Extremal values of eigenvalues of Sturm–Liouville operators with potentials in balls (original) (raw)
Related papers
A Survey on Extremal Problems of Eigenvalues
Abstract and Applied Analysis, 2012
Given an integrable potential q ∈ L 1 0, 1 , R , the Dirichlet and the Neumann eigenvalues λ D n q and λ N n q of the Sturm-Liouville operator with the potential q are defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when the L 1 metric for q is given; q L 1 r. Note that the L 1 spheres and L 1 balls are nonsmooth, noncompact domains of the Lebesgue space L 1 0, 1 , R , · L 1 . To solve these extremal problems, we will reveal some deep results on the dependence of eigenvalues on potentials. Moreover, the variational method for the approximating extremal problems on the balls of the spaces L α 0, 1 , R , 1 < α < ∞ will be used. Then the L 1 problems will be solved by passing α ↓ 1. Corresponding extremal problems for eigenvalues of the one-dimensional p-Laplacian with integrable potentials have also been solved. The results can yield optimal lower and upper bounds for these eigenvalues. This paper will review the most important ideas and techniques in solving these difficult and interesting extremal problems. Some open problems will also be imposed.
On the first two eigenvalues of Sturm-Liouville operators
Proceedings of the American Mathematical Society
Among the Schrödinger operators with single-well potentials defined on (0, π) with transition point at π 2 , the gap between the first two eigenvalues of the Dirichlet problem is minimized when the potential is constant. This extends former results of Ashbaugh and Benguria with symmetric singlewell potentials. An analogous result is given for the Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.
On the first two eigenvalues of the Sturm-Liouville operators
2008
Among the Schrödinger operators with single-well potential defined on (0, π) with transition point at π 2 , the gap between the first two eigenvalues of the Dirichlet problem is being investigated. We also show how this extends former results with symmetric potential. Finally we will consider an analogous Dirichlet problem of vibrating strings with single-barrier densities for the ratio of the first two eigenvalues.
Continuity in weak topology and extremal problems of eigenvalues of the ppp-Laplacian
Transactions of the American Mathematical Society, 2011
We will study the dependence of eigenvalues of the one-dimensional p-Laplacian on potentials or weights. Two results are obtained. One is the continuity of eigenvalues in potentials with respect to the weak topologies of L γ spaces, 1 ≤ γ ≤ ∞, and the other is the continuous differentiability of eigenvalues in potentials with respect to L γ norms. As applications, we will study some extremal problems of eigenvalues by developing some analytical methods.
Hacettepe Journal of Mathematics and Statistics, 2019
The spectral problem\[-y''+q(x)y=\lambda y,\ \ \ \ 0<x<1\]\[y(0)=0, \quad y'(0)=\lambda(ay(1)+by'(1)),\]is considered, where lambda\lambdalambda is a spectral parameter, q(x)inL1(0,1)q(x)\in{{L}_{1}}(0,1)q(x)inL1(0,1) is a complex-valued function, aaa and bbb are arbitrary complex numbers which satisfy the condition ∣a∣+∣b∣ne0|a|+|b|\ne 0∣a∣+∣b∣ne0. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in Lp(0,1){{L}_{p}}(0,1)Lp(0,1)) of the above-mentioned Sturm-Liouville problem.
On two-point boundary value problems for the Sturm-Liouville operator
arXiv: Spectral Theory, 2015
In this paper, we study spectral problems for the Sturm-Liouville operator with arbitrary complexvalued potential q(x) and two-point boundary conditions. All types of mentioned boundary conditions are considered. We ivestigate in detail the completeness property and the basis property of the root function system.