eigenvalues of this problem are simple and the corresponding eigenfunctions $y_{-n}$ have $n-1$ zeros in $(0,1)$.

math.SP↗

arXiv2010

Inverse problems for Sturm--Liouville operators with potentials from Sobolev spaces. Uniform stability

The paper deals with two inverse problems for Sturm--Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,π]$. The first one is the problem of recovering of a potential by two spectra. We associate with this problem the map $F:\, W^θ_2\to l_B^θ,\ F(σ) =\{s_k\}_1^\infty$, where $W^θ_2 = W^θ_2[0,π]$ are Sobolev spaces with $θ\geqslant 0$, $σ=\int q$ is a primitive of the potential $q$ and $l_B^θ$ are special Hilbert spaces which we construct to place in the regularized spectral data $\bold s = \{s_k\}_1^\infty$. The properties of the map $F$ are studied in details. The main result is the theorem on uniform stability. It gives uniform estimates from above and below of the norm of the difference $\|σ-σ_1\|_θ$ by the norm of the difference of the regularized spectral data $\|\bold s -\bold s_1\|_θ$ where the last norm is taken in $l_B^θ$. A similar result is obtained for the second inverse problem when the potential is recovered by the spectral function of the operator $L$ generated by Dirichlet boundary conditions. The results are new for classical case $q\in L_2$ which corresponds to the value $θ=1$.

math.SP↗

arXiv2010

Properties of the map associated with recovering of the Sturm-Liouville operator by its spectral function. Uniform stability in the scale of Sobolev spaces

Denote by $L_D$ the Sturm-Liouville operator $Ly=-y" +q(x)y$ on the finite interval $[0,π]$ with Dirichlet boundary conditions $y(0)=y(π)=0$. Let $\{λ_k\}_1^\infty$ and $\{α_k\}_1^\infty$ be the sequences of the eigenvalues and norming constants of this operator. For all $θ\geqslant 0$ we study the map $F: W_2^θ \to l_D^θ$ defined by $F(σ) =\{s_k\}_1^\infty$. Here $σ= \int q $ is the primitive of $q$, $\bold s = \{s_k\}_1^\infty$ be regularized spectral data defined by $s_{2k} =\sqrt{λ_k}-k,\ s_{2k-1}=α_k-π/2$ and $l_D^θ$ are special Hilbert spaces which are constructed in the paper as finite dimensional extensions of the usual weighted $l_2$ spaces. We give a complete characterization of the image of this nonlinear operator, show that it is locally invertible analytic map, find explicit form of its Frechet derivative. The main result of the paper are the uniform estimates of the form $\|σ-σ_1\|_θ\asymp \|\bold s -\bold s_1\|_θ$, provided that the spectral data $\bold s$ and $\bold s_1$ run through special convex sets in the spaces $l_D^θ$.

math.SP↗

arXiv2008

On the Riesz Basis Property of the Eigen- and Associated Functions of Periodic and Antiperiodic Sturm-Liouville Problems

The paper deals with the Sturm-Liouville operator $$ Ly=-y^{\prime\prime}+q(x)y,\qquad x\in\lbrack0,1], $$ generated in the space $L_{2}=L_{2}[0,1]$ by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator $L$ are proved. One of the main results is the following. \textsl{Let $q$ belong to Sobolev space $W_{1}^{p}[0,1]$ with some integer $p\geq0$ and satisfy the conditions $q^{(k)}(0)=q^{(k)}(1)=0$ for $0\leq k\leq s-1$, where s}$\leq p.$ \textsl{Let the functions $Q$ and $S$ be defined by the equalities $Q(x)=\int_{0}^{x}q(t) dt, S(x)=Q^{2}(x)$ and let $q_{n}%, Q_{n},S_{n}$ be the Fourier coefficients of $q,Q,S$ with respect to the trigonometric system $\{e^{2πinx}\}_{-\infty}^{\infty}$. Assume that the sequence $q_{2n}-S_{2n}+2Q_{0}Q_{2n}$ decreases not faster than the powers $n^{-s-2}$. Then the system of eigen and associated functions of the operator $L$ generated by periodic boundary conditions forms a Riesz basis in the space $L_{2}[0,1]$ (provided that the eigenfunctions are normalized) if and only if the condition $$ q_{2n}-S_{2n}+Q_{0}Q_{2n}\asymp q_{-2n}-S_{-2n}+2Q_{0}Q_{-2n},\quad n>1, $$ holds.

math.SP↗