SOME SPECTRAL PROPERTIES OF THE NON-SELF-ADJOINT FRIEDRICHS MODEL OPERATOR

Abstract 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.

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Description of the structure of singular spectrum for Friedrichs model operator near singular point

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201011668 ◽

2001 ◽

Vol 28 (1) ◽

pp. 25-40

Author(s):

Serguei I. Iakovlev

Keyword(s):

Analytic Function ◽

Singular Point ◽

Uniqueness Theorem ◽

Analytic Functions ◽

Point Spectrum ◽

Positive Imaginary Part ◽

Model Operator ◽

Real Roots ◽

Singular Spectrum ◽

Friedrichs Model

The study of the point spectrum and the singular continuous one is reduced to investigating the structure of the real roots set of an analytic function with positive imaginary partM(λ). We prove a uniqueness theorem for such a class of analytic functions. Combining this theorem with a lemma on smoothness ofM(λ)near its real roots permits us to describe the density of the singular spectrum.

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On the spectral properties of the matrix-valued Friedrichs model

Many-Particle Hamiltonians: Spectra and Scattering - ADVSOV ◽

10.1090/advsov/005/01 ◽

1991 ◽

pp. 1-38 ◽

Cited By ~ 2

Author(s):

Zh. Abdullaev ◽

S. Lakaev

Keyword(s):

Spectral Properties ◽

The Matrix ◽

Friedrichs Model

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About the Spectral Properties of One Three-Partial Model Operator

Russian Mathematics ◽

10.3103/s1066369x20050011 ◽

2020 ◽

Vol 64 (5) ◽

pp. 1-7

Author(s):

G. P. Arzikulov ◽

Yu. Kh. Eshkabilov

Keyword(s):

Spectral Properties ◽

Model Operator ◽

Partial Model

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On the Discrete Spectrum of a Model Operator in Fermionic Fock Space

Abstract and Applied Analysis ◽

10.1155/2013/875194 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Zahriddin Muminov ◽

Fudziah Ismail ◽

Zainidin Eshkuvatov ◽

Jamshid Rasulov

Keyword(s):

Bound States ◽

Fock Space ◽

Critical Value ◽

Efimov Effect ◽

Model Operator ◽

The Third ◽

Direct Sum ◽

Friedrichs Model ◽

Number Of Particles ◽

Generalized Friedrichs Model

We consider a model operatorHassociated with a system describing three particles in interaction, without conservation of the number of particles. The operatorHacts in the direct sum of zero-, one-, and two-particle subspaces of thefermionic Fock space ℱa(L2(𝕋3))overL2(𝕋3). We admit a general form for the "kinetic" part of the HamiltonianH, which contains a parameterγto distinguish the two identical particles from the third one. (i) We find a critical valueγ*for the parameterγthat allows or forbids the Efimov effect (infinite number of bound states if the associated generalized Friedrichs model has a threshold resonance) and we prove that only forγ<γ*the Efimov effect is absent, while this effect exists for anyγ>γ*. (ii) In the caseγ>γ*, we also establish the following asymptotics for the numberN(z)of eigenvalues ofHbelowz<Emin=infσessH:limz→EminNz/logEmin-z=𝒰0γ 𝒰0γ>0, for allγ>γ*.

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The Detectable Subspace for the Friedrichs Model

Integral Equations and Operator Theory ◽

10.1007/s00020-019-2548-9 ◽

2019 ◽

Vol 91 (5) ◽

Author(s):

B. M. Brown ◽

M. Marletta ◽

S. Naboko ◽

I. G. Wood

Keyword(s):

Operator Theory ◽

Complex Analysis ◽

Hankel Operators ◽

Precise Description ◽

Model Operator ◽

Finite Rank Perturbation ◽

Boundary Measurements ◽

Independent Variable ◽

Friedrichs Model ◽

Perturbation Problems

Abstract This paper discusses how much information on a Friedrichs model operator can be detected from ‘measurements on the boundary’. We use the framework of boundary triples 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’. The Friedrichs model, a finite rank perturbation of the operator of multiplication by the independent variable, is a toy model that is used frequently in the study of perturbation problems. We view the Friedrichs model as a key example for the development of the theory of detectable subspaces, because it is sufficiently simple to allow a precise description of the structure of the detectable subspace in many cases, while still exhibiting a variety of behaviours. The results also demonstrate an interesting interplay between modern complex analysis, such as the theory of Hankel operators, and operator theory.

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