Francesco Cannata

Schrodinger operators with complex potential but real spectrum

Abstract Several aspects of complex-valued potentials generating a real and positive spectrum are discussed. In particular, we construct complex-valued potentials whose corresponding Schrodinger eigenvalue problem can be solved analytically.

Supersymmetry without hermiticity within PT symmetric quantum mechanics

Abstract A new model of supersymmetry between bosons and fermions is proposed. Its representation space is spanned by states with PT symmetry and real energies but the inter-related partner Hamiltonians themselves remain complex and non-Hermitian. The formalism admits vanishing Witten index.

Smooth dynamical crossing of the phantom divide line of a scalar field in simple cosmological models

Simple scalar field cosmological models are considered describing gravity assisted crossing of the phantom divide line. This crossing or (de)-phantomization characterized by the change of the sign of the kinetic term of the scalar field is smooth and driven dynamically by the Einstein equations. Different cosmological scenarios, including the phantom phase of matter are sketched.

Scattering in PT-symmetric quantum mechanics

Abstract A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P , T or PT invariance of the Hamiltonian. Applications to some solvable PT -symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with Hermiticity, T invariance and PT invariance.

PT-symmetric sextic potentials

Abstract The family of complex PT-symmetric sextic potentials is studied to show that for various cases the system is essentially quasi-solvable and possesses real, discrete energy eigenvalues. For a particular choice of parameters, we find that under supersymmetric transformations the underlying potential picks up a reflectionless part.

New methods for the two-dimensional Schrödinger equation: SUSY-separation of variables and shape invariance

Two new methods for the investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. The first one—SUSY-separation of variables—is based on the intertwining relations of higher-order SUSY quantum mechanics (HSUSY QM) with supercharges allowing separation of variables. The second one is a generalization of shape invariance. While in one dimension shape invariance allows us to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been explored yet. Here we provide a formal framework in HSUSY QM for two-dimensional quantum mechanical systems for which shape invariance holds. Given the knowledge of one eigenvalue and eigenfunction, shape invariance allows us to construct a chain of new eigenfunctions and eigenvalues. These methods are applied to a two-dimensional quantum system, and partial explicit solvability is achieved in the sense that only part of the spectrum is found analytically and a limited set of eigenfunctions is constructed explicitly.

Algebraic and scattering aspects of a -symmetric solvable potential

We study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions. We determine the transmission and reflection coefficients for the -symmetric case and also formulate the problem in terms of an SU(1,1) potential group, which allows unified treatment of the discrete and the continuous spectra in a natural way. We find that (bound and scattering) states of the -symmetric problem supply a basis for the unitary irreducible representations of the SU(1,1) potential group, and this gives a straightforward group theoretical interpretation of the fact that the (complex) -invariant potential has a real energy spectrum.

A new class of PT-symmetric Hamiltonians with real spectra

Abstract We investigate complex PT-symmetric potentials, associated with quasi-exactly solvable non-Hermitian models involving polynomials and a class of rational functions. We also look for special solutions of intertwining relations of SUSY quantum mechanics providing a partnership between a real and a complex PT-symmetric potential of the kind mentioned above. We investigate conditions sufficient to ensure the reality of the full spectrum or, for the quasi-exactly solvable systems, the reality of the energy of the finite number of levels.

PT-Symmetric Potentials and the so(2,2) Algebra

Starting from a differential realization of the generators of the so(2, 2) algebra we connect the eigenvalue equation of the Casimir invariant either with the hypergeometric equation, or the Schr¨ odinger equation. In the latter case we consider problems for whichso(2, 2) appears as a potentia la lgebra, connecting states with the same energy in different potentials. We analyse the role of the two so(2, 1) subalgebras and point out their importance for PT -symmetric problems, where the doubling of bound states is known to occur. We present two mechanisms for this and illustrate them with the example of the Scarf and the P¨ oschl–Teller II potentials. We also analyse scattering states, transmission

PT symmetry breaking and explicit expressions for the pseudo-norm in the Scarf II potential

Closed expressions are derived for the pseudo-norm, norm and orthogonality relations for arbitrary bound states of the PT symmetric and the Hermitian Scarf II potential for the first time. The pseudo-norm is found to have indefinite sign in general. Some aspects of the spontaneous breakdown of PT symmetry are analyzed.