Inverse spectral problems for left-definite Sturm-Liouville equations with indefinite weight
The authors consider the Sturm-Liouville problem of the form \[ -(py')'+qy=\lambda ry \quad \text{on }[a,b], \tag{1} \] \[ y(a)\cos\alpha+(py')(a)\sin\alpha = 0, \quad \alpha\in \left(-{\pi\over 2}, {\pi\over 2}\right],\tag{2} \] \[ y(b)\cos\beta+(py')(b)\sin\beta = 0, \quad \beta\in \left(-{\pi\over 2}, {\pi\over 2}\right], \tag{3} \] assuming that \(p\) is a \(C^1\) positive function with absolutely continuous derivative on \([a,b]\) with \(a<0<b\); \(r\) is a \(C^1\) positive function with absolutely continuous derivative on \((0,b]\) and a negative function with absolutely continuous derivative on \([a,0)\). It is assumed also that \(-\infty<r(0^-)<0<r(0^+)<\infty\) or that \(r\) is differentiable at \(0\) with \(r'(0)>0\), and \(q\) and \(\widetilde{q}\) are real \(L^1\) functions on \([a,b]\). The main result is as follows: Let \(\lambda_n\) and \(\omega_n\) denote the eigenvalues and the eigenfunctions of problem (1)-(3), where \(\omega_n\) is normalized by \(\omega_n(a)=\sin\alpha\), \(\omega'_n(a)=-\cos\alpha\). Let \(\rho_n=|||\omega_n|||=\sqrt{\int_a^b \overline{\omega_n}(-(p\omega'_n)'+q\omega_n)dx}\). Define \(\widetilde{\rho}_n\) and \(\widetilde{\lambda}_n\) in an analogous manner but with \(q\) replaced by \(\widetilde{q}\). Suppose that \(\lambda_n=\widetilde{\lambda}_n\) and \(\rho_n=\widetilde{\rho}_n\) for all \(n=\pm 1, \pm 2,\dots\). Then \(q=\widetilde{q}\).
scientific article; zbMATH DE number 1836761
scientific article