Comment on: 'A simple analytical expression for bound state energies for an attractive Gaussian confining potential'

We discuss a recently proposed analytical formula for the eigenvalues of the Gaussian well and compare it with the analytical expression provided by the variational method with the simplest trial function. The latter yields considerably more accurate results than the former for the energies and critical parameters

Eigenvalues and eigenfunctions of the anharmonic oscillator V(x,y)=x2y2V(x,y)=x^{2}y^{2}

We obtain sufficiently accurate eigenvalues and eigenfunctions for the anharmonic oscillator with potential $V(x,y)=x^{2}y^{2}$ by means of three different methods. Our results strongly suggest that the spectrum of this oscillator is discrete in agreement with early rigorous mathematical proofs and against a recent statement that cast doubts about it

On two different kinds of resonances in one-dimensional quantum-mechanical models

We apply the Riccati-Pad\'{e} method and the Rayleigh-Ritz method with complex rotation to the study of the resonances of a one-dimensional well with two barriers. The model exhibits two different kinds of resonances and we calculate them by means of both approaches. While the Rayleigh-Ritz method reveals each set at a particular interval of rotation angles the Riccati Pad\'{e} method yields both of them as roots of the same Hankel determinants

Rayleigh-Ritz variational method with suitable asymptotic behaviour

We discuss Rayleigh-Ritz variational calculations with nonorthogonal basis sets that exhibit the correct asymptotic behaviour. We construct the suitable basis sets for general one-dimensional models and illustrate the application of the approach on two double-well oscillators proposed recently by other authors. The rate of convergence of the variational method proves to be considerably greater than the one exhibited by the recently developed orthogonal polynomial projection quantization

Non-hermitean hamiltonians with unitary and antiunitary symmetry

We analyze several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. It enables us to predict the degeneracy of the energy levels and to reduce the dimension of the matrices necessary for the diagonalization of the Hamiltonian in a given basis set. We can also classify the solutions according to the irreducible representations of the point group and thus analyze their properties separately. One of the main results of this paper is that some PT-symmetric Hamiltonians with point-group symmetry $C_{2v}$ exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit which suggests that the phenomenon is not robust. Point-group symmetry enables us to explain such anomalous behaviour and to choose a suitable antiunitary operator for the PT symmetry