Essential spectra of singular matrix differential operators of mixed order in the limit circle case

2010 ◽  
Vol 284 (2-3) ◽  
pp. 342-354 ◽  
Author(s):  
Jiangang Qi ◽  
Shaozhu Chen
Keyword(s):  
Differential Operators ◽  
Singular Matrix ◽  
Essential Spectra ◽  
Limit Circle ◽  
Matrix Differential Operators ◽  
Mixed Order ◽  
Limit Circle Case ◽  
Matrix Differential ◽  
Circle Case

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).

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

The spectra of well-posed operators

1995 ◽  
Vol 125 (6) ◽  
pp. 1331-1348 ◽  
Author(s):  
Sobhy El-sayed Ibrahim
Keyword(s):  
Adjoint Equation ◽  
Integral Operators ◽  
Essential Spectra ◽  
Limit Circle ◽  
All Solutions ◽  
Limit Circle Case ◽  
Complex Coefficients ◽  
Well Posed ◽  
Circle Case ◽  
Quasidifferential Expression

In this paper, the general ordinary quasidifferential expression M of nth order, with complex coefficients, and its formal adjoint M− are considered. It is shown in the case of two singular endpomts and when all solutions of the equation and the adjoint equation are in (the limit-circle case) that all well-posed extensions of the minimal operator T0(M) have resolvents which are Hilbert Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all of the standard essential spectra to be empty. These results extend those for the formally symmetric expression M studied in [1] and [14], and also extend those proved in [8] for one singular endpoint.

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