Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

Spectra of self-adjoint extensions and applications to solvable Schroedinger operators

We give a self-contained presentation of the theory of self-adjoint extensions using the technique of boundary triples. A description of the spectra of self-adjoint extensions in terms of the corresponding Krein maps (Weyl functions) is given. Applications include quantum graphs, point interactions, hybrid spaces, singular perturbations.Comment: 81 pages, new references added, subsection 1.3 extended, typos correcte

Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions

© 2020 The Authors. Mathematische Nachrichten published by Wiley‐VCH Verlag GmbH & Co. KGaA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed

Cantor and band spectra for periodic quantum graphs with magnetic fields

We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

Experimental and morphological justification of the prevention of wound complications during fixation of the mesh to the abdominal wall tissues with composite

The aim of the study is to justify the prevention of wound complications during fixation of the mesh to abdominal wall tissues with cross-linked polyurethane (CP) composite in experimental animals. Маterials and methods. An experimental study was conducted on 60 male laboratory rats, which were divided into 3 groups. In group I (n = 21), the mesh and wound edges were fixed with CP composite with an antiseptic, in group II (n = 24) – with ligatures, in group III (n = 15) muscle incision was performed and the wound was sutured with ligatures. The animals were observed for 30 days, wound healing and the presence of complications were evaluated. Morphological examination was performed on days 7 and 30. Signs of mesh germination by connective tissue and presence of tissue inflammation around the mesh were evaluated. Results. The observation results showed that in group I, the incidence of complications and the duration of wound healing were lower comparing to groups II and III. In group I, seroma was detected in 1 (5.3 %) animal, in group II – in 7 (33.3 %), in group III – in 3 (23.1 %), infection of the wound in 1 (5.3 %) animal of group I, in 4 (19.1 %) of group II and in 2 (15.4 %) of group III. The term of wound healing in group I was 7 (6; 8) days, in group II – 13 (12; 14), in group III –11 (10; 12) days (p < 0.05). The results of observation were confirmed morphologically, namely, in animals of group I, faster process of mesh germination with collagen fibers and wound healing were revealed, which was confirmed by the predominance of a fibrous component over the cellular one (р < 0.05), and the formation of a connective tissue capsule around the CP with the germination of collagen fibers deep into the capsule. Conclusions. The use of the cross-linked polyurethane composite with an antiseptic to fix the mesh implant to abdominal wall tissues in laboratory animals confirms its higher efficiency compared to traditional ligature fixation, by increasing the fibrous component of connective tissue and reducing the cellular component and hemodynamic disorders in dynamics, by the germination of collagen fibers deep into the capsule in the areas of mesh fixation, reduced occurrence of seroma and duration of wound healing, which justifies its use in clinical practice for prevention of wound complications in allogernioplasty

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144