Deformed Canonical (anti‐)commutation relations and non‐self‐adjoint hamiltonians

Weak pseudo-bosons

“…This is related to the fact that the eigenvalues of, say, N and N † , coincide and that their eigenvectors are related by the operators S ϕ and S ψ , in agreement with what is known on intertwining operators, [14,15]. Many more results and examples on D-quasi bosons can be found in [1].…”

“…Here 1 1 is the identity operator on H. Notice that this situation extends that of ordinary bosons, which is recovered if b = a † , and that of pseudo-bosons, for which it is assumed that a dense subspace of H exists, D, which is left invariant by the action of a, b, and of their (Hilbert-)adjoints a † and b † . In this paper the relevant aspect is that this set D ⊂ H is replaced by a set of distributions, and that the biorthogonality of the eigenstates of the two adjoint number operators, see [1], will be replaced by a weak biorthogonality, i.e. by a biorthogonality between distributions which, of course, should be defined properly.…”

Gibbs states defined by biorthogonal sequences

“…In particular, in [7] several triples of operators of this kind have been discussed and the following eigenvalues equations have been found in concrete quantum mechanical models…”

“…basis for the Hilbert space H, while F ϕ = {ϕ n , n ≥ 0} and F ψ = {ψ n , n ≥ 0} are two biorthogonal sets, ϕ n , ψ m = δ n,m , but not necessarily bases for H. However, quite often, F ϕ and F ψ are complete (or total: the only vector which is orthogonal to all the ϕ n 's, or to all the ψ n 's, is the zero vector) in H and, see [7], they are also D-quasi bases, i.e., they produce a weak resolution of the identity in a suitable set D, dense in H: n f, ϕ n ψ n , g = n f, ψ n ϕ n , g = f, g , for all f, g ∈ D.…”

Symmetrized exponential oscillator

“…Now, let us only mention the feature of non-analyticity of the potential in the origin. Indeed, the analyticity along the whole real line is precisely what makes harmonic oscillators so popular, say, as toy models in quantum field theory [4] as well as in rigorous functional analysis [5,6].…”