On the Barcilon formula for the string equation with a piecewise continuous density function
Chao-Liang Shen · Inverse Problems · 2005
Let p(x) be a positive PC-function defined in the interval [0, 1], and let y(x; λ) denote the solution of the initial value problem y''(x) + λp(x)y(x) = 0, y(0) = 0, y'(0) = 1. Then for . If the density function p(x) is a PC1-function, then , where λn(a) is the nth eigenvalue of the equation y''(x) + λp(x)y(x) = 0, y(0) = y(a) = 0, and νn(a) is the nth eigenvalue of the equation u''(x) + νp(x)u(x) = 0, u(0) = u'(a) = 0.