Spectrum of a Differential Operator with Periodic Generalized Potential

We study some spectral problems for a second-order differential operator with periodic potential. Notice that the given potential is a sum of zero- and first-order generalized functions. It is shown that the spectrum of the investigated operator consists of infinite number of gaps whose length limit unlike the classic case tends to nonzero constant in some place and to infinity in other place.

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First Order Operators on Manifolds With a Group Action

Canadian Journal of Mathematics ◽

10.4153/cjm-1996-039-6 ◽

1996 ◽

Vol 48 (4) ◽

pp. 758-776 ◽

Cited By ~ 1

Author(s):

H. D. Fegan ◽

B. Steer

Keyword(s):

Differential Operator ◽

Group Action ◽

Lie Group ◽

Differential Operators ◽

Order Differential Operator ◽

Compact Lie Group ◽

First Order ◽

Rank 2 ◽

Homogeneous Operators

AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

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Ger-type and Hyers-Ulam stabilities for the first-order linear differential operators of entire functions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204304333 ◽

2004 ◽

Vol 2004 (22) ◽

pp. 1151-1158 ◽

Cited By ~ 5

Author(s):

Takeshi Miura ◽

Go Hirasawa ◽

Sin-Ei Takahasi

Keyword(s):

Entire Function ◽

Differential Operator ◽

Stability Problem ◽

Differential Operators ◽

Entire Functions ◽

Ulam Stability ◽

Nonzero Constant ◽

First Order ◽

Linear Differential Operators ◽

Linear Differential

Lethbe an entire function andTha differential operator defined byThf=f′+hf. We show thatThhas the Hyers-Ulam stability if and only ifhis a nonzero constant. We also consider Ger-type stability problem for|1−f′/hf|≤ϵ.

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To the eigenvalue problems of a special-loaded first-order differential operator

International Journal of Mathematical Analysis ◽

10.12988/ijma.2014.48263 ◽

2014 ◽

Vol 8 ◽

pp. 2247-2254

Author(s):

Nurlan Imanbaev ◽

Burkhan Kalimbetov ◽

Zhanibek Khabibullayev

Keyword(s):

Differential Operator ◽

Eigenvalue Problems ◽

Order Differential Operator ◽

First Order

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Solvability of a first order differential operator on the two-torus

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2014.02.006 ◽

2014 ◽

Vol 416 (1) ◽

pp. 166-180

Author(s):

A.P. Bergamasco ◽

P.L. Dattori da Silva ◽

A. Meziani

Keyword(s):

Differential Operator ◽

Order Differential Operator ◽

First Order

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Singularities of the eta function of first order differential operators

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/v10309-012-0040-5 ◽

2012 ◽

Vol 20 (2) ◽

pp. 59-70

Author(s):

Paul Loya ◽

Sergiu Moroianu

Keyword(s):

Differential Operator ◽

Differential Operators ◽

Closed Manifold ◽

Order Differential Operator ◽

First Order ◽

Positive Integers

Abstract We report on a particular case of the paper [7], joint with Raphaël Ponge, showing that generically, the eta function of a first-order differential operator over a closed manifold of dimension n has first-order poles at all positive integers of the form n - 1; n - 3; n - 5;. . . .

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Reducibility of differential operators some: examples

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100000608 ◽

1979 ◽

Vol 86 (1) ◽

pp. 29-33 ◽

Cited By ~ 1

Author(s):

B. Fishel ◽

N. Denkel

Keyword(s):

Differential Operator ◽

Differential Operators ◽

Volterra Operator ◽

Second Order ◽

Order Differential Operator ◽

Order Operator ◽

Minimal Operator ◽

Deficiency Indices ◽

First Order ◽

Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

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