Delta-Function Potential with a Complex Coupling
arXiv:quant-ph/0606198 · doi:10.1088/0305-4470/39/43/008
Abstract
We explore the Hamiltonian operator H=-d^2/dx^2 + z δ(x) where x is real, δ(x) is the Dirac delta function, and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a (real) spectral singularity at E=-z^2/4. For \Re(z)<0, H has an eigenvalue at E=-z^2/4. For the case that \Re(z)>0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to address the intriguing question of the nonlocal aspects of the equivalent Hermitian Hamiltonian for the system. In particular, we compute the energy expectation values for various Gaussian wave packets to show that the non-Hermiticity effect diminishes rapidly outside an effective interaction region.
Published version, 14 pages, 2 figures
References in corpus (3)
Cited by in corpus (12)
- Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator
- Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics
- Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
- Scattering theory with localized non-Hermiticities
- Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
- Discrete PT-symmetric models of scattering
- Non-Hermitian Hamiltonians of Lie algebraic type
- Scattering theory using smeared non-Hermitian potentials
- Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem
- Fundamental length in quantum theories with PT-symmetric Hamiltonians
- PT-symmetric deformations of Calogero models
- Exact Isospectral Pairs of PT-Symmetric Hamiltonians