16,733 research outputs found
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
We obtain sufficiently accurate eigenvalues and eigenfunctions for the
anharmonic oscillator with potential 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 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
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