Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4616038 | Journal of Mathematical Analysis and Applications | 2014 | 9 Pages |
Abstract
We provide lower bounds on the eigenvalue gap for vibrating strings with fixed endpoints depending only on qualitative properties of the density function. For example, if the density ρ is symmetric on the interval [0,a][0,a], and if λ1λ1 and λ2λ2 are the first two eigenvalues of u″(x)+λρ(x)u(x)=0u″(x)+λρ(x)u(x)=0 in (0,a)(0,a) with u(0)=u(a)=0u(0)=u(a)=0 boundary conditions, thenλ2−λ1>max{1∫0a/2(a2−x)ρ(x)dx,π2ρMa2}, where ρM=max0⩽x⩽aρ(x). The ideas used also lead to applications in the case of monotone densities.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Duo-Yuan Chen, Min-Jei Huang,