Inverse Problems for Sturm–Liouville-Type Differential Equation with a Constant Delay Under Dirichlet/Polynomial Boundary Conditions

The topic of this paper are non-self-adjoint second-order differential operators with constant delay generated by -y′′+q(x)y(x-τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-y''+q(x)y(x-\tau )$$\end{document} where potential q is complex-valued function, q∈L2[0,π]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\in L^{2}[0,\pi ]$$\end{document}. We study inverse problems of these operators for τ∈2π5,π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in \left[ \frac{2\pi }{5},\pi \right) $$\end{document}. We investigate the inverse spectral problems of recovering operators from their two spectra, firstly under Dirichlet–Dirichlet and second under Dirichlet/Polynomial boundary conditions. We will prove theorem of uniqueness, and we will give procedure for constructing potential. In the first case, for τ∈π2,π:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in \left[ \frac{\pi }{2},\pi \right) :$$\end{document} we will show that Fourier coefficients of a potential are uniquely0 determined by spectra. In the second case for τ∈2π5,π2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in \left[ \frac{2\pi }{5},\frac{\pi }{2}\right) ,$$\end{document} we will construct integral equation under potential and we will prove that this integral equation has a unique solution. Also, we will show that other parameters are uniquely determined by spectra.