A test for the absence of the singular continuous spectrum in the Friedrichs model

For the proof of the absence of the singular continuous spectrum in the manybody scattering problem, we suggest a new method using the analogue of the triangular interlacing operators in the inverse scattering problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 61–71, 1982.     

On infinity of the discrete spectrum of operators in the Friedrichs model

The discrete spectrumof selfadjoint operators in the Friedrichs model is studied. Necessary and sufficient conditions of existence of infinitely many eigenvalues in the Friedrichs model are presented. A discrete spectrum of a model three-particle discrete Schrödinger operator is described.     

Operators with singular continuous spectrum, V. Sparse potentials

By presenting simple theorems for the absence of positive eigenvalues for certain one-dimensional Schrödinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum.

     

Nonphysical sheet for the Friedrichs model

A method of deducing an expansion theorem in resonance states for the two-dimensional Friedrichs model is suggested. The construction is based on detailed Lax-Phillips analogs of the corresponding unitary Friedrichs model , perturbed by the projector P_ϕ, generated by an element ϕ possessing special analytic properties. Bibliography: 11 titles. To Olga A. Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 149–155. Translated by V. I. Ochkur.     

Imbedded singular continuous spectrum for Schrödinger operators

We construct examples of potentials satisfying where the function is growing arbitrarily slowly, such that the corresponding Schrödinger operator has an imbedded singular continuous spectrum. This solves one of the fifteen ``twenty-first century" problems for Schrödinger operators posed by Barry Simon. The construction also provides the first example of a Schrödinger operator for which Möller wave operators exist but are not asymptotically complete due to the presence of a singular continuous spectrum. We also prove that if the singular continuous spectrum is empty. Therefore our result is sharp.

     

Embedded singular continuous spectrum for one-dimensional Schrödinger operators

We investigate one-dimensional Schrödinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.

     

Cantor singular continuous spectrum for operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.

     

A continuous path of singular masas in the hyperfinite II1 factor

Using methods of Tauer, we exhibit an uncountable family ofsingular masas in the hyperfinite II₁ factor R all with Pukánszkyinvariant {1}, no pair of which is conjugate by an automorphismof R. This is done by introducing an invariant (A) for a masaA in a II₁ factor N as the maximal size of a projection eA forwhich A e contains non-trivial centralizing sequences for eNe. The masas produced give rise to a continuous map from theinterval [0, 1] into the singular masas in R equipped with thed_{, 2}-metric. A result is also given showing that the Pukánszky invariantis d_{, 2}-upper semi-continuous. As a consequence, the sets ofmasas with Pukánszky invariant {n} are all closed.     

Absolutely continuous spectrum of a nondissipative operator and a functional model. II

The second part of the paper (the first is published in J. Sov. Math.,16, No. 3 (1981)), is devoted to the study of nondissipative operators in Hilbert space, which are nearly self-adjoint. In the model representation, generalizing the familiar model of B. S. Nagy-C. Foias for dissipative operators, formulas are obtained for spectral projectors on a segment of the absolutely continuous spectrum and conditions for their boundedness are studied. Questions of linear similarity for a generally nondissipative operator and its parts to self-adjoint and dissipative operators are considered. New proofs are found for the similarity theorems of L. A. Sakhnovich and Davis-Foias. Some of the results are new even in the dissipative case which is not excluded.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 118–135, 1977.In conclusion the author expresses gratitude to B. S. Pavlov for interest in the work and helpful discussions.     

Conditions for the finiteness of the singular spectrum in the self-adjoint friedrichs model

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 4, pp. 88–89, October–December, 1990.     

Operators with singular continuous spectrum, VI. Graph Laplacians and Laplace-Beltrami operators

Examples are constructed of Laplace-Beltrami operators and graph Laplacians with singular continuous spectrum.

     

Spectral properties of the generalized Friedrichs model

We consider the self-adjoint generalized Friedrichs model with small values of the “coupling parameter.” In this case, we completely investigate the spectrum of the model and the structure of its eigenvectors (both ordinary and generalized). The constructions we use are based on an analysis of the resolvent of the Friedrichs operator and on the corresponding scattering theory.     

Expression for the number of eigenvalues of a Friedrichs model

We consider the self-adjoint operator of a generalized Friedrichs model whose essential spectrum may contain lacunas. We obtain a formula for the number of eigenvalues lying on an arbitrary interval outside the essential spectrum of this operator. We find a sufficient condition for the discrete spectrum to be finite. Applying the formula for the number of eigenvalues, we show that there exist an infinite number of eigenvalues on the lacuna for a particular Friedrichs model and obtain the asymptotics for the number of eigenvalues.