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Register a new userOn perturbation theory for the Sturm-Liouville problem, Part II
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Added by
Vladimir Kalitvianski
Publication date
2018
fields
Physics
and research's language is
English
Authors
Vladimir Kalitvianski
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I study some possibilities of analytically solving a particular Sturm-Liouville problem with step-wise (piece-constant) coefficients with help of an iterative procedure mentioned in my previous paper (Greens function sum rules). I construct short, simple, but very accurate analytical formulae for calculating the ground state eigenvalue and eigenfunction as well as for calculating the first eigenfunction. I study numerical precision of the obtained approximations together with the perturbation theory results.
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We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {it Dirichlet type} boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter $alpha$, $0<alpha<1$, there is a finite set of real eigenvalues and that, for $alpha$ near $1/2$, there may be none at all. As $alpha to 1^-$ we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.
The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.
We consider Sturm-Liouville problems on the finite interval. We show that spectral data for the case of Dirichlet boundary conditions are equivalent to spectral data for Neumann boundary conditions. In particular, the solution of the inverse problem for the first one is equivalent to the solution of the inverse problem for the second one. Moreover, we discuss similar results for other Sturm-Liouville problems, including a periodic case.
This article considers the eigenvalue problem for the Sturm-Liouville problem including $p$-Laplacian begin{align*} begin{cases} left(vert uvert^{p-2}uright)+left(lambda+r(x)right)vert uvert ^{p-2}u=0,,, xin (0,pi_{p}), u(0)=u(pi_{p})=0, end{cases} end{align*} where $1<p<infty$, $pi_{p}$ is the generalized $pi$ given by $pi_{p}=2pi/left(psin(pi/p)right)$, $rin C[0,pi_{p}]$ and $lambda<p-1$. Sharp Lyapunov-type inequalities, which are necessary conditions for the existence of nontrivial solutions of the above problem are presented. Results are obtained through the analysis of variational problem related to a sharp Sobolev embedding and generalized trigonometric and hyperbolic functions.
Sturm-Liouville spectral problem for equation $-(y/r)+qy=lambda py$ with generalized functions $rge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $requiv 1$. The case of $q=0$ and self-similar $r$ and $p$ is considered as an example.
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