Scattering theory for a class of non-selfadjoint extensions of symmetric operators

Cherednichenko, Kirill; Kiselev, Alexander V.; Silva, Luı́s O.

doi:10.1007/978-3-030-31531-3_14

Cited by 7 publications

(18 citation statements)

References 24 publications

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“…Henceforth in this section we follow closely the analysis of the named two papers, facilitated by the fact that essentially this way to construct the functional model only relies upon the characteristic function S of the maximal dissipative operator and an estimate of the type claimed in Lemma 4.1 above. A similar argument for extensions of symmetric operators, based on the theory of boundary triples, was developed in[51],[21].…”

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confidence: 92%

“…The present paper is a development of the recent activity [16][17][18][19] aimed at implementing the above strategy in the context of problems of materials science and wave propagation in inhomogeneous media. Our recent papers [20,21] have shown that the language of boundary triples is particularly fitting for direct and inverse scattering problems on quantum graphs, as one of the key challenges to their analysis stems from the presence of interfaces through which energy exchange between different components of the medium takes place. In the present work, we continue the research initiated in these papers, adapting the technology so that BVP, especially those stemming from materials sciences, become within reach.…”

mentioning

confidence: 99%

“…In the present work, we continue the research initiated in these papers, adapting the technology so that BVP, especially those stemming from materials sciences, become within reach. As in [20,21], the ideas of [40,47] concerning the functional model allow one to efficiently incorporate into the analysis information about the mentioned energy exchange, by employing a suitable Dirichlet-to-Neumann map. In our analysis of BVP, we adopt the approach to the operatortheoretic treatment of BVP suggested by [53], which appears to be particularly convenient for obtaining sharp quantitative information about scattering properties of the medium, cf., for example, [17], where this same approach is used as a framework for the asymptotic analysis of homogenisation problems in resonant composites.…”

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confidence: 99%

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Functional Model for Boundary‐value Problems

2021

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We develop a functional model for operators arising in the study of boundary-value problems of materials science and mathematical physics. We then provide explicit formulae for the resolvents of the associated extensions of symmetric operators in terms of appropriate Dirichletto-Neumann maps, which can be utilised in the analysis of the properties of parameter-dependent problems, including the study of their spectra. §1. Introduction. The need to understand and quantify the behaviour of solutions to problems of mathematical physics has been central in driving the development of theoretical tools for the analysis of boundary-value problems (BVP). On the other hand, the second part of the last century witnessed several substantial advances in the abstract methods of spectral theory in Hilbert spaces, stemming from the groundbreaking achievement of John von Neumann in laying the mathematical foundations of quantum mechanics. Some of these advances have made their way into the broader context of mathematical physics [22,36,44]. In spite of these obvious successes of spectral theory applied to concrete problems, the operator-theoretic understanding of BVP has been lacking. However, in models of short-range interactions, the idea of replacing the original complex system by an explicitly solvable one, with a zero-radius potential (possibly with an internal structure), has proved to be highly valuable [5, 7, 14, 33, 34, 48, 57]. This facilitated an influx of methods of the theory of extensions (both self-adjoint and non-selfadjoint) of symmetric operators to problems of mathematical physics, culminating in the theory of boundary triples.The theory of boundary triples introduced in [23, 25, 31, 32] has been successfully applied to the spectral analysis of BVP for ordinary differential operators and related setups, for example, that of finite "quantum graphs", where the Dirichlet-to-Neumann maps act on finitedimensional "boundary" spaces, see [20] and references therein. However, in its original form this theory is not suited for dealing with BVP for partial differential equations (PDE), see [12, Section 7] for a relevant discussion. The key obstacle to such analysis is the lack of boundary traces 0 u and 1 u for functions u : → R (where is a bounded open set with a smooth boundary) in the domain of the maximal operator A corresponding to the differential expression considered (e.g., the operator − on the domain of L 2 ( )-functions u such that u is in L 2 ( )) entering the Green identity

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Functional Model for Boundary‐value Problems

2021

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“…The present paper is a development of the recent study [5,6,8,7] aimed at implementing the ideas of the Birman-Kreȋn-Višik theory [3,15,16,24] as proposed by Ryzhov [20], in the context of problems of materials science and wave propagation in inhomogeneous media. Our recent papers [9,10] have shown that the language of boundary triples is particularly fitting for direct and inverse scattering problems on quantum graphs, as one of the key difficulties in their analysis stems from the presence of interfaces through which an exchange of energy between different components of the medium takes place. In the present work, we adapt approach taken by these earlier works so that BVP, especially those stemming from materials sciences, become within reach.…”

Section: Introductionmentioning

confidence: 99%

“…In the present work, we adapt approach taken by these earlier works so that BVP, especially those stemming from materials sciences, become within reach. As in [9,10], the framework of the functional model of [19,18] allows one to efficiently incorporate the information about this energy exchange, by employing a suitable Dirichlet-to-Neumann map.…”

Section: Introductionmentioning

confidence: 99%

Operator-norm resolvent asymptotic analysis of continuous media with high-contrast inclusions

Cherednichenko,

Kiselev,

Silva

2020

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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 problem of "electrostatic" type for functions that are constant on the inclusion. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates.

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Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

2022

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Using a generalisation of the classical notion of the Weyl m-function 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 problem of "electrostatic" type. In particular, our results imply the convergence of the spectra of high-contrast problems to the spectrum of the limit operator, with order-sharp convergence estimates. The approach developed in the paper is of a general nature and can thus be successfully applied in the study of other problems of the same type.

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Scattering theory for a class of non-selfadjoint extensions of symmetric operators

Cited by 7 publications

References 24 publications

Functional Model for Boundary‐value Problems

Functional Model for Boundary‐value Problems

Operator-norm resolvent asymptotic analysis of continuous media with high-contrast inclusions

Operator-Norm Resolvent Asymptotic Analysis of Continuous Media with High-Contrast Inclusions

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