Inverse spectral problems for Sturm--Liouville operators in impedance form
The paper presents a study of the direct and inverse spectral problems for a class of Sturm-Liouville equations on an interval in the so-called impedance form \[ -(a^2(x)y'(x))'=\lambda^2 a^2(x)y(x), \quad x\in[0,1], \] subjected to certain boundary conditions, namely, the Neumann boundary conditions \(y'(0)=y'(1)=0\) and the Neumann--Dirichlet ones \(y'(0)=y(1)=0\). The impedance function \(a(x)\) is assumed to be positive and to belong to the Sobolev space \(W_p^1(0,1)\) for some \(p\in[1,\infty)\). The presented approach is actually aimed at recovery of the logarithmic derivative \(\tau=(\log a)'\in L_p(0,1)\). The authors give a complete solution of the direct and inverse problems for an arbitrary \(p\in[1,\infty)\), i.e., they give a complete description of all possible spectra for the class of operators under consideration (basically, it is the description of the asymptotics of the eigenvalues) and for any such spectral data, they provide an effective algorithm of recovery of the function \(\tau\).
scientific article; zbMATH DE number 2172990
scientific article