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

2005 ◽  
Vol 78 (1-2) ◽  
pp. 134-139 ◽  
Author(s):  
A. S. Kostenko
Keyword(s):  
Singular Potential ◽  
Adjoint Operator ◽  
Sturm Liouville ◽  
Liouville Operators

scholarly journals Spectral analysis of singular Sturm-Liouville operators on time scales

2018 ◽  
Vol 72 (1) ◽  
pp. 1 ◽  
Author(s):  
Bilender P. Allahverdiev ◽  
Huseyin Tuna
Keyword(s):  
Spectral Analysis ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Time Scales ◽  
Liouville Operator ◽  
Adjoint Operator ◽  
The Self ◽  
Sturm Liouville ◽  
Liouville Operators

In this paper, we consider properties of the spectrum of a Sturm-Liouville<br />operator on time scales. We will prove that the regular symmetric<br />Sturm-Liouville operator is semi-bounded from below. We will also give some<br />conditions for the self-adjoint operator associated with the singular<br />Sturm-Liouville expression to have a discrete spectrum. Finally, we will<br />investigate the continuous spectrum of this operator.

Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation

2020 ◽  
Vol 28 (5) ◽  
pp. 651-658
Author(s):  
Shavkat Alimov ◽  
Ravshan Ashurov
Keyword(s):  
Inverse Problem ◽  
Fractional Derivative ◽  
Adjoint Operator ◽  
Fourier Method ◽  
Fixed Time ◽  
Sturm Liouville ◽  
Fractional Differential ◽  
Time Fractional Derivative ◽  
Subdiffusion Equation ◽  
Liouville Operators

AbstractAn inverse problem for determining the order of the Caputo time-fractional derivative in a subdiffusion equation with an arbitrary positive self-adjoint operator A with discrete spectrum is considered. By the Fourier method it is proved that the value of {\|Au(t)\|}, where {u(t)} is the solution of the forward problem, at a fixed time instance recovers uniquely the order of derivative. A list of examples is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm–Liouville operators, and many others.

scholarly journals Spectral properties of singular Sturm—Liouville operators with indefinite weight sgn x

2009 ◽  
Vol 139 (3) ◽  
pp. 483-503 ◽  
Author(s):  
Illya Karabash ◽  
Carsten Trunk
Keyword(s):  
Critical Point ◽  
Spectral Properties ◽  
Krein Space ◽  
Adjoint Operator ◽  
Real Spectrum ◽  
Real Point ◽  
Indefinite Weight ◽  
Sturm Liouville ◽  
Regular Critical Point ◽  
Liouville Operators

We consider a singular Sturm—Liouville expression with the indefinite weight sgn x. There is a self-adjoint operator in some Krein space associated naturally with this expression. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d2/dx2 + q) with non-real spectrum accumulating to a real point. The results obtained are applied to several classes of Sturm—Liouville operators.

scholarly journals Partial inverse problems for quadratic differential pencils on a graph with a loop

2020 ◽  
Vol 28 (3) ◽  
pp. 449-463 ◽  
Author(s):  
Natalia P. Bondarenko ◽  
Chung-Tsun Shieh
Keyword(s):  
Inverse Problems ◽  
A Priori ◽  
Spectral Characteristics ◽  
The Other ◽  
Boundary Edge ◽  
Partial Inverse ◽  
Quadratic Pencil ◽  
Sturm Liouville ◽  
Differential Pencils ◽  
Liouville Operators

AbstractIn this paper, partial inverse problems for the quadratic pencil of Sturm–Liouville operators on a graph with a loop are studied. These problems consist in recovering the pencil coefficients on one edge of the graph (a boundary edge or the loop) from spectral characteristics, while the coefficients on the other edges are known a priori. We obtain uniqueness theorems and constructive solutions for partial inverse problems.

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