Invariant subspaces of a dissipative operator and divisors of its characteristic function

Abstract In this paper, maximal dissipative fourth order operators with equal deficiency indices are investigated. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which based on the method of Pavlov and define its characteristic function. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative fourth order operators.

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Spectral theory of dissipative q-Sturm-Liouville problems

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.3.1289 ◽

2014 ◽

Vol 51 (3) ◽

pp. 366-383

Author(s):

Aytekin Eryilmaz ◽

Hüseyin Tuna

Keyword(s):

Hilbert Space ◽

Characteristic Function ◽

Spectral Theory ◽

Difference Operator ◽

Functional Model ◽

Dissipative Operator ◽

Eigenvalues And Eigenvectors ◽

Maximal Dissipative Operator ◽

Sturm Liouville ◽

Liouville Operators

This paper is devoted to studying a q-analogue of Sturm-Liouville operators. We formulate a dissipative q-difference operator in a Hilbert space. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which is based on the method of Pavlov and define its characteristic function. Finally, we prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative q-Sturm-Liouville difference operator.

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Spectral analysis of dissipative Schrödinger operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026962 ◽

1997 ◽

Vol 127 (6) ◽

pp. 1113-1121 ◽

Cited By ~ 15

Author(s):

B. P. Allahverdiev ◽

Ahmet Canoǧlu

Keyword(s):

Characteristic Function ◽

Schrödinger Operators ◽

Functional Model ◽

Dissipative Operator ◽

Schrodinger Operators ◽

Associated Functions ◽

Index 2 ◽

Eigenfunctions And Associated Functions ◽

Spectral Representations ◽

Symmetric Operators

Dissipative Schrodinger operators are studied in L2(0, ∞) which are extensions of symmetric operators with defect index (2, 2). We construct a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix according to the scheme of Lax and Phillips. With the help of the incoming spectral representation, we construct a functional model of the dissipative operator and construct its characteristic function in terms of solutions of the corresponding differential equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operator.

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