On the recovery of a differential equation from its spectral functions (Q5952271)

An inverse spectral problem associated with the differential equation NEWLINE\[NEWLINEy''+({\mu}-q(x))y=0, \quad x{\in}[0,{\infty}),\tag{1}NEWLINE\]NEWLINE with the boundary condition NEWLINE\[NEWLINEy(0)\cos{\alpha}+y'(0)\sin{\alpha}=0,\;{\alpha}{\in}[0,{\pi}),\tag{2}NEWLINE\]NEWLINE is studied. Here, \(q\in L_{\text{loc}}^1[0,{\infty})\) is defined and finite on \([0,{\infty}).\) The equation above is also considered on the interval \([X,{\infty})\), for \(X \geq 0,\) with the boundary condition NEWLINE\[NEWLINEy(X)\cos{\alpha}+y'(X) \sin{\alpha}=0. \tag{3}NEWLINE\]NEWLINE Denote by \(\rho_{\alpha,X}(\mu)\) the spectral function associated with (1), (3). The Riccati equation is used to determine \(q(x)\) knowing the spectral derivatives \(\rho_{\alpha,X}'(\mu)\) for a given \(\mu\) and variable \(\alpha\) and \(X.\) Some examples are given.