ASYMPTOTIC ESTIMATES FOR THE STURM-LIOUVILLE SPECTRUM

Abstract

It is shown that the differential equation y'' + + (x) y = 0 can, under suitable conditions, be solved by assuming a solution of the form y = A(x) sin (x) where '(x) = + (x) + 1/4 + (x)'(x) sin 2 (x)A'(x) = -A(x) 2 + 2 (x)'(x) cos2 (x). Use of the first equation leads, when boundary conditions are applied, to asymptotic estimates of the eigenvalues. In particular, in the case of Hill's equation, it is shown that the instability intervals vanish faster than any inverse power of k, k being the order of the corresponding eigenvalues, when (x) is an analytic function (Author)

Document Details

Document Type
Technical Report
Publication Date
Feb 01, 1961
Accession Number
AD0257782

Entities

People

  • Harry Hochstadt

Organizations

  • New York University

Tags

DTIC Thesaurus Topics

  • Analytic Functions
  • Boundaries
  • Differential Equations
  • Eigenvalues
  • Equations
  • Instability
  • Intervals
  • Mathematical Analysis
  • Mathematics
  • Partial Differential Equations
  • Real Variables

Fields of Study

  • Mathematics

Readers

  • Approximation Theory.
  • Calculus or Mathematical Analysis
  • Linear Algebra