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

This work deals with the functional model for a class of 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 with equal deficiency indices. On the basis of these formulae, we are able to construct wave operators and derive a new representation for the scattering matrix for pairs of such extensions in both self-adjoint and non-self-adjoint situations.Comment: 32 pages; This is the continuation of arXiv:1703.06220 (and formerly contained in v1); this version is as accepted by the journal (Operator Theory: Advances and Applications

Functional model for boundary-value problems and its application to the spectral analysis of transmission problems

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 properties of parameter-dependent problems as well as in the study of their spectra.Comment: 30 pages, 1 figur

Norm-resolvent convergence for Neumann Laplacians on manifolds thinning to graphs

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

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

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.Comment: 15 pages, 1 figure. Continuation of: arXiv:1907.08144. As accepted by: Math. Note

СПОСОБИ ПІДРОБКИ СУЧАСНИХ ПАСПОРТНИХ ДОКУМЕНТІВ КРАЇН ЄС ТА ОЗНАКИ ЇХ ВИЯВЛЕННЯ

The development of modern social relations associated with the process of European integration of Ukraine has led to an increase in the number of travelers crossing the state border, including the EU’s external borders. For this reason there becomes urgent to study the problems concerning protection of travel documents against counterfeiting and the ways to detect them. A modern analysis of the forensic-expert provision of the State border service of Ukraine shows a consistent trend of increase in detections offorgery passports and other documents. The purpose of this paper is to present the results of studies regarding the counterfeiting methods and characteristics of detection in passport documents of countries - members of EU citizens and to identify the ways of increasing efficiency of realizing tasks in the field of forensic examination and expert researches ofpassports and other documents in the context of European integration of Ukraine. In this perspective, with the purpose to increase the efficiency of tasks realization in the field offorensic examination and expert researches of passports and other documents, according to the authors, it’s necessary: to guarantee the development of interagency and international cooperation; to realize effective and rapid mechanism for information exchange on detected forged passport documents with expert research institutions, law enforcement bodies of Ukraine. This will help to develop an effective system to counter existing and potential threats to national security of Ukraine in the context of liberalization of the EU visa regime for Ukraine. The study of counterfeiting methods and characteristics of their detection in the passport documents of countries - EU members citizens and citizens of Ukraine with the contactless electronic carrier of biometric data, as well as in the form ofID-cards may be considered as the prospects for further research.Розглянуто способи підробок і ознаки їх виявлення в паспортних документах громадян країн - членів ЄС. Визначено шляхи збільшення ефективності виконання завдань у сфері судової експертизи та експертних досліджень паспортних та інших документів у контексті євроінтеграції України

Asymptotic analysis of operator families and applications to resonant media

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 the links of this rich area, on the one hand, to functional frameworks developed by S. N. Naboko and his students, and on the other hand, to concrete applications of current interest in the physics and engineering communities.Comment: 60 pages, 2 figures; a survey of recent results in the area, see also arXiv:2010.13318, arXiv:1808.03961, arXiv:1703.06220, arXiv:1510.0336

Functional model for extensions of symmetric operators and applications to scattering theory

On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with $\delta$-type vertex conditions.Comment: 28 page

High-frequency homogenization for periodic media

This article is available open access through the publisher’s website at the link below. Copyright @ 2010 The Royal Society.An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.NSERC (Canada) and the EPSRC