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Sergey Simonov

Publications and source records attributed to Sergey Simonov.

16 recordsLinked to original sources
arXiv2025

Smoothness of solutions to the initial-boundary value problem for the telegraph equation on the half-line with a locally summable potential

We study solutions to the system $u_{tt}-u_{xx}+q(x)u=0, x>0,t>0$; $u|_{t=0}=u_t|_{t=0}=0, x>0$; $u|_{x=0}=g(t), t>0$, with a locally summable Hermitian matrix-valued potential $q$ and a $C^{\infty}$-smooth $\mathbb C^n$-valued boundary control $g$ vanishing near the origin. We prove that the solution $u^{g}(\cdot,T)$ is a function from $W^2_1([0,T];\mathbb C^n)$ and that the control operator $W^T:g\mapsto u^g(\cdot,T)$ is an isomorphism in $L_2([0,T];\mathbb C^n)$, and, in the case that $q$ is from $L_2([0,T];\mathbb C^n)$, also an isomorphism in $H^2([0,T];\mathbb C^n)$.

math.AP↗
arXiv2023

Spectral multiplicity of selfadjoint Schroedinger operators on star-graphs with general interface conditions

We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schrödinger operator on a star-graph with a finite number of finite or infinite edges and an interface condition at the common vertex. A wide class of "selfadjoint" interface conditions, subject to a assumption which is generically satisfied, is considered. We investigate properties of spectral multiplicity of singular spectrum (continuous as well as point) in terms of the spectral data of decoupled operators.

math.SP↗
arXiv2019

The wave model of the Sturm-Liouville operator on an interval

In the paper we construct the wave functional model of a symmetric restriction of the regular Sturm-Liouville operator on an interval. The model is based upon the notion of the wave spectrum and is constructed according to an abstract scheme which was proposed earlier. The result of the construction is a differential operator of the second order on an interval, which differs from the original operator only by a simple transformation.

math-ph↗
arXiv2018

Wave model of the regular Sturm-Liouville operator

We describe the wave functional model for the minimal (symmetric) Sturm-Liouville operator on the finite interval. We construct the wave spectrum of this operator, then, following the abstract scheme, we construct the model space of functions on the wave spectrum and introduce in that space the model operator. The latter is a matrix Sturm-Liouville operator which is unitarily equivalent to the original.

math-ph↗
arXiv2016

Zeroes of the spectral density of the Schroedinger operator with the slowly decaying Wigner-von Neumann potential

We consider the Schrödinger operator $\mathcal L_α$ on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner--von Neumann potential $\frac{c\sin(2ωx+δ)}{x^γ}$, where $γ\in(\frac12,1)$. The continuous spectrum of this operator has the same band-gap structure as the continuous spectrum of the unperturbed periodic operator. In every band there exist two points, called critical, where the eigenfunction equation has square summable solutions. Every critical point $ν_{cr}$ is an eigenvalue of the operator $\mathcal L_α$ for some value of the boundary parameter $α=α_{cr}$, specific to that particular point. We prove that for $α\neqα_{cr}$ the spectral density of the operator $\mathcal L_α$ has a zero of the exponential type at $ν_{cr}$.

math.SP↗
arXiv2014

Spectral analysis of the half-line Kronig-Penney model with Wigner-von Neumann perturbations

The spectrum of the self-adjoint Schrödinger operator associated with the Kronig-Penney model on the half-line has a band-gap structure: its absolutely continuous spectrum consists of intervals (bands) separated by gaps. We show that if one changes strengths of interactions or locations of interaction centers by adding an oscillating and slowly decaying sequence which resembles the classical Wigner-von Neumann potential, then this structure of the absolutely continuous spectrum is preserved. At the same time in each spectral band precisely two critical points appear. At these points "instable" embedded eigenvalues may exist. We obtain locations of the critical points and discuss for each of them the possibility of an embedded eigenvalue to appear. We also show that the spectrum in gaps remains discrete.

math.SP↗
arXiv2013

Superconducting fluctuations in organic molecular metals enhanced by Mott criticality

Unconventional superconductivity typically occurs in materials in which a small change of a parameter such as bandwidth or doping leads to antiferromagnetic or Mott insulating phases. As such competing phases are approached, the properties of the superconductor often become increasingly exotic. For example, in organic superconductors and underdoped high-$T_\mathrm{c}$ cuprate superconductors a fluctuating superconducting state persists to temperatures significantly above $T_\mathrm{c}$. By studying alloys of quasi-two-dimensional organic molecular metals in the $κ$-(BEDT-TTF)$_2$X family, we reveal how the Nernst effect, a sensitive probe of superconducting phase fluctuations, evolves in the regime of extreme Mott criticality. We find strong evidence that, as the phase diagram is traversed through superconductivity towards the Mott state, the temperature scale for superconducting fluctuations increases dramatically, eventually approaching the temperature at which quasiparticles become identifiable at all.

cond-mat.str-el↗
arXiv2012

Spectral multiplicity of selfadjoint Schroedinger operators on star-graphs with standard interface conditions

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schroedinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of Theorem of I.S. Kac.

math.SP↗
arXiv2012

Zeroes of the spectral density of discrete Schroedinger operator with Wigner-von Neumann potential

We consider a discrete Schroedinger operator whose potential is the sum of a Wigner-von Neumann term and a summable term. The essential spectrum of this operator equals to the interval [-2,2]. Inside this interval, there are two critical points where eigenvalues may be situated. We prove that, generically, the spectral density of the operator has zeroes of the power type at these points.

math.SP↗
arXiv2011

Zeroes of the spectral density of the periodic Schroedinger operator with Wigner-von Neumann potential

We consider the Schroedinger operator L_α on the half-line with a periodic background potential and the Wigner-von Neumann potential of Coulomb type: csin(2ωx+d)/(x+1). It is known that the continuous spectrum of the operator L_α has the same band-gap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (so-called critical or resonance) where the operator L_α has a subordinate solution, which can be either an eigenvalue or a `half-bound' state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator L_α has power-like zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the above-mentioned sense.

math.SP↗
arXiv2010

Weyl-Titchmarsh type formula for Hermite operator with small perturbation

Small perturbations of the Jacobi matrix with weights \sqrt n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical Weyl-Titchmarsh formula for the Schroedinger operator on the half-line with summable potential. Additionally a base of generalized eigenvectors for "free" Hermite operator is studied and asymptotics of Plancherel-Rotach type are obtained.

math.SP↗
arXiv2010

An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights

We consider self-adjoint unbounded Jacobi matrices with diagonal q_n=n and weights λ_n=c_n n, where c_n is a 2-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum is either purely absolutely continuous or discrete. This constitutes an example of the spectral phase transition of the first order. We study the lines where the spectral phase transition occurs, obtaining the following main result: either the interval (-\infty;1/2) or the interval (1/2;+\infty) is covered by the absolutely continuous spectrum, the remainder of the spectrum being pure point. The proof is based on finding asymptotics of generalized eigenvectors via the Birkhoff-Adams Theorem. We also consider the degenerate case, which constitutes yet another example of the spectral phase transition.

math.SP↗
arXiv2010

Spectral analysis of a class of hermitian Jacobi matrices in a critical (double root) hyperbolic case

We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic ("double root") situation. For the model with "non-smooth" matrix entries we obtain the asymptotics of generalized eigenvectors and analyze the spectrum. In addition, we reformulate a very helpful theorem from a paper of Janas and Moszynski in its full generality in order to serve the needs of our method.

math.SP↗