An isospectral problem related to the Dirichlet eigenvalues and the Neumann eigenvalues of a string equation and some related problems

作者：Shen Chao Liang^*

来源：Inverse Problems, 2011, 27(7): 075005.

DOI：10.1088/0266-5611/27/7/075005

Let p(x) be a positive continuous function defined in [0, 1] with integral(1)(0) p(t) dt = 1. Denote S(p)(x) = integral(1)(x) p(t) dt. We prove that the string equation, with the density function p(x), satisfies the Dirichlet-Neumann (DN)-isospectral property if and only if the function S(p) satisfies the relation S(p)(S(p)(x)) = x for every x is an element of [0, 1]. Applying this result, with the help of a technique developed by Kac and Krein for transforming a potential equation into a string equation, we investigate the DN-isospectral problem for the potential equation with a continuous potential function defined in [0, 1]. We prove among others that a potential equation, with a continuous potential function q, satisfies the condition nu(0)(q) = 0 and the DN-isospectral property if and only if q(x) = (f' (x))(2) + f ''(x), x is an element of [0, 1], where f (x) is a periodic C(2)-function of period 2, whose Fourier series expansion is of the form Sigma(infinity)(m=0) a(2m+1) cos(2m + 1)pi x.

• 出版日期2011-7
• 单位清华大学