Periodic ``weighted'' operators
The author and his collaborators, specifically M. Krein and P. Kargaev, extensively studied problems leading to second-order differential equations with periodic coefficients. These problems arise in studies of propagation of waves in a periodic medium, mechanics of the hydrogen atom in a periodic magnetic field, or vibration of crystals with periodic dielectric coefficients. This study, initiated by Lyapunov, was continued in the 1950s by M. Krein, who extended Lyapunov results to the \(2\times 2\) matrix case. It considers equations of the type \[ -f''(t)=\lambda b^2(t) f(t),\quad\lambda\in \mathbb{C},\;t\in\mathbb{R},\tag{1} \] with \(b(t)\geq 0\) and \(1\)-periodic, and \(\int^1_0 b(t) dt= 1\). The investigation of the inverse problem leads to the study of the weighted operator given by \(T_b f= -b^{-2}f''(t)\) posed in the \(L^2\) setting. This was studied by Lyapunov and later by M. Krein. They found that the spectrum of \(T_b f\) is absolutely continuous and consists of intervals \(\sigma_n\), which are separated by gaps \(\gamma_n= (\lambda^-_n, \lambda^+_n)\). If the gap is degenerate, then these intervals merge. A Lyapunov function is defined in terms of the elementary solutions \(y_1\), \(y_2\) to system (1), with initial conditions imposed on these functions at \(t= 0\). The Lyapunov function is defined as \(D(\lambda)= {1\over 2}(y_2'(1, \lambda)+ y_1(1,\lambda))\). The author rewrites system (1) by introducing the variable \(x(t)= \int^t_0 b(\tau) d\tau\) and variable \(\rho(x)= b(t(x))^{{1\over 2}}\). The well-known substitution \(q(t)= \rho'/\rho\) completes this introduction of new variables. The corresponding operator \(T_\rho f= -\rho^{-2}(\rho^2 f')'= -f''- 2qf'\) is even, general with nonsmooth coefficients. The author considers the case when these coefficients are smooth, and derives asymptotic estimates on the lengths of the gaps in the spectrum. His proofs use the analysis of a conformal map called quasi-momentum. The Lyapunov function plays an essential part in the proofs of the estimates at high-energy levels.
second-order ordinary differential equations
scientific article; zbMATH DE number 1915522