Inverse spectral problem with partial information given on the potential and norming constants (Q2884403)

Consider the regular Sturm-Liouville operator \(L:=L(q,h_0,h_1)\) defined in \(L^2[0,1]\) by NEWLINE\[NEWLINE Lu:=-u''+q(x) NEWLINE\]NEWLINE and subject to the separated boundary conditions NEWLINE\[NEWLINE u'(0)-h_0u(0)=u'(1)-h_1 u(1)=0. NEWLINE\]NEWLINE Here, the potential \(q\in L^1[0,1]\) is real valued and \(h_0,h_1\in \mathbb{R}\cup\{\infty\}\).NEWLINENEWLINEIt is well known that the operator \(L\) is self-adjoint in \(L^2[0,1]\), lower semibounded and that its spectrum is discrete and consists of simple real eigenvalues \(\sigma(L)=\{\lambda_j\}_{j=0}^\infty\), \(\lambda_j\uparrow +\infty\).NEWLINENEWLINEThe knowledge of \(\sigma(L)\) is, in general, insufficient for recovering the potential \(q\), and one usually requires additional spectral data in order to uniquely recover \(q\) (the set of norming constants, or the second spectrum corresponding to \(\tilde{h}_0=h_0\) and \(\tilde{h}_1\neq h_1\), or the knowledge of \(q\) for \(x\in [0,1/2]\)).NEWLINENEWLINEIn the paper under review, the authors obtain some uniqueness results which imply that the potential \( q\) can be completely determined even if only partial information is given on \( q\) together with partial information on the spectral data, consisting of either one full spectrum and a subset of norming constants or a subset of pairs of eigenvalues and the corresponding norming constants.