Nonselfadjoint spectral problems for linear pencilsN-?P of ordinary differential operators with ?-linear boundary conditions: Completeness results

1996 ◽  
Vol 26 (2) ◽  
pp. 222-248 ◽  
Author(s):  
Christiane Tretter
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Linear Boundary ◽  
Spectral Problems ◽  
Ordinary Differential Operators

Semibounded Extensions of Singular Ordinary Differential Operators

1988 ◽  
Vol 31 (4) ◽  
pp. 432-438
Author(s):  
Allan M. Krall
Keyword(s):  
Differential Operator ◽  
Lower Bound ◽  
Boundary Conditions ◽  
Differential Operators ◽  
The Self ◽  
Limit Circle ◽  
Singular Differential Operator ◽  
Ordinary Differential Operators

AbstractThe self-adjoint extensions of the singular differential operator Ly = [(py’)’ + qy]/w, where p < 0, w > 0, q ≧ mw, are characterized under limit-circle conditions. It is shown that as long as the coefficients of certain boundary conditions define points which lie between two lines, the extension they help define has the same lower bound.

Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter

1980 ◽  
Vol 86 (1-2) ◽  
pp. 1-27 ◽  
Author(s):  
Aalt Dijksma
Keyword(s):  
Boundary Conditions ◽  
Differential Expression ◽  
Eigenfunction Expansion ◽  
Differential Operators ◽  
Real Interval ◽  
Eigenfunction Expansions ◽  
Selfadjoint Operators ◽  
Ordinary Differential Operators ◽  
Eigenvalue Parameter ◽  
Ordinary Differential Expression

SynopsisIn provided with a J-innerproduct we characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι. For a subclass of these operators we prove eigenfunction expansion results using Hilbertspace-techniques.

Pencils of differential operators containing the eigenvalue parameter in the boundary conditions

2003 ◽  
Vol 133 (4) ◽  
pp. 893-917 ◽  
Author(s):  
Marco Marletta ◽  
Andrei Shkalikov ◽  
Christiane Tretter
Keyword(s):  
Boundary Conditions ◽  
Spectral Properties ◽  
Differential Operators ◽  
Finite Interval ◽  
Linear Operators ◽  
Riesz Bases ◽  
Ordinary Differential Operators ◽  
Associated Functions ◽  
Eigenvalue Parameter

The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.

Inverse spectral problems for differential operators with non-separated boundary conditions

2020 ◽  
Vol 28 (4) ◽  
pp. 567-616
Author(s):  
Vjacheslav A. Yurko
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Short Review ◽  
Inverse Spectral Problems ◽  
Spectral Problems ◽  
Separated Boundary Conditions ◽  
Inverse Spectral ◽  
Type Boundary

AbstractWe give a short review of results on inverse spectral problems for second-order differential operators on an interval with non-separated boundary conditions. We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics. In the first part of the review, we provide the main results and methods related to inverse problems for Sturm–Liouville operators with non-separated boundary conditions: periodic, quasi-periodic and Robin-type boundary conditions. At the end, we present the main results on inverse problems for differential pencils with non-separated boundary conditions.

Spectral properties of ordinary differential operators with involuton

2019 ◽  
Vol 484 (1) ◽  
pp. 12-17 ◽  
Author(s):  
V. E. Vladykina ◽  
A. A. Shkalikov
Keyword(s):  
Boundary Conditions ◽  
Unconditional Basis ◽  
Spectral Properties ◽  
Differential Operators ◽  
Root Functions ◽  
Ordinary Differential Operators ◽  
Normal Boundary ◽  
The Space L2

Let P and Q be ordinary differential operators of order n and m generated by s = max{n; m} boundary conditions on a nite interval [a; b]. We study operators of the form L = JP + Q, where J is the involution operator in the space L2[a; b]. We consider three cases n > m, n < m, and n = m, for which we dene concepts of regular, almost regular, and normal boundary conditions. We announce theorems on unconditional basis and completeness of the root functions of operator L depending on the type of boundary conditions from selected classes.

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