Delta-Function Potential with a Complex Coupling

Details Delta-Function Potential with a Complex Coupling Delta-Function Potential with a Complex Coupling

2006-06-23

arxiv.org/abs/quant-ph/0606198v2

J. Phys. A: Math. Gen. 39, 13495-13506 (2006)

Physics

Quantum Physics

Published version, 14 pages, 2 figures

Scientific paper

10.1088/0305-4470/39/43/008

We explore the Hamiltonian operator H=-d^2/dx^2 + z \delta(x) where x is real, \delta(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.

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