scholarly journals Asymptotic estimates for spectral functions connected with hypoelliptic differential operators

1965 ◽  
Vol 5 (6) ◽  
pp. 527-540 ◽  
Author(s):  
Nils Nilsson
Keyword(s):  
Differential Operators ◽  
Asymptotic Estimates ◽  
Spectral Functions

18.—Asymptotic Estimates for the Lengths of the Gaps in the Essential Spectrum of Self-adjoint Differential Operators

1976 ◽  
Vol 74 ◽  
pp. 239-252 ◽  
Author(s):  
M. S. P. Eastham ◽  
W. N. Everitt
Keyword(s):  
Essential Spectrum ◽  
Differential Operators ◽  
Second Order ◽  
Asymptotic Estimates ◽  
Image Position ◽  
Differentiability Properties

SynopsisThe paper gives asymptotic estimates of the formas λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.

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.

Eigenvalues of regular differential operators

1978 ◽  
Vol 79 (3-4) ◽  
pp. 299-305 ◽  
Author(s):  
Harold E. Benzinger
Keyword(s):  
Fine Structure ◽  
Large Class ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Ordinary Differential Operators

SynopsisIt is shown that the fine structure of the asymptotic estimates for the eigenvalues of a large class of ordinary differential operators, can be described in terms of the Fourier coefficients of a function of class L2.

scholarly journals The Calderon-Zygmund Operator and Its Relation to Asymptotic Estimates for Ordinary Differential Operators

2017 ◽  
Vol 63 (4) ◽  
pp. 689-702 ◽  
Author(s):  
A M Savchuk
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Carleson Measure ◽  
Differential Operators ◽  
Fundamental System ◽  
Asymptotic Estimates ◽  
Zygmund Operator ◽  
Ordinary Differential Operators ◽  
Upper Half Plane

We consider the problem of estimating of expressions of the kind Υ(λ)=supx∈[0,1]∣∣∫x0f(t)eiλtdt∣∣. In particular, for the case f∈Lp[0,1], p∈(1,2], we prove the estimate ∥Υ(λ)∥Lq(R)≤C∥f∥Lp for any q>p′, where 1/p+1/p′=1. The same estimate is proved for the space Lq(dμ), where dμ is an arbitrary Carleson measure in the upper half-plane C+. Also, we estimate more complex expressions of the kind Υ(λ) arising in study of asymptotics of the fundamental system of solutions for systems of the kind y′=By+A(x)y+C(x,λ)y with dimension n as |λ|→∞ in suitable sectors of the complex plane.

On bounds for Titchmarsh–Weyl m-coefficients and for spectral functions for second-order differential operators

1984 ◽  
Vol 97 ◽  
pp. 1-7 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Weight Function ◽  
Differential Operators ◽  
Order Term ◽  
Weyl Function ◽  
Second Order ◽  
Spectral Functions ◽  
Positive Weight ◽  
Nevanlinna Function ◽  
Fixed Sign ◽  
Order Of Magnitude

SynopsisOrder-of-magnitude results are extended to the case of general second-order term, with coefficient not necessarily of fixed sign, with general positive weight-function. The bounds are used to establish the expression for the Titchmarsh–Weyl function m(λ) as a Nevanlinna function in terms of the spectral function.

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