The one-dimensional Coulomb-like potential with a real coupling constant beta, and a centrifugal-like core of strength G = alpha^2 - {1/4}, viz. V(x) = {alpha^2 - (1/4)}/{(x-ic)^2} + beta/|x-ic|, is discussed in the framework of PT-symmetry. The PT-invariant exactly solvable model so formed, is found to admit a double set of real and discrete energies, numbered by a quasi-parity q = +/- 1.
The one-dimensional Coulomb-like potential with a real coupling constant beta, and a centrifugal-like core of strength G = alpha^2 - {1/4}, viz. V(x) = {alpha^2 - (1/4)}/{(x-ic)^2} + beta/|x-ic|, is discussed in the framework of PT-symmetry. The PT-invariant exactly solvable model so formed, is found to admit a double set of real and discrete energies, numbered by a quasi-parity q = +/- 1.
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form H=p2+V(x), where V(x) is odd in x, pure imaginary, and vanishes as ∣x∣→∞. Five PT-symmetric