Results and problems in the spectral theory of periodic differential equations

Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces

“…This implies that our eigenfunctions ϕ are 2π-periodic and satisfy the Neumann boundary conditions. Therefore, the eigenvalues λ of P | W coincide with (λ N 2n ) n which are known to be simple and satisfy (1.2) (see for instance chapter 4 of [7]). …”

“…Then, the spectrum of P | W coincides with the spectrum of P on [0, π] with Dirichlet boundary conditions. Therefore, the spectrum of P | W consists of simple eigenvalues and (1.2) holds (see for example chapter 4 of [7]). Thus, we have the following corollary: …”

“…Let (λ P n ) n denote the periodic eigenvalues, (λ A n ) n the anti-periodic eigenvalues, (λ D n ) n the Dirichlet eigenvalues and (λ N n ) n the Neumann eigenvalues. It is known (see for instance chapters 2 and 3 of [7]…”

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

“…Then, since W (u 2 ,ũ 2 ) is constant, we can assume 0 < ∆ũ 2 ,u2 (x) < π. This implies ∆ũ 2,u0 (c) = ∆ũ 2,u2 (c) < π and ∆ũ 2 …”

“…For convenience of the reader, we recall some basic facts of the theory of periodic differential operators first (see for example [2] or [42]). …”