Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and l...

Abstract. We generalize earlier results of [2], [3], [6], [13], [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space B ∞,1(R ), then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then ‖f(A1 · · · , An)− f(B1, · · · , Bn)‖ ≤ constmax1≤j≤n ‖Aj −Bj‖ α for all n...

Interpolation inequalities of Gagliardo-Nirenberg type and compactness results for self-adjoint trace-class operators with finite kinetic energy are established. Applying these results to the minimization of various free energy functionals, we determine for instance stationary states of the Hartree problem with temperature corresponding to various statistics. Key-words. Compact self-adjoint ope...

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...