On the location of the essential spectra and regularity fields of complex Sturm—Liouville operators

1980 ◽  
Vol 85 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
David Race
Keyword(s):  
Essential Spectrum ◽  
Essential Spectra ◽  
Linearly Independent ◽  
Complete Determination ◽  
Associated Operators ◽  
Sturm Liouville ◽  
Selfadjoint Extensions ◽  
Complex Valued ◽  
Liouville Operators

SynopsisIn this paper the Sturm-Liouville expression τy= −(py′)′ +qy, with complex-valued coefficients is considered, and a number of results concerning the location of the essential spectrum of associated operators are obtained. Some of these are extensions or generalizations of results due to Birman, and Glazman, whilst others are new. These lead to criteria for the non-emptiness of the regularity field of the corresponding minimal operator—a condition which is needed in the theory ofJ-selfadjoint extensions. A complete determination of the regularity field is made when the equation τy= λ0yhas two linearly independent solutions inL2[a,∞) for some complex λ0.

On the essential spectra of linear 2nth order differential operators with complex coefficients

1982 ◽  
Vol 92 (1-2) ◽  
pp. 65-75 ◽  
Author(s):  
David Race
Keyword(s):  
Differential Expression ◽  
Essential Spectrum ◽  
Differential Operators ◽  
Essential Spectra ◽  
Associated Operators ◽  
Linear Differential ◽  
Selfadjoint Extensions ◽  
Complex Valued ◽  
Complex Coefficients ◽  
Liouville Operators

SynopsisIn this paper, a formally J-symmetric, linear differential expression of 2nth order, with complex-valued coefficients, is considered. A number of results concerning the location of the essential spectrum of associated operators are obtained. These are extensions of earlier work dealing with complex Strum-Liouville operators, and include results which, in the real case, are due to Birman, Glazman and others. They lead to criteria, for the non-emptiness of the regularity field, of the corresponding minimal operator-a condition which is needed in the theory of J-selfadjoint extensions.

Spectra of a class of non-symmetric operators in Hilbert spaces with applications to singular differential operators

2019 ◽  
Vol 150 (4) ◽  
pp. 1769-1790
Author(s):  
Huaqing Sun ◽  
Bing Xie
Keyword(s):  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Homogeneous Equation ◽  
Fundamental Theory ◽  
Linearly Independent ◽  
Sturm Liouville ◽  
Symmetric Operators ◽  
Complex Valued

AbstractThis paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

2021 ◽  
Author(s):  
Abdizhahan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Order Differential Operator ◽  
Spectral Expansions ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

Dissipative Sturm-Liouville operators

1981 ◽  
Vol 88 (3-4) ◽  
pp. 329-343 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Minimal Operator ◽  
Sturm Liouville ◽  
Image Position ◽  
Complex Valued ◽  
Liouville Operators

SynopsisConsider the differential expressionwherepandw> 0 are real-valued andqis complex-valued onI. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert spaceto be maximal dissipative.

An inverse problem for Sturm–Liouville operators on the half-line with complex weights

2019 ◽  
Vol 27 (3) ◽  
pp. 439-443
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Half Line ◽  
Sturm Liouville ◽  
The Given ◽  
Complex Valued ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Abstract Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

Lower Bounds for the Essential Spectrum of Fourth-Order Differential Operators

1969 ◽  
Vol 21 ◽  
pp. 460-465
Author(s):  
Kurt Kreith
Keyword(s):  
Lower Bound ◽  
Essential Spectrum ◽  
Lower Bounds ◽  
Fourth Order ◽  
Differential Operators ◽  
Intimate Connection ◽  
Sturm Liouville ◽  
Image Position ◽  
Oscillation Properties ◽  
Liouville Operators

In this paper, we seek to determine the greatest lower bound of the essential spectrum of self-adjoint singular differential operators of the form1where 0 ≦ x < ∞. In the event that this bound is + ∞, our results will yield criteria for the discreteness of the spectrum of (1).Such bounds have been established by Friedrichs (3) for Sturm-Liouville operators of the formand our techniques will be closely related to those of (3). However, instead of studying the solutions of2directly, we shall exploit the intimate connection between the infimum of the essential spectrum of (1) and the oscillation properties of (2).

m (λ)-functions for complex Sturm-Liouville operators

1980 ◽  
Vol 86 (3-4) ◽  
pp. 275-289 ◽  
Author(s):  
David Race
Keyword(s):  
Limit Point ◽  
Limit Circle ◽  
Limit Circle Case ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Limit Point Case ◽  
Complex Valued ◽  
Well Posed ◽  
Circle Case ◽  
Liouville Operators

SynopsisIn this paper, the formally J-symmetric Sturm-Liouville operator with complex-valued coefficients is considered and a generalisation of the Weyl limit-point, limit-circle dichotomy is sought by means of m (λ )-functions. These functions are then used to give an explicit description of all the associated J-selfadjoint operators with separated boundary conditions in the limit-circle case. A formulation of the eigenvalues of these operators, and a characterisation of which extensions are non-well-posed, are also found. Finally, the limit-point case is studied, mainly by means of an example.

scholarly journals Asymptotic Behavior of Solutions of the Dirac System with an Integrable Potential

2021 ◽  
Vol 93 (5) ◽  
Author(s):  
Łukasz Rzepnicki
Keyword(s):  
Asymptotic Behavior ◽  
Spectral Parameter ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Dirac System ◽  
Eigenvalues And Eigenfunctions ◽  
Integrable Potential ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

AbstractWe consider the Dirac system on the interval [0, 1] with a spectral parameter $$\mu \in {\mathbb {C}}$$ μ ∈ C and a complex-valued potential with entries from $$L_p[0,1]$$ L p [ 0 , 1 ] , where $$1\le p$$ 1 ≤ p . We study the asymptotic behavior of its solutions in a strip $$|\mathrm{Im}\,\mu |\le d$$ | Im μ | ≤ d for $$\mu \rightarrow \infty $$ μ → ∞ . These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system.

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