dorsal/arxiv
View SchemaDelta-Function Potential with a Complex Coupling
| Authors | Ali Mostafazadeh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0606198 |
| URL | https://arxiv.org/abs/quant-ph/0606198 |
| DOI | 10.1088/0305-4470/39/43/008 |
| Journal | J. Phys. A: Math. Gen. 39, 13495-13506 (2006) |
Abstract
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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"abstract": "We explore the Hamiltonian operator H=-d^2/dx^2 + z \\delta(x) where x is\nreal, \\delta(x) is the Dirac delta function, and z is an arbitrary complex\ncoupling constant. For a purely imaginary z, H has a (real) spectral\nsingularity at E=-z^2/4. For \\Re(z)\u003c0, H has an eigenvalue at E=-z^2/4. For the\ncase that \\Re(z)\u003e0, H has a real, positive, continuous spectrum that is free\nfrom spectral singularities. For this latter case, we construct an associated\nbiorthonormal system and use it to perform a perturbative calculation of a\npositive-definite inner product that renders H self-adjoint. This allows us to\naddress the intriguing question of the nonlocal aspects of the equivalent\nHermitian Hamiltonian for the system. In particular, we compute the energy\nexpectation values for various Gaussian wave packets to show that the\nnon-Hermiticity effect diminishes rapidly outside an effective interaction\nregion.",
"arxiv_id": "quant-ph/0606198",
"authors": [
"Ali Mostafazadeh"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/39/43/008",
"journal_ref": "J. Phys. A: Math. Gen. 39, 13495-13506 (2006)",
"title": "Delta-Function Potential with a Complex Coupling",
"url": "https://arxiv.org/abs/quant-ph/0606198"
},
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