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.
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.
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 systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, , in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1 + 1)-dimensions. For the case that v ( x ) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v ( x ) = i x 3 . This reveals an infinite class of previously unknown - as well as non- -inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in that is consistent with the usual choice for the quantum Hamiltonian. Finally, we discuss subtleties involved with higher dimensions and systems having a fixed length scale.
We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modeling the propagation of electromagnetic waves traveling in a waveguide half and half filled with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter ζ, we show that the equivalent Hermitian Hamiltonian has the form p2∕2m+(ζ2∕2)∑n=0∞{αn(x),p2n} with αn(x) vanishing outside an interval that is three times larger than the support of v(x), i.e., in 2∕3 of the physical interaction region the potential v(x) vanishes identically. We provide a physical interpretation for this unusual behavior and comment on the classical limit of the system.
We consider a couple of examples to study the pseudo-Hermitian interaction in relativistic quantum mechanics. Rasbha interaction, commonly used to study the spin Hall effect, is considered with imaginary coupling. The corresponding Dirac Hamiltonian is shown to be parity pseudo-Hermitian. In the other example we consider parity pseudo-Hermitian scalar interaction with arbitrary parameter in Dirac theory. In both cases we show that the energy spectrum is real and all the other features of nonrelativistic pseudo-Hermitian formulation are present. Using the spectral method, the positive definite metric operator (η) has been calculated explicitly for both the models to ensure positive definite norms for the state vectors.
We introduce a Hamiltonian H α, β depending on the two complex parameters α and β and develop a through analysis of the spectral properties of this Hamiltonian and its adjoint. We determine the regions in the space of the complex parameters α and β, where H α, β is (i) Hermitian, (ii) non-Hermitian with real spectrum and (iii) non-Hermitian but PT -symmetric. In the case when H α, β is non-Hermitian having real spectrum, we derive a closed formula for a family of the metric operators, depending on two arbitrary real positive parameters, which render the Hamiltonian H α, β Hermitian. In a particular case we calculate the Hermitian counterpart of H α, β .