# Alexander V. KiselevUniversity of Bath | UB · Department of Mathematical Sciences

Alexander V. Kiselev

PhD

## About

44

Publications

2,534

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283

Citations

Introduction

At present, my research interests have somewhat shifted from the non-self-adjoint spectral theory to various types of inverse spectral problems on quantum graphs and scattering theory approach to both metamaterials and the theory of homogenization.

Additional affiliations

September 2021 - March 2022

May 2021 - March 2022

August 2019 - August 2020

Education

June 1997 - June 2000

September 1994 - December 1996

September 1990 - May 1994

## Publications

Publications (44)

Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum
are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple.
Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy–Foiaş model...

A novel approach to critical contrast homogenisation is proposed. This allows to see homogenisation limits restricted to the "stiff" component of the composite as a class of time-dispersive media. By an inversion of this argument, we also offer a recipe for the construction of such media with prescribed dispersive properties from periodic composite...

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homoge...

The link to public view-only published version (as provided by Springer): https://rdcu.be/cMbxv

We present a mathematically rigorous procedure for obtaining spectrograms of tectonic plate eigenmodes at extremely low frequencies, corresponding to oscillation periods of 15 minutes to 8 hours. The data is sourced from superconducting gravimeters of International Geodynamics and Earth Tide Service (IGETS) network. The motivation for this research...

For the system of equations of linear elasticity with periodic coefficients displaying high contrast, in the regime of resonant scaling between the material properties of the ``soft" part of the medium and the spatial period, we construct an asymptotic approximation exhibiting time-dispersive properties of the medium and prove associated order-shar...

We give an overview of operator-theoretic tools that have recently proved useful in the analysis of boundary-value and transmission problems for second-order partial differential equations, with a view to addressing, in particular, the asymptotic behaviour of resolvents of physically motivated parameter-dependent operator families. We demonstrate t...

The paper surveys the area of functional models for dissipative and non-dissipative operators, and in particular the contributions made in this area by Sergey Naboko, to include: an explicit model construction, spectral analysis of the absolutely continuous subspace, the functional model approach to the scattering theory, and the work on the singul...

We consider an $\varepsilon$-periodic ($\varepsilon\to 0$) tubular structure, modelled as a magnetic Laplacian on a metric graph, which is periodic along a single axis. We show that the corresponding Hamiltonian admits norm-resolvent convergence to an ODE on $\mathbb{R}$ which is fourth order at a discrete set of values of the magnetic potential (\...

We present a mathematically rigorous procedure to obtain spectrograms of tectonic plate eigenmodes in an extremely low-frequency band, corresponding to oscillation periods of 15 min to 8 hours. The data is sourced from superconducting gravimeters (IGETS network). The motivation for this research stems from the proposal by Pavlov et al to use this s...

Norm-resolvent convergence with order-sharp error estimate is established for Neumann Laplacians on thin domains in $\mathbb{R}^2$ and $\mathbb{R}^3$, converging to metric graphs in the limit of vanishing thickness parameter in the resonant case.

This is an overview of mathematical heritage of Sergey Naboko in the area of functional models of non-self-adjoint operators. It covers the works by Sergey in model construction, the analysis of absolutely continuous and singular spectra and the construction of the scattering theory in model terms.

We give an overview of operator-theoretic tools that have recently proved useful in the analysis of boundary-value and transmission problems for second-order partial differential equations, with a view to addressing, in particular, the asymptotic behaviour of resolvents of physically motivated parameter-dependent operator families. We demonstrate t...

С использованием обобщения понятия классической вейлевской $m$-функции, а также связанных с ним формул для резольвент операторов краевых задач изучается асимптотическое поведение решений "задачи трансмиссии" для высококонтрастного включения в непрерывной среде. Для указанной задачи показана сходимость по норме разности резольвент к предельной задач...

Using a generalisation of the classical notion of Dirichlet-to-Neumann map and the related formulae for the resolvents of boundary-value problems, we analyse the asymptotic behaviour of solutions to a "transmission problem" for a high-contrast inclusion in a continuous medium, for which we prove the operator-norm resolvent convergence to a limit pr...

We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We provide explicit formulae for the resolvents of the associated extensions of symmetric operators in terms of the associated generalised Dirichlet-to-Neumann maps, which can be utilised in the analysis of the...

A novel approach to critical-contrast homogenisation is proposed. Norm-resolvent asymptotics are explicitly constructed. An essential feature of our approach is that it relates homogenisation limits to a class of time-dispersive media. References: 50 entries. UDK: 517.98. MSC2010: Primary 34E13; Secondary 34E05, 35P20, 47A20, 81Q35.

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed. Norm-resolvent asymptotics are explicitly constructed. An essential feature of our approach is that it relates homogenisation limits to a class of time-dispersive media.

