The extension theory of Hermitian operators and the moment problem

Derkach, Vladimir; Malamud, M. M.

doi:10.1007/bf02367240

Cited by 275 publications

(603 citation statements)

References 32 publications

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“…Proposition 2.2 below is taken from [3], where its proof was omitted. For the convenience of the reader a short proof of the resolvent formula with the help of boundary triples and their γ-fields and Weyl functions (see [13][14][15]) is given. From now on we will suppose that the following assumption is satisfied.…”

Section: Operators With Finitely Many Negative Squares and Rank One Pmentioning

confidence: 99%

Eigenvalue estimates for operators with finitely many negative squares

Behrndt

Möws²,

Trunk

2016

OpMath

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Abstract. Let A and B be selfadjoint operators in a Krein space. Assume that the resolvent difference of A and B is of rank one and that the spectrum of A consists in some interval I ⊂ R of isolated eigenvalues only. In the case that A is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of B in the interval I. The general results are applied to singular indefinite Sturm-Liouville problems.

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Section: Operators With Finitely Many Negative Squares and Rank One Pmentioning

confidence: 99%

Eigenvalue estimates for operators with finitely many negative squares

Behrndt

Möws²,

Trunk

2016

OpMath

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“…To calculate the spectrum and the resolvent of the operator A we will use the concepts of boundary triplets and abstract Weyl functions (see [7,8]). Let us briefly recall basic notions and facts.…”

Section: Applications To J-nonnegative Operatorsmentioning

confidence: 99%

“…A connection between the Krein-Najmark formula (see, for example, [2]) and boundary triplets has been established in [7,8]. We use the corresponding result in the form given in [36].…”

Section: ) That the Above Implicit Definition Of The Weyl Function Ismentioning

confidence: 99%

Similarity of Indefinite Sturm-Liouville Operators with Singular Potential to a Self-Adjoint Operator

Kostenko

2005

Math Notes

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We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh m-functions. Also we obtain necessary conditions for regularity of the critical points 0 and ∞ of J-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular critical point zero. In particular, it is shown that 0 is a singular critical point of the operator − (sgn x)dx 2 acting in the Hilbert space L 2 (R, (3|x| + 1) −4/3 dx) and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type (sgn x)(−d 2 /dx 2 + q(x)) with the same properties.

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“…In the present paper, we suggest a model of point-like interaction between a relativistic fermion (the Dirac operator) and bosons (infinite matrix operator in the Fock space). We use the boundary triplet approach to describe extensions of symmetric operators (see, e.g., [8][9][10][11][12][13]15]). …”

Section: Introductionmentioning

confidence: 99%

Dirac operator coupled to bosons

Boitsev¹,

Neidhardt

Popov³

2016

Nanosystems: Phys. Chem. Math.

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We consider a model of point-like interaction between electrons and bosons in a cavity. The electrons are relativistic and are described by a Dirac operator on a bounded interval while the bosons are treated by second quantization. The model fits into the extension theory of symmetric operators. Our main technical tool to handle the model is the so-called boundary triplet approach to extensions of symmetric operators. The approach allows explicit computation of the Weyl function.

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The extension theory of Hermitian operators and the moment problem

Cited by 275 publications

References 32 publications

Eigenvalue estimates for operators with finitely many negative squares

Eigenvalue estimates for operators with finitely many negative squares

Similarity of Indefinite Sturm-Liouville Operators with Singular Potential to a Self-Adjoint Operator

Dirac operator coupled to bosons

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