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

Abstract

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. Mathematics Subject Classification (2010): 47A45 34L25 81Q35

Spectral enclosures for non-self-adjoint extensions of symmetric operators

“…Furthermore, for the treatment of quantum graphs via boundary triples and similar techniques we refer to, e.g. [46,59,61,109,120,123]. Let G be a finite graph consisting of a finite set V of vertices and a finite set E of edges, where we allow infinite edges, i.e.…”

Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I

“…ε , which cannot be additively decomposed into "independent" parts pertaining to the soft and stiff components, owing to the transmission interface conditions on the common boundary Γ of the two, the M -function turns out to be additive (cf., e.g., [27], where this additivity was observed and exploited in an independent, but closely related, setting of scattering). In what follows we will observe that the resolvent (A Alongside the transmission problem (9), whose boundary conditions can now be (so far, formally) represented as Γ 1 u = 0, u ∈ dom A 0 ran Π, in what follows we consider a wider class of problems, formally given by transmission conditions of the type…”

“…andM soft is the M -operator defined in accordance with (25), (27) relative to the triple (H,Π soft ,Λ soft ). 2.…”

Unified approach to critical-contrast homogenisation with explicit links to time-dispersive media

“…Note that (14) defines a Hermitian matrix for real values of k away from a discrete set of k. The next statement, which is commonly used in both ODE and PDE contexts, proves to be valuable for our analysis. Its proof can be obtained, e.g., by minor modifications, due to the presence of Datta-Das Sarma weights, of the related proof in [11].…”