Spectral problem for a self-adjoint third-order differential operator with non-local potential on finite interval is studied. Elementary functions that are analogues of sines and cosines such operators described. Direct inverse problems solved.

In quantum theory, real degrees of freedom are usually described by operators which are self-adjoint. There are, however, exceptions to the rule. This is because, in infinite dimensional Hilbert spaces, an operator is not necessarily self-adjoint even if its expectation values are real. Instead, the operator may be merely symmetric. Such operators are not diagonalizable and as a consequence the...

A wide class of problems in the study of the spectral and orbital stability of dispersive waves in Hamiltonian systems can be reduced to understanding the so-called “energy spectrum,” that is the spectrum of the second variation of the Hamiltonian evaluated at the wave shape, which is constrained to act on a closed subspace of the underlying Hilbert space. We present a substantially simplified ...

This agrees with the definition of the spectrum in the matrix case, where the resolvent set comprises all complex numbers that are not eigenvalues. In terms of its spectrum, we will see that a compact operator behaves like a matrix, in the sense that its spectrum is the union of all of its eigenvalues and 0. We begin with the eigenspaces of a compact operator. We start with two lemmas that we w...

Let V be a finite-dimensional vector space, either real or complex, and equipped with an inner product 〈· , ·〉. Let A : V → V be a linear operator. Recall that the adjoint of A is the linear operator A : V → V characterized by 〈Av, w〉 = 〈v, Aw〉 ∀v, w ∈ V (0.1) A is called self-adjoint (or Hermitian) when A = A. Spectral Theorem. If A is self-adjoint then there is an orthonormal basis (o.n.b.) o...