On the location of eigenvalues of second order linear differential operators

1978 ◽  
Vol 80 (1-2) ◽  
pp. 15-22 ◽  
Author(s):  
Ian Knowles
Keyword(s):  
Boundary Condition ◽  
Hilbert Space ◽  
Differential Expression ◽  
Differential Operators ◽  
Homogeneous Boundary Condition ◽  
Upper Bounds ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Linear Differential ◽  
Image Position

SynopsisThis paper is concerned with finding upper bounds on the set of eigenvalues of self-adjoint differential operators generated in the Hilbert space L2[0, ∞) by the differential expressionon [0,∞), together with a real homogeneous boundary condition at t = 0.

Completeness properties in L2 of the eigenfunctions of two semi-linear differential operators

1980 ◽  
Vol 88 (3) ◽  
pp. 451-468 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Hilbert Space ◽  
Product Space ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Real Hilbert Space ◽  
The Real ◽  
Linear Differential Operators ◽  
Linear Problems ◽  
Linear Differential ◽  
Image Position

This paper concerns the boundary-value problemsin which λ is a real parameter, u is to be a real-valued function in C2[0, 1], and problem (I) is that with the minus sign. (The differential operators are called semi-linear because the non-linearity is only in undifferentiated terms.) If we linearize the equations (for ‘ small’ solutions u) by neglecting , there result the eigenvalues λ = n2π2 (with n = 1,2,…) and corresponding normalized eigenfunctionsand it is well known ((2), p. 186) that the sequence {en} is complete in that it is an orthonormal basis for the real Hilbert space L2(0, 1). We shall be concerned with possible extensions of this result to the non-linear problems (I) and (II), for which non-trivial solutions (λ, u) bifurcate from the trivial solution (λ, 0) at the points {n2π2,0) in the product space × L2(0, 1). (Here denotes the real line.)

The Oscillation of Fourth Order Linear Differential Operators

1975 ◽  
Vol 27 (1) ◽  
pp. 138-145 ◽  
Author(s):  
Roger T. Lewis
Keyword(s):  
Fourth Order ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Oscillatory Behavior ◽  
The Self ◽  
Order Operator ◽  
Order Linear ◽  
Linear Differential Operators ◽  
Even Order ◽  
Linear Differential

Define the self-adjoint operatorwhere r(x) > 0 on (0, ∞) and q and p are real-valued. The coefficient q is assumed to be differentiate on (0, ∞) and r is assumed to be twice differentia t e on (0, ∞).The oscillatory behavior of L4 as well as the general even order operator has been considered by Leigh ton and Nehari [5], Glazman [2], Reid [7], Hinton [3], Barrett [1], Hunt and Namb∞diri [4], Schneider [8], and Lewis [6].

Regularly solvable extensions of non-self-adjoint ordinary differential operators

1984 ◽  
Vol 97 ◽  
pp. 79-95 ◽  
Author(s):  
W. D. Evans
Keyword(s):  
Hilbert Space ◽  
Weighted Space ◽  
Differential Operators ◽  
Linear Operators ◽  
Order Linear ◽  
Adjoint Pair ◽  
Closed Operators ◽  
Ordinary Differential Operators ◽  
Linear Differential ◽  
Densely Defined

SynopsisLetL0,M0be closed densely defined linear operators in a Hilbert spaceHwhich form an adjoint pair, i.e.. In this paper, we study closed operatorsSwhich satisfyand are regularly solvable in the sense of Višik. The abstract results obtained are applied to operators generated by second-order linear differential expressions in a weighted spaceL2(a, b; w).

Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)

1999 ◽  
Vol 129 (1) ◽  
pp. 165-179
Author(s):  
Yurii B. Orochko
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Differential Expression ◽  
Symmetric Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Image Position ◽  
Symmetric Differential Operators

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.

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