Spectral analysis of dissipative Schrödinger operators

1997 ◽  
Vol 127 (6) ◽  
pp. 1113-1121 ◽  
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.

scholarly journals Functional Model of Dissipative Fourth Order Differential Operators

2013 ◽  
Vol 21 (2) ◽  
pp. 237-252 ◽  
Author(s):  
Hüseyin Tuna
Keyword(s):  
Characteristic Function ◽  
Fourth Order ◽  
Differential Operators ◽  
Functional Model ◽  
Dissipative Operator ◽  
Eigenvalues And Eigenvectors ◽  
Deficiency Indices ◽  
Maximal Dissipative Operator

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.

Spectral theory of dissipative q-Sturm-Liouville problems

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.

scholarly journals Dissipative Dirac operator with general boundary conditions on time scales

2020 ◽  
Vol 72 (5) ◽  
Author(s):  
B. P. Allahverdiev ◽  
H. Tuna
Keyword(s):  
Dirac Operator ◽  
Boundary Conditions ◽  
Time Scales ◽  
Functional Model ◽  
Scattering Matrix ◽  
The Other ◽  
General Boundary ◽  
Dissipative Operator ◽  
General Boundary Conditions ◽  
Symmetric Operators

UDC 517.9 In this paper, we consider the symmetric Dirac operator on bounded time scales. With general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.

scholarly journals Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Kun Li ◽  
Maozhu Zhang ◽  
Jinming Cai ◽  
Zhaowen Zheng
Keyword(s):  
Boundary Condition ◽  
Dirac Operator ◽  
Completeness Theorem ◽  
Scattering Matrix ◽  
Dissipative Operator ◽  
Limit Circle ◽  
Associated Functions ◽  
Limit Circle Case ◽  
Eigenfunctions And Associated Functions ◽  
Circle Case

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

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