A novel approach to critical-contrast homogenisation is proposed. Norm-resolvent asymptotics are explicitly constructed. An essential feature of our approach is that it relates homogenisation limits to a class of time-dispersive media.

This work deals with the functional model for extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator...

This work deals with the functional model for extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator...

We prove operator-norm resolvent convergence estimates for one-dimensional
periodic differential operators with rapidly oscillating coefficients in the
non-uniformly elliptic high-contrast setting, which has been out of reach of
the existing homogenisation techniques. Our asymptotic analysis is based on a
special representation of the resolvent of...

Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. An infinite series of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrodin...

Laplacian operators on finite compact metric graphs are considered under the
assumption that matching conditions at graph vertices are of $\delta$ and
$\delta'$ types. An infinite series of trace formulae is obtained which link
together two different quantum graphs under the assumption that their spectra
coincide. The general case of graph Schrodin...

Laplace operators on finite compact metric graphs are considered under the
assumption that matching conditions at graph vertices are of $\delta$ and
$\delta'$ types. Assuming rational independence of edge lengths, necessary and
sufficient conditions of isospectrality of two Laplacians defined on the same
graph are derived and scrutinized. It is pro...

Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of type. Under one additional assumption, the inverse topology problem is treated. Using the apparatus of boundary triples, we generalize and extend existing results on necessary conditions of isospectrality of two...

The paper is a continuation of the study started in our previous joint paper (MFAT vol. 18(4), 2012).
Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at
the graph vertices are of $\delta$ type.
Either an infinite series of trace formulae (provided that edge potentials are infini...

Graph Laplacians on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of either $\delta$ or $\delta'$ type.
In either case, an infinite series of trace formulae
which link together two different graph Laplacians provided that their spectra coincide is derived.
Applications are gi...

We prove an analogue to the Cayley identity for an arbitrary self-adjoint
operator in a Hilbert space. We also provide two new ways to characterize
vectors belonging to the singular spectral subspace in terms of the analytic
properties of the resolvent of the operator, computed on these vectors. The
latter are analogous to those used routinely in t...

The similarity problem for restrictions of a non-selfadjoint operator with absolutely continuous spectrum to its spectral
subspaces corresponding to arbitrary Borel subsets d of the spectrum is considered, generalizing the results of [7, 11]. Necessary and sufficient conditions of such similarity
are obtained in the form of a pair of integral estim...

The similarity problem for non-self-adjoint extensions of a symmetric operator having equal deficiency indices is studied.
Necessary and sufficient conditions for a wide class of such operators to be similar to self-adjoint ones are obtained. The
paper is based on the construction of functional model for the operators of the class considered due to...

We discuss the definition of a rank one singular perturbation of a non-self-adjoint operator L in Hilbert space H. Provided that the operator L is a non-self-adjoint perturbation of a self-adjoint operator A and that the spectrum of the operator L is absolutely continuous we are able to establish a concise resolvent formula for the singular perturb...

Nonself-adjoint, nondissipative perturbations of bounded self-adjoint operators with real purely singular spectrum are considered. Using a functional model of a nonself-adjoint operator as a principal tool, spectral properties of such operators are investigated. In particular, in the case of rank two perturbations the pure point spectral component...

A non-self-adjoint, rank-one Friedrichs model operator in L 2 (R) is considered in the case where the determinant of perturbation is an outer function in the half-planes C ± . Its spectral structure is investigated. The impact of the linear resolvent growth condition on its spectral properties (including the similarity problem) is studied.

We consider nonself-adjoint nondissipative trace class additive perturbations L=A+iV of a bounded self-adjoint operator A in a Hilbert space ,H. The main goal is to study the properties of the singular spectral subspace N
i
0 of L corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint no...

C
0
-groups of Hilbert space operators with polynomial and more general functional growth are considered. We investigate how the polynomial growth condition can be expressed in terms of integral estimates for the resolvent of the group generator. In the case of C
0
-groups with functional growth such necessary and sufficient conditions are given wh...

The similarity problem for one-dimensional non-selfadjoint perturbations of the operator of multiplication by an independent variable in the space L 2 (ℝ) is considered. Some sufficient conditions for similarity of this operator to a selfadjoint one are given under some constraints on the functions defining the perturbation. In particular, under th...

The similarity to a self-adjoint operator for a class of non-selfadjoint operators is considered, namely, for the one-dimensional perturbations of the operator of multiplication by an independent variable (Au)(x)=xu(x)+ψ(x)∫ -∞ +∞ u(t)φ(t) ¯dμ(t) in the space L 2 (ℝ,μ) under the condition that φ(x)ψ(x)=0 for μ-almost all x. A criterion for this typ...

We consider nonself-adjoint, non-dissipative operators acting in a Hilbert space. Our main aim is the spectral analysis of the singular spectral subspace N 0 i , which possible presence separates non-dissipative operators from dissipative ones. In par-ticular, we single out the class of operators with almost Hermitian spectrum in which the spectral...