The Detectable Subspace for the Friedrichs Model: Applications of Toeplitz Operators and the Riesz–Nevanlinna Factorisation Theorem

We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used to introduce the generalised Titchmarsh-Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. In this paper we choose functions arising as parameters in the Friedrichs model in certain Hardy classes. This allows us to determine the detectable subspace by using the canonical Riesz-Nevanlinna factorisation of the symbol of a related Toeplitz operator

Simplicity of eigenvalues in Anderson-type models

We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schr\"odinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.Comment: 20 page

Smilansky's model of irreversible quantum graphs, I: the absolutely continuous spectrum

In the model suggested by Smilansky one studies an operator describing the interaction between a quantum graph and a system of $K$ one-dimensional oscillators attached at several different points in the graph. The present paper is the first one in which the case $K>1$ is investigated. For the sake of simplicity we consider K=2, but our argument is of a general character. In this first of two papers on the problem, we describe the absolutely continuous spectrum. Our approach is based upon scattering theory

Exhaust of Underexpanded Jets from Finite Reservoirs

We examine the response of an underexpanded jet to a depleting, finite reservoir with experiments and simulations. An open-ended shock tube facility with variable reservoir length is used to obtain images of nitrogen and helium jet structures at successive instances during the blowdown from initial pressure ratios of up to 250. The reservoir and ambient pressures are simultaneously measured to obtain the instantaneous pressure ratio. We estimate the time-scales for jet formation and reservoir depletion as a function of the specific heat ratio of the gas and the initial pressure ratio. The jet structure formation time-scale is found to become approximately independent of pressure ratio for ratios greater than 50. In the present work, no evidence of time-dependence in the Mach disk shock location is observed for rates of pressure decrease associated with isentropic blowdown of a finite reservoir while the pressure ratio is greater than 15. The shock location in the finite- reservoir jet can be calculated from an existing empirical fit to infinite-reservoir jet data evaluated at the instantaneous reservoir pressure. For pressure ratios below 15, however, the present data deviate from a compilation of data for infinite-reservoir jets. A new fit is obtained to data in the lower pressure regime. The self-similarity of the jet structure is quantified and departure from similarity is noted to begin at pressure ratios lower than about 15, approximately the same ratio which limits existing empirical fits

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

On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated "charge conjugation" operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.Comment: 27 page

ЭФФЕКТ ВЛИЯНИЯ ОРИЕНТАЦИИ И НАПРЯЖЕННОСТИ ВНЕШНЕГО МАГНИТНОГО ПОЛЯ НА ПРОЦЕСС КОРРОЗИИ ФЕРРОМАГНИТНЫХ ТОНКИХ ПЛЕНОК ЖЕЛЕЗА

The results of a study on the rate of galvanic corrosion of thin ferromagnetic iron films in an external uniform magnetic field are presented. It is shown that the direction of the external magnetic field has different influence on the corrosion rate in the presence of a magnetic field. The shape of the micro-relief surface iron film subjected to a short-term treatment with a solution of acetic acid was studied. The relief is formed on the surface due to the heterogeneity of the corrosion rate on the surface of the film. Areas of high corrosion rates correspond to areas of the domain boundaries of the ferromagnetic film. When the orientation of the external magnetic field is changed, the domain structure changes, which is reflected in the change of the average rate of corrosion.Представлены результаты исследования скорости электрохимической коррозии тонких ферромагнитных пленок железа в присутствии внешнего однородного магнитного поля. Показано, что направление внешнего магнитного поля по-разному влияет на характер зависимости скорости коррозии от напряженности магнитного поля