VII.—On the Spectrum of Ordinary Second Order Differential Operators.

1969 ◽  
Vol 68 (2) ◽  
pp. 95-119 ◽  
Author(s):  
Jyoti Chaudhuri ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Analytic Function ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Space Theory ◽  
Alternative Approach

SynopsisThis paper considers properties of the spectrum of differential operators derived from differential expressions of the second order. The object is to link the spectral properties of these differential operators with the analytic, function-theoretic properties of the solutions of the differential equation. This provides an alternative approach to the spectral theory of these differential operators but one which is consistent with the standard definitions used in Hilbert space theory. In this way the approach may be of interest to applied mathematicians and theoretical physicists.

Spectral Theory for the Differential Equation Lu= λ Mu

1958 ◽  
Vol 10 ◽  
pp. 431-446 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Spectral Theory ◽  
Real Line ◽  
Differential Operators ◽  
Identity Operator ◽  
The Real ◽  
Ordinary Differential Operators ◽  
Extensive Bibliography

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L 2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

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 Positive functionals and oscillation criteria for second order differential systems

1979 ◽  
Vol 22 (3) ◽  
pp. 277-290 ◽  
Author(s):  
Garret J. Etgen ◽  
Roger T. Lewis
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Second Order ◽  
Linear Operators ◽  
The Self ◽  
Differential Systems ◽  
Bounded Linear ◽  
Positive Functionals ◽  
Image Position ◽  
Adjoint Operators

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation

scholarly journals Multiparameter spectral theory of singular differential operators

1988 ◽  
Vol 31 (1) ◽  
pp. 49-66 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Spectral Theory ◽  
Unbounded Domain ◽  
Differential Operators ◽  
Separation Of Variables ◽  
Singular Case ◽  
Ordinary Differential Operators ◽  
Adjoint Operators ◽  
Unbounded Intervals

In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].

On the study of the spectral properties of differential operators with a smooth weight function

2020 ◽  
pp. 25-47
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Third Order ◽  
Indicator Diagram ◽  
Estimates Of Solutions ◽  
Smooth Coefficients

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

Existence for Second Order Differential Inclusions on ℝ+ Governed by Monotone Operators

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Gheorghe Moroşanu
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Differential Inclusions ◽  
Real Hilbert Space ◽  
Monotone Operators ◽  
Second Order

AbstractConsider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t) + q(t)u′(t) ∈ Au(t) + f (t) for a.a. t ∈ ℝ

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