Inverse spectral theory for a singular Sturm-Liouville operator on \([0,1]\)
Let us consider the Schrödinger operator \(\overset\circ H=-\Delta +q(| x|),\) \(x\in\mathbb{R}^ 3\) acting on functions in \(C^\infty\) (\(| x| \leq 1)\) vanishing on \(| x| =1\) and \(q\in L^ 2[0,1]\). One may prove [\textit{M. Reed} and \textit{B. Simon}, ``Methods of modern mathematical physics. II: Fourier analysis, self-adjointness'' (1975; Zbl 0308.47002)] that the closure H of \(\overset\circ H\) is an unitarily equivalent operator to a direct sum of Sturm-Liouville operators acting on \(L^ 2[0,1]\). The first operator in this sum is \(A_ 0(q)=-d^ 2/dx^ 2+q(x)\) with domain \(D_ 0=\{u\in H^ 2[0,1]:\) \(u(0)=u(1)=0\}\) and the second is \(A_ 1(q)=-d^ 2/dx^ 2+2/x^ 2+q(x)\) with domain \(D_ 1=\{u\in L^ 2[0,1]:\) u, du/dx are absolutely continuous on [0,1], \(A_ 1(q)u\) belongs to \(L^ 1[0,1]\) and \(u(1)=0\}\). In this paper the authors study the inverse spectral problem for \(A_ 1(q)\) extending the results obtained for \(A_ 0(q)\) by \textit{J. Pöschel} and \textit{E. Trubowitz} [Inverse spectral theory (1987; Zbl 0623.34001)].
scientific article; zbMATH DE number 4127635