A Mapping Method in Inverse Sturm–Liouville Problems with Singular Potentials

In the space $L_2[0,\pi]$, the Sturm–Liouville operator $L_\mathrm D(y)=-y''+q(x)y$ with the Dirichlet boundary conditions $y(0)=y(\pi)=0$ is analyzed. The potential $q$ is assumed to be singular; namely, $q=\sigma'$, where $\sigma\in L_2[0,\pi]$, i.e., $q\in W_2^{-1}[0,\pi]$. The inverse problem of reconstructing the function $\sigma$ from the spectrum of the operator $L_\mathrm D$ is solved in the subspace of odd real functions $\sigma(\pi/2-x)=-\sigma(\pi/2+x)$. The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.