A finiteness result for inverse three spectra Sturm-Liouville problems
The inverse problem for the Sturm-Liouville equation \[ -u'' + q(x) u = \lambda u \] is studied, which consists in recovering the real-valued potential \(q \in L^1[0, 1]\) and the coefficients \(h_0\), \(h_1\) by three spectra of the boundary value problems \(L\), \(L^-\) and \(L^+\) with the following boundary conditions \[ L : \quad u'(0) + h_0 u(0) = 0 = u'(1) + h_1 u(1), \] \[ L^- : \quad u'(0) + h_0 u(0) = 0 = u'(a) + h_- u(a), \] \[ L^+ : \quad u'(a) + h_+ u(a) = 0 = u'(1) + h_1 u(1), \] where \(h_0\), \(h_1\), \(h_+\), \(h_-\) are real numbers, \(a \in (0, 1)\). The authors prove that, if \(a\) is rational, \(h_- \neq h_+\) and the following condition is satisfied \[ \max \{ h_-, h_+\} < B(h_0, h_1; q) \quad \text{or} \quad \min\{ h_-, h_+ \} > B(h_0, h_1; q), \] where \[ B(h_0, h_1; q) = a h_1 + (1 - a) h_0 + \frac{a}{2} \int_a^1 q(t) \, dt - \frac{1 - a}{2} \int_0^a q(t) \, dt, \] the inverse problem by three spectra has at most a finite number of solutions. Note that the authors do not require the interlacing property of the spectra, which is usually used for such kind of problems.
scientific article; zbMATH DE number 6693696