On quasihomogeneous strings (Q2577374)

Consider the boundary value problem \[ -y''= \lambda^2 \rho(x)y\qquad\text{for }0< x< b,\quad y(0)= y(b)= 0, \] describing the oscillations of a string, under the assumptions \[ \rho(x)= a^2_k\quad\text{for }x\in (c_{k-1}, c_k),\;k= 1,\dots, n, \] \[ 0= c_0< c_1<\cdots< c_n= b, \] \[ a_k> 0,\;a_k\,(c_k- c_{k-1})= \alpha\quad\text{for }k= 1,\dots, n. \] A string is called quasihomogeneous if its spectrum coincides with that of a homogeneous string \((\rho(x)\equiv a)\). The authors prove a criterion for a string to be quasihomogeneous.