A spectral theorem for reversible second order equations with periodic coefficients
The differential operator \(L=-d^ 2/dt^ 2+Q(t)\) is considered where \(Q\) is a symmetric \(n\) by \(n\) matrix over the Sobolev space of functions \(x:[0,T]\mapsto \mathbb{R}^ n\) satisfying \(x(0)=x(T)\). The Morse index of \(L\) is the number of its negative eigenvalues; the system \(Lx=0\) is said to be reversible if \(Q(-t)=Q(t)\). The main theorem proved in this paper says that if \(Lx=0\) is reversible, the index of \(L\) is zero, and \(Lx=0\) has no symmetric \(T\)-periodic solution then the monodromy matrix \(M(T)\) has no eigenvalues on the unit circle. This means that if, in particular, \(Lx=0\) is the variational system of an autonomous nonlinear system of differential equations with respect to a periodic solution then the latter is hyperbolic, i.e. the system exhibits an exponential dichotomy. The results can be applied to the characterization of periodic solutions of certain Hamiltonian systems with respect to the manifold of solutions with fixed total energy.
second order linear differential operator
scientific article; zbMATH DE number 68018