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arXiv:quant-ph/0606198 · Journal of Physics A Mathematical and General

Delta-function potential with a complex coupling

Ali Mostafazadeh

Published Oct 10, 2006• 79 citations
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Abstract

We 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 = − 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.

DOI
10.1088/0305-4470/39/43/008

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We 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 = − 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.
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.
We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p 2 /2 m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h . η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ + ) = −ℑ(ζ − ). This in particular contains the -symmetric case: ζ + = ζ* − . We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H . We show that these physical quantities are not directly sensitive to the presence of the -symmetry.
A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., − d2 dx 2 +v(x), and its adjoint. We establish the equivalence of this description with the mathematicians ’ definition of spectral singularities for the potential v(x) = z−δ(x + a) + z+δ(x − a), where z ± and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry. PACS numbers: 03.65.-w
A curious feature of complex scattering potentials v ( x ) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., , and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential v ( x ) = z − δ( x + a ) + z + δ( x − a ), where z ± and a are respectively complex and real parameters and δ( x ) is the Dirac delta function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z ± where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing -symmetry.
The axioms of quantum mechanics require that the Hamiltonian of any closed system is self-adjoint, so that energy levels are real and time evolution preserves probability. On the other hand, non-Hermitian Hamiltonians with PT\mathcal{PT}PT symmetry can have both real spectra and unitary time evolution. In this paper, we study in detail a pair of quantum oscillators coupled by an imaginary bilinear term, both in quantum mechanics and in quantum field theory. We discuss explicitly how such Hamiltonians lead to perfectly sound physical theories with real spectra and unitary time evolution, in spite of their non-Hermiticity. We also analyze two-point correlation functions and their associated K\"allen-Lehmann representation. In particular, we discuss the intimate relation between positivity violation of the spectral functions and the nonobservability of operators in a given correlation function. Finally, we conjecture that positivity violation of some spectral functions of the theory could be a generic sign of the existence of complex pairs of energy eigenvalues (i.e., a PT\mathcal{PT}PT-broken phase) somewhere in its parameter space.
We consider the interaction between the Hermitian world, represented by a real delta-function potential $\ensuremath{-}\ensuremath{\alpha}\ensuremath{\delta}(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $i\ensuremath{\beta}(\ensuremath{\delta}(x\ensuremath{-}L)\ensuremath{-}\ensuremath{\delta}(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave function is small for LLL large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be studied instead in the context of quasi-Hermiticity, whereby quantum mechanics is endowed with a new metric $\ensuremath{\eta}$, and consequently a new wave function $\ensuremath{\Psi}(x)$, defined in terms of the original wave function $\ensuremath{\psi}(x)$ by means of $\ensuremath{\eta}$. In this picture, working perturbatively in $\ensuremath{\beta}$, the bound-state wave function is actually unchanged from its unperturbed form for $|x|\ensuremath{\ll}L$. However, the scattering wave function, for $|x|\ensuremath{\gg}L$, is changed in a significant manner. In particular, there are incoming and outgoing waves on both sides of the potential. One can then no longer talk in terms of reflection and transmission coefficients, but the total right-moving flux is now conserved.