2,603 research outputs found
Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally
challenging task, especially when a large set of eigenvalues is computed, or
just when particularly large eigenvalues are sought. This is a consequence of
the highly oscillatory behaviour of the solutions corresponding to high
eigenvalues, which forces a naive integrator to take increasingly smaller
steps. We will discuss some techniques that yield uniform approximation over
the whole eigenvalue spectrum and can take large steps even for high
eigenvalues. In particular, we will focus on methods based on coefficient
approximation which replace the coefficient functions of the Sturm-Liouville
problem by simpler approximations and then solve the approximating problem. The
use of (modified) Magnus or Neumann integrators allows to extend the
coefficient approximation idea to higher order methods
Surface shear waves in a half-plane with depth-variant structure
We consider the propagation of surface shear waves in a halfplane, whose shear modulus and density depend continuously on the depth coordinate. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency and wave number as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum in the wave number/frequency plane for the corresponding surface wave. We establish the criteria for nonexistence of surface waves and for the existence of a finite number of surface wave solutions; the number grows and tends to infinity with the wave number. The most intriguing result is a possibility of the existence of infinite number of solutions for any given wave number. These three options are conditioned by the asymptotic behaviour of the shear modulus and density close to infinite depth
The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality
We study two problems. The first one is the similarity problem for the
indefinite Sturm-Liouville operator A=-(\sgn\, x)\frac{d}{wdx}\frac{d}{rdx}
acting in . It is assumed that w,r\in L^1_{\loc}(-b,b) are
even and positive a.e. on .
The second object is the so-called HELP inequality
where the coefficients \tilde{w},\tilde{r}\in L^1_{\loc}[0,b) are
positive a.e. on .
Both problems are well understood when the corresponding Sturm-Liouville
differential expression is regular. The main objective of the present paper is
to give criteria for both the validity of the HELP inequality and the
similarity to a self-adjoint operator in the singular case. Namely, we
establish new criteria formulated in terms of the behavior of the corresponding
Weyl-Titchmarsh -functions at 0 and at . As a biproduct of this
result we show that both problems are closely connected. Namely, the operator
is similar to a self-adjoint one precisely if the HELP inequality with
and is valid.
Next we characterize the behavior of -functions in terms of coefficients
and then these results enable us to reformulate the obtained criteria in terms
of coefficients. Finally, we apply these results for the study of the two-way
diffusion equation, also known as the time-independent Fokker-Plank equation.Comment: 42 page
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