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

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 δ-type vertex conditions.</p

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

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

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

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