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.

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.

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.

Essential Spectra of General Second-Order Differential Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Essential Spectrum ◽  
Distance Function ◽  
Differential Operators ◽  
Second Order ◽  
Compact Perturbation ◽  
Essential Spectra ◽  
Sectorial Operators

In this chapter, the operators considered are those m-sectorial operators discussed in Chapter VII, and the essential spectra are the sets defined in Chapter IX that remain invariant under compact perturbation. A generalization of a result of Persson is used to determine the least point of the essential spectrum. Davies’ mean distance function is introduced and consequences investigated.

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.

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.

On the location of eigenvalues of second order linear differential operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 15-22 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Boundary Condition ◽  
Hilbert Space ◽  
Differential Expression ◽  
Differential Operators ◽  
Homogeneous Boundary Condition ◽  
Upper Bounds ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expressionon [0,∞), together with a real homogeneous boundary condition at t = 0.

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).

scholarly journals ON THE ESSENTIAL SPECTRA OF GENERAL DIFFERENTIAL OPERATORS

1999 ◽  
Vol 30 (2) ◽  
pp. 105-126
Author(s):  
SOBHY EL-SAYED IBRAHIM
Keyword(s):  
Discrete Spectrum ◽  
Differential Operators ◽  
Integral Operators ◽  
Essential Spectra ◽  
Limit Circle ◽  
All Solutions ◽  
Limit Circle Case ◽  
Complex Coefficients ◽  
Well Posed ◽  
Circle Case

In this paper, it is shown in the cases of one and two singular end-points and when all solutions of the equation $M[u]-\lambda uw=0$, and its adjoint $M^+[v] -\lambda wv = 0$ are in $L_w^2 (a, b)$ (the limit circle case) with $f\in L^2_w(a,b)$ for $M[u]-\lambda wu=wf$ that all well-posed extensions of the minimal operator $T_0(M)$ generated by a general ordinary quasi-differential expression $M$ of $n$-th order with complex coefficients have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly slovable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression $M$ studied in [1] and [12], and also extend those proved in [8] in the case of one singular end-point of the interval [a,b).

Perturbation of Direct Sum Differential Operators

1978 ◽  
Vol 30 (03) ◽  
pp. 600-630 ◽  
Author(s):  
S. J. Lee
Keyword(s):  
Differential Expression ◽  
Differential Operators ◽  
Direct Sum ◽  
Linear Differential ◽  
Image Position

Let I be an interval, and let for 1 ≦ j ≦ I &lt; ∞ be abutted subintervals such that . Let τ j be a linear differential expression defined on I j . In this paper we study densely defined operators associated with (0.1)

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