96 research outputs found
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
Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach
We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schrödinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues
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 one-dimensional
oscillators attached at several different points in the graph. The present
paper is the first one in which the case 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
Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von Neumann approach
The Wigner-von Neumann method, which has previously been used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary T-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, ?, into the operator’s absolutely continuous spectrum. Introducing a new rational function, C(?;T), related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of C(?;T)); in particular showing that there are only finitely many of them
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
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