On an inverse Regge problem for the Sturm–Liouville operator with deviating argument

Boundary value problem of the form $Ly=\rho^2 y$, $y(0)=y'(\pi)+i\rho y(\pi)=0$, where $L$ is the Sturm–Liouville operator with constant delay $a$ is studied. The boundary value problem can be considered as a generalization of the classical Regge problem. The potential $q({}\cdot{})$ is assumed to be a real-valued function from $L_2(0,\pi)$ equal to $0$ a.e. on $(0,a)$. No other restrictions on the potential are imposed, in particular, we make no additional assumptions regarding an asymptotical behavior of $q(x)$ as $x\to\pi$. In this general case, the asymptotical expansion of the characteristic function of the boundary value problem as $\rho\to\infty$ contains no leading term. Therefore, no explicit asymptotics of the spectrum can be obtained using the standard methods. We consider the inverse problem of recovering the operator from given subspectrum of the boundary value problem. Inverse problems for differential operators with deviating argument are essentially more difficult with respect to the classical inverse problems for differential operators. “Non-local” nature of such operators is insuperable obstacle for classical methods of inverse problem theory. We consider the inverse problem in case of delay, which is not less than the half length of the interval and establish that the specification of the subspectrum of the boundary value problem determines, under certain conditions, the potential uniquely. Corresponding subspectra are characterized in terms of their densities. We also provide a constructive procedure for solving the inverse problem.