Emanuela Caliceti

Spectra of PT-symmetric operators and perturbation theory

A criterion is formulated for existence and another for the non-existence of complex eigenvalues for a class of non-self-adjoint operators in Hilbert space invariant under a particular discrete symmetry. Applications to the PT-symmetric Schrodinger operators are discussed.

symmetric non-self-adjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum

Consider in L(R), d ≥ 1, the operator family H(g) := H0 + igW . H0 = a∗1a1 + . . .+ a∗ d ad + d/2 is the quantum harmonic oscillator with rational frequencies , W a P symmetric bounded potential, and g a real coupling constant. We show that if |g| < ρ, ρ being an explicitly determined constant, the spectrum of H(g) is real and discrete. Moreover we show that the operator H(g) = a∗ 1 a1 + a ∗ 2 a2 + iga ∗ 2 a1 has real discrete spectrum but is not diagonalizable.Consider in , the operator family . H0 = a*1a1 + + a*dad + d/2 is the quantum harmonic oscillator with rational frequencies, W is a symmetric bounded potential, and g is a real coupling constant. We show that if |g| < ?, ? being an explicitly determined constant, the spectrum of H(g) is real and discrete. Moreover we show that the operator has a real discrete spectrum but is not diagonalizable.

Distributional Borel summability of odd anharmonic oscillators

It is proved that the divergent Rayleigh-Schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are Borel summable in the distributional sense to the resonances naturally associated with the system.

Perturbation theory of symmetric Hamiltonians

In the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric Hamiltonians is proved using stability techniques. We apply this method to symmetric unperturbed Hamiltonians perturbed by symmetric additional interactions.

Quadratic

It is established that a -symmetric elliptic quadratic differential operator with real spectrum is similar to a self-adjoint operator precisely when the associated fundamental matrix has no Jordan blocks.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

{\mathcal P}{\mathcal T}

We prove the reality of the perturbed eigenvalues of some PT symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one- dimensional x 2 (ix) for 1 < < 0.

-symmetric operators with real spectrum and similarity to self-adjoint operators

We study nonlinear perturbative expansions for -symmetric local Schrodinger operators. The Schrodinger operator is a sum of the harmonic oscillator Hamiltonian and a local -symmetric potential depending, in general, nonlinearly on the perturbation parameter. A specific class of models having real spectrum for any value of the parameter is proposed.

PT Symmetric Schrodinger Operators: Reality of the Perturbed Eigenvalues ?

Abstract We prove reality of the spectrum for a class of PT − symmetric, non self-adjoint quantum nonlinear oscillators of the form H=p 2 + P(q) + igQ(q). Here P(q) is an even polynomial of degree 2p positive at infinity, Q(q) an odd polynomial of degree 2r − 1, and the conditions p > 2r, |g| < for some > 0 hold.

An analytic family of -symmetric Hamiltonians with real eigenvalues

Let H be any -symmetric Schrodinger operator of the type −2Δ + (x21 + + x2d) + igW(x1, ..., xd) on , where W is any odd homogeneous polynomial and . It is proved that is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of H, i.e., the eigenvalues of . Moreover we explicitly construct the canonical expansion of H and determine the singular values μj of H through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues λj of H by Weyls inequalities.

On a Class of Non Self-Adjoint Quantum Nonlinear Oscillators with Real Spectrum

It is proved that the divergent perturbation expansion for the vacuum polarization by an external constant electric field in the pair production sector is Borel summable in the distributional sense.