Limit Operators and Their Applications in Operator Theory

Rabinovich, Vladimir; Roch, Steffen; Silbermann, Bernd

doi:10.1007/978-3-0348-7911-8

Cited by 212 publications

(386 citation statements)

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“…Some of these results are obtained for the scalar case, some others for the vector case, under the assumption that the coefficients of the operator stabilize at infinity or without this assumption. This theory is also developed for some classes of pseudodifferential operators [12], [19], [30]- [34], [37], [38] and discrete operators [5], [35]. A survey of this literature is presented in the recent monograph [35].…”

Section: Introductionmentioning

confidence: 99%

Fredholm property of general elliptic problems

Volpert

2006

Trans. Moscow Math. Soc.

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Abstract. Linear elliptic problems in bounded domains are normally solvable with a finite-dimensional kernel and a finite codimension of the image, that is, satisfy the Fredholm property, if the ellipticity condition, the condition of proper ellipticity and the Lopatinskii condition are satisfied. In the case of unbounded domains these conditions are not sufficient any more. The necessary and sufficient conditions of normal solvability with a finite-dimensional kernel are formulated in terms of limiting problems. Adjoint operators to elliptic operators in unbounded domains are studied and the conditions in order for them to be normally solvable with a finite-dimensional kernel are also formulated by means of limiting problems. The properties of the direct and of the adjoint operators are used to prove the Fredholm property of elliptic problems in unbounded domains. Some special function spaces introduced in this work play an important role in the study of elliptic problems in unbounded domains.

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Section: Introductionmentioning

confidence: 99%

Fredholm property of general elliptic problems

Volpert

2006

Trans. Moscow Math. Soc.

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“…The proof of Theorem 1.2 is based on the method of limit operators, which was essentially developed by V. S. Rabinovich (see, e.g., [1,12,10] and the references therein), and on the Allan-Douglas localization (see [3]). The paper is organized as follows.…”

Section: Theorem 12 (Main Result) Suppose a B C D ∈ So(r + ) Andmentioning

confidence: 99%

“…Abstract approach. In our previous work [5] the techniques of limit operators (see, e.g., [1,10,12]) was successfully used to study the invertibility of binomial functional operators that are now the coefficients of the singular integral operator with shift N given by (1.1). In what follows we make use of such techniques to obtain a necessary condition for the Fredholmness of the operator N. Let us recall the abstract version of such techniques.…”

Section: Limit Operatorsmentioning

confidence: 99%

Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Karlovich

Lebre

2011

Integr. Equ. Oper. Theory

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Abstract. Suppose α is an orientation-preserving diffeomorphism (shift) of R + = (0, ∞) onto itself with the only fixed points 0 and ∞. In [6] we found sufficient conditions for the Fredholmness of the singular integral operator with shiftacting on L p (R + ) with 1 < p < ∞, where P ± = (I ± S)/2, S is the Cauchy singular integral operator, and W α f = f • α is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α ′ of the shift are bounded and continuous on R + and may admit discontinuities of slowly oscillating type at 0 and ∞. Now we prove that those conditions are also necessary.

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“…Our presentation will follow the papers [20,21]. A comprehensive account of this material can be found in [22]. In Section 3, we give applications of the limit operators method to the description of the essential spectrum of the discrete Schrödinger operator (2) under different assumptions on the behavior of magnetic and electric potentials at infinity.…”

Section: Introductionmentioning

confidence: 99%

The essential spectrum of Schrödinger operators on lattices

Rabinovich¹,

Roch²

2006

J. Phys. A: Math. Gen.

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The paper is devoted to the study of the essential spectrum of discrete Schrödinger operators on the lattice Z N by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schrödinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well.We consider the following classes of the discrete Schrödinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schrödinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schrödinger operators which describe the motion of two particles around a heavy nuclei on the lattice Z 3 .

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Limit Operators and Their Applications in Operator Theory

Cited by 212 publications

References 0 publications

Fredholm property of general elliptic problems

Fredholm property of general elliptic problems

Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

The essential spectrum of Schrödinger operators on lattices

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