Asymptotic formulas with arbitrary order for nonselfadjoint differential operators

2007 ◽  
Vol 44 (3) ◽  
pp. 391-409 ◽  
Author(s):  
Melda Duman ◽  
Alp Kiraç ◽  
Oktay Veliev
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Arbitrary Order ◽  
Ordinary Differential Operator ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Order Of Accuracy ◽  
Strongly Regular

We obtain asymptotic formulas with arbitrary order of accuracy for the eigenvalues and eigenfunctions of a nonselfadjoint ordinary differential operator of order n whose coefficients are Lebesgue integrable on [0, 1] and the boundary conditions are strongly regular. The orders of asymptotic formulas are independent of smoothness of the coefficients.

scholarly journals On the Differential Operators with Periodic Matrix Coefficients

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
O. A. Veliev
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
The Self ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Periodic Matrix ◽  
Matrix Coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then by using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.

Analyticity of the eigenvalues and eigenfunctions of an ordinary differential operator with respect to a parameter

1974 ◽  
Vol 336 (1607) ◽  
pp. 475-486 ◽  
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Arbitrary Order ◽  
General Theorem ◽  
Order Differential Operator ◽  
Eigenvalues And Eigenfunctions ◽  
Linear Problems ◽  
Homogeneous Linear ◽  
Conditions At Infinity

We present a general theorem, with simple proof, on the analyticity (with respect to a parameter λ ) of the eigenvalues and eigenfunctions of a linear homogeneous second-order differential operator H(λ) . The theorem is more general than commonly used ones (Newton 1960) in so far as the boundary conditions may depend explicitly on the parameter λ and eigenvalue E . We discuss analogous theorems, the meaning of the condition ( C ) required in the theorem, and boundary conditions at infinity. Finally we extend the theorem to cover homogeneous and non-homogeneous linear problems of arbitrary order , and general non-linear eigenvalue problems .

A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer†

1978 ◽  
Vol 82 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Ordinary Differential Operators ◽  
Half Axis ◽  
Complex Coefficients

SynopsisFormulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

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.

Asymptotic formulas for Dirichlet boundary value problems

2005 ◽  
Vol 42 (2) ◽  
pp. 153-171 ◽  
Author(s):  
Bülent Yilmaz ◽  
O. A. Veliev
Keyword(s):  
Boundary Conditions ◽  
Main Part ◽  
Dirichlet Boundary ◽  
Arbitrary Order ◽  
Dirichlet Boundary Conditions ◽  
Asymptotic Formulas ◽  
Summable Function ◽  
Special Cases ◽  
Sturm Liouville ◽  
Liouville Operators

In this article we obtain asymptotic formulas of arbitrary order for eigenfunctions and eigenvalues of the nonselfadjoint Sturm-Liouville operators with Dirichlet boundary conditions, when the potential is a summable function. Then using these we compute the main part of the eigenvalues in special cases.

Singular differential operators with spectra discrete and bounded below

1979 ◽  
Vol 84 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisA weighted, formally self-adjoint ordinary differential operator l of order 2n is considered, and conditions are given on the coefficients of l which ensure that all self-adjoint operators associated with l have a spectrum which is discrete and bounded below. Both finite and infinite singularities are considered. The results are obtained by the establishment of certain conditions which imply that l is non-oscillatory.

scholarly journals Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
Davide Bianchi
Keyword(s):  
Differential Operator ◽  
Isogeometric Analysis ◽  
Differential Operators ◽  
Continuous Operator ◽  
Uniform Grid ◽  
Discretization Scheme ◽  
Order Of Accuracy ◽  
Discretization Schemes ◽  
Symbol Function ◽  
Relative Approximation

AbstractGiven a linear self-adjoint differential operator $$\mathscr {L}$$ L along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator $$\mathscr {L}\,^{(n)}$$ L ( n ) is, compared to the spectrum of the continuous operator $$\mathscr {L}$$ L . The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function $$\omega $$ ω associated to the discrete matrix $$\mathscr {L}\,^{(n)}$$ L ( n ) . Inspired by a recent work by T. J. R. Hughes and coauthors, we prove that the symbol $$\omega $$ ω can measure, asymptotically, the maximum spectral relative error $$\mathscr {E}\ge 0$$ E ≥ 0 . It measures how the scheme is far from a good relative approximation of the whole spectrum of $$\mathscr {L}$$ L , and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees $$\mathscr {E}=0$$ E = 0 .

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