The point-continuous spectrum of second-order, ordinary differential operators

1979 ◽  
Vol 84 (3-4) ◽  
pp. 197-211 ◽  
Author(s):  
Christine Thurlow
Keyword(s):  
Boundary Condition ◽  
Continuous Spectrum ◽  
Differential Operators ◽  
Second Order ◽  
Ordinary Differential Operators ◽  
Isolated Points ◽  
Infinite Sets ◽  
Countably Infinite ◽  
Infinite Set

SynopsisGiven any countably infinite set of isloated points on the ℷ -axis, it is shown that there is a continuous q(x) such that these points constitute exactly the point-continuous spectrum for the equation yn″(x) + (ℷ —q(x))y(x) = 0(0≦x<∞) with some homogenous boundary condition at x = 0. This extends a result given by Eastham and McLeod for countably infinite sets of isolated points on the positive ℷ-axis.

The existence of eigenvalues embedded in the continuous spectrum of ordinary differential operators

1977 ◽  
Vol 79 (1-2) ◽  
pp. 25-34 ◽  
Author(s):  
M. S. P. Eastham ◽  
J. B. McLeod
Keyword(s):  
Boundary Condition ◽  
Continuous Spectrum ◽  
Differential Operators ◽  
Homogeneous Boundary Condition ◽  
Ordinary Differential Operators ◽  
Isolated Points ◽  
Countably Infinite ◽  
Set Of Points ◽  
Infinite Set

SynopsisIn answer to two questions raised by W. N. Everitt, we show that, givenp> l and any countably infinite set of isolated points on the positive λ-axis, there is aq(x)inLp(0, ∞) for which the set of points constitutes the point-continuous spectrum associated with the equationy”(x) + {λ − q(x)}y(x) = 0 (0≦x<∞)and some homogeneous boundary condition atx= 0.

V.— On the Spectrum of Second Order Partial Differential Operators.

1970 ◽  
Vol 69 (1) ◽  
pp. 77-84
Author(s):  
K. J. Brown ◽  
I. M. Michael
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

scholarly journals Multiple solutions for eigenvalue problems involving an indefinite potential and with (p1(x), p2(x)) balanced growth

2019 ◽  
Vol 27 (1) ◽  
pp. 289-307 ◽  
Author(s):  
Vasile-Florin Uţă
Keyword(s):  
Boundary Condition ◽  
Continuous Spectrum ◽  
Dirichlet Boundary Condition ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
The Other ◽  
Balanced Growth ◽  
Indefinite Potential

Abstract In this paper we are concerned with the study of the spectrum for a class of eigenvalue problems driven by two non-homogeneous differential operators with different variable growth and an indefinite potential in the following form $$\eqalign{ & - {\rm{div}}\left[ {{\cal H}(x,|\nabla u|)\nabla u + \Im (x,|\nabla u|)\nabla u} \right] + V(x)|u{|^{m(x) - 2}}u = \cr & = \lambda \left( {|u{|^{{q_1}(x) - 2}} + |u{|^{{q_2}(x) - 2}}} \right)u\;{\rm{in}}\;\Omega , \cr}$$ which is subjected to Dirichlet boundary condition. The proofs rely on variational arguments and they consist in finding two Rayleigh-type quotients, which lead us to an unbounded continuous spectrum on one side, and the nonexistence of the eigenvalues on the other.

Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum

1988 ◽  
Vol 109 (1-2) ◽  
pp. 127-144 ◽  
Author(s):  
F. Fiedler
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Oscillation Criteria ◽  
Elliptic Differential Equations ◽  
Ordinary Differential Operators ◽  
Elliptic Differential Operators ◽  
Sturm Liouville ◽  
Lower Spectrum ◽  
Liouville Operators

SynopsisSufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

Inverse-monotone nonlinear differential operators of the second order

1978 ◽  
Vol 79 (3-4) ◽  
pp. 193-226 ◽  
Author(s):  
Johann Schröder
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Monotone Operators ◽  
Second Order ◽  
Differential Inequalities ◽  
Partial Differential ◽  
Nonlinear Differential Operators ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisThis paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.

Sign in / Sign up

Export Citation Format

Share Document