Canonical systems (Q1938654)

The author considers the equation of a string with variable density and damped endpoints of the form \[ Jy'=\lambda B(x)y,\quad x\in [0,d],\quad y(0)= y(d), \] where \[ J= \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix},\quad B(x)= \begin{pmatrix} a(x) & b(x)\\ b(x) & c(x)\end{pmatrix} \] and the function \(x\to B(x)\) is piecewise constant. Thus, there are points \(0= x_0< x_1< x_2<\dotsb< x_n= d\) and matrices \(B_k= \begin{pmatrix} a_k & b_k\\ b_k & c_k\end{pmatrix}\) such that \(B(x)= B_k\) for \(x\in (x_{k-1}, x_k]\) for \(k= 1,2,\dotsc, n\). The main problem is to find conditions on \(\{x_k\}\) and \(\{B_k\}\) so that the spectrum \(\{\lambda_i: i\in\mathbb{N}\}\) of this problem coincides with the spectrum of a problem where the matrix \(B\) is constant. In this case, the original problem is called quasihomogeneous.