Ali Mostafazadeh 2006 J. Phys. A: Math. Gen. 39 13495 doi:10.1088/0305-4470/39/43/008
Delta-function potential with a complex coupling
Ali Mostafazadeh
Show affiliationsWe explore the Hamiltonian operator 
, where 
is the Dirac delta function and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a spectral singularity at 
. For Re(z) < 0, H has an eigenvalue at E = −z2/4. 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.
- PACS
- MSC
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81Rxx Groups and algebras in quantum theory
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
- Subjects
- Dates
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Issue 43 (27 October 2006)
Received 23 June 2006, in final form 11 September 2006
Published 11 October 2006
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Delta-function potential with a complex coupling
Ali Mostafazadeh 2006 J. Phys. A: Math. Gen. 39 13495
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