Odd-order differential expressions with positive supporting coefficients

1987 ◽  
Vol 105 (1) ◽  
pp. 167-192 ◽  
Author(s):  
Bernd Schultze
Keyword(s):  
Essential Spectrum ◽  
Deficiency Index ◽  
Deficiency Indices ◽  
Compact Perturbations ◽  
Odd Order ◽  
Independent Variable

SynopsisThe deficiency indices (mean deficiency index) and the essential spectrum for a class of odd order ordinary differential expressions are determined. The considered expressions are relatively bounded or relatively compact perturbations of symmetric expressions with odd order terms having as coefficients real powers of the independent variable.

The deficiency indices of powers of second order expressions with large leading coefficient

1985 ◽  
Vol 101 (3-4) ◽  
pp. 227-235
Author(s):  
Thomas T. Read
Keyword(s):  
Large Class ◽  
Differential Expression ◽  
Limit Point ◽  
Second Order ◽  
Simple Function ◽  
Deficiency Index ◽  
Deficiency Indices

SynopsisThe deficiency index of each power of the differential expression M[y] = w−1(−(py′)′ + qy), defined on [a, ∞), is calculated exactly in terms of the behaviour of a simple function of p and w for a large class of expressions satisfying a hypothesis which requires that p be large compared with w and q. In general, not all powers of M are limit-point.

scholarly journals On the Solvability Complexity Index for Unbounded Selfadjoint and Schrödinger Operators

2019 ◽  
Vol 91 (6) ◽  
Author(s):  
Frank Rösler
Keyword(s):  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Selfadjoint Operator ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Selfadjoint Operators ◽  
Complexity Index ◽  
Compact Perturbations ◽  
Complex Valued

AbstractWe study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential spectrum of a selfadjoint operator is convex, then the SCI for computing its spectrum is equal to 1. This result is then extended to relatively compact perturbations of such operators and applied to Schrödinger operators with (complex valued) potentials decaying at infinity to obtain $${\text {SCI}}=1$$SCI=1 in this case, as well.

scholarly journals Approximate point spectrum and commuting compact perturbations

1986 ◽  
Vol 28 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Vladimir Rakočević
Keyword(s):  
Banach Space ◽  
Essential Spectrum ◽  
Point Spectrum ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Approximate Point Spectrum ◽  
Infinite Dimensional ◽  
Compact Perturbations ◽  
Image Position

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B (X) (K(X)). Let σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Setσem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

Deficiency indices of an odd-order differential operator

1984 ◽  
Vol 97 ◽  
pp. 223-231 ◽  
Author(s):  
R. B. Paris ◽  
A. D. Wood
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Asymptotic Solutions ◽  
Order Differential Operator ◽  
Deficiency Indices ◽  
Odd Order ◽  
Associated Operators

SynopsisWe obtain asymptotic solutions of odd-order formally self-adjoint differential equations with power coefficients and discuss possible values for the deficiency indices of the associated operators.

Stability of deficiency indices

1977 ◽  
Vol 78 (1-2) ◽  
pp. 119-127 ◽  
Author(s):  
Horst Behncke ◽  
Heinz Focke
Keyword(s):  
Hilbert Space ◽  
Deficiency Index ◽  
Deficiency Indices ◽  
The Stability ◽  
Symmetric Operators

SynopsisMany known results about the stability of selfadjointness are extended to results about the stability of the deficiency index of closed symmetric operators on Hilbert space under perturbation.

Asymptotic theory and deficiency indices for differential equations of odd order

1981 ◽  
Vol 90 (3-4) ◽  
pp. 263-279 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Equations ◽  
Asymptotic Theory ◽  
Linear Differential Equations ◽  
Deficiency Indices ◽  
Odd Order ◽  
Linear Differential ◽  
General Conditions

SynopsisAn asymptotic theory is developed for linear differential equations of odd order. The theory is applied to the evaluation of the deficiency indicesN+andN−associated with symmetric differential expressions of odd order. General conditions on the coefficients are given under which all possible values ofN+andNsubject to |N+−N| ≦ 1 are realized.

On the L2 nature of solutions of nth order symmetric differential equations and McLeod's conjecture

1981 ◽  
Vol 90 (3-4) ◽  
pp. 209-236 ◽  
Author(s):  
R. B. Paris ◽  
A. D. Wood
Keyword(s):  
Differential Equations ◽  
Homogeneous Equation ◽  
Deficiency Index ◽  
Suitable Choice ◽  
Positive Real ◽  
Variation Of Parameters ◽  
Linearly Independent ◽  
Hypergeometric Type ◽  
Independent Variable ◽  
Real Multiple

SynopsisWe investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2m, whose coefficients involve a real multiple ɑr of certain positive real powers β of the independent variable x. Information on the L2 nature is obtained by variation of parameters from Meijer function solutions of an associated homogeneous equation of hypergeometric type. When the coefficients of the differential expressions are positive, it is possible, by a suitable choice of ɑr, β and m, to obtain between m and 2m —1 linearly independent solutions in L2(0, ∞). This proves a conjecture of J. B. McLeod that the deficiency index can take values between m and 2m —1 for such operators.

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