The Null Space Pursuit Algorithm Based on an Arbitrary Order Differential Operator

2016 ◽  
pp. 10-15
Author(s):  
Weiwei Xiao ◽  
Taotao Xing
Keyword(s):  
Differential Operator ◽  
Arbitrary Order ◽  
Null Space ◽  
Order Differential Operator

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 .

scholarly journals On the Sampson Laplacian

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1059-1070 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok ◽  
Josef Mikes
Keyword(s):  
Differential Operator ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Analytical Method ◽  
Null Space ◽  
Second Order ◽  
Order Differential Operator ◽  
Vanishing Theorems ◽  
Symmetric Tensors ◽  
Lowest Eigenvalue

In the present paper we consider the little-known Sampson operator that is strongly elliptic and self-adjoint second order differential operator acting on covariant symmetric tensors on Riemannian manifolds. First of all, we review the results on this operator. Then we consider the properties of the Sampson operator acting on one-forms and symmetric two-tensors. We study this operator using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenb?ck decomposition. Further we estimate operator?s lowest eigenvalue.

scholarly journals Gap opening in two-dimensional periodic systems

2019 ◽  
Vol 23 (01) ◽  
pp. 1950080
Author(s):  
D. I. Borisov ◽  
P. Exner
Keyword(s):  
Differential Operator ◽  
Finite Number ◽  
Distinctive Feature ◽  
Perturbation Parameter ◽  
Periodic Systems ◽  
Order Differential Operator ◽  
Two Dimensional ◽  
Gap Opening ◽  
Gap Control ◽  
Waveguide System

We present a new method of gap control in two-dimensional periodic systems with the perturbation consisting of a second-order differential operator and a family of narrow potential “walls” separating the period cells in one direction. We show that under appropriate assumptions one can open gaps around points determined by dispersion curves of the associated “waveguide” system, in general any finite number of them, and to control their widths in terms of the perturbation parameter. Moreover, a distinctive feature of those gaps is that their edge values are attained by the corresponding band functions at internal points of the Brillouin zone.

Stability of Solutions of Differential-operator and Operator-difference Equations in the Sense of Perturbation of Operators

2006 ◽  
Vol 6 (3) ◽  
pp. 269-290 ◽  
Author(s):  
B. S. Jovanović ◽  
S. V. Lemeshevsky ◽  
P. P. Matus ◽  
P. N. Vabishchevich
Keyword(s):  
Differential Operator ◽  
Difference Equations ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Order Differential Operator ◽  
Stability Of Solutions ◽  
Operator Equations ◽  
The Difference ◽  
Coefficient Stability

Abstract Estimates of stability in the sense perturbation of the operator for solving first- and second-order differential-operator equations have been obtained. For two- and three-level operator-difference schemes with weights similar estimates hold. Using the results obtained, we construct estimates of the coefficient stability for onedimensional parabolic and hyperbolic equations as well as for the difference schemes approximating the corresponding differential problems.

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.

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