Lower bounds on the eigenvalue gap for vibrating strings

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:417

Article

Chen, Duo-Yuan¹  Huang, Min-Jei¹ 

[1] Natl Tsing Hua Univ, Dept Math, Hsinchu 30043, Taiwan

关键词: Lower bound;    Eigenvalue gap;    Vibrating string;    Symmetric density;    Monotone density;    Bessel function;   

DOI  :  10.1016/j.jmaa.2014.03.045

来源: Elsevier

【 摘 要 】

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 rho is symmetric on the interval [0, a], and if lambda(1) and lambda(2) are the first two eigenvalues of u ''(x) + lambda rho(x)u(x) = 0 in (0, a) with u(0) = u(a) = 0 boundary conditions, then lambda(2)-lambda(1) > max{1/integral(a/2)(0) (a/2 - x)rho(x)dx, pi(2)/rho M-a2}, where rho(M) = max 0(<= x <= a) rho(x). The ideas used also lead to applications in the case of monotone densities. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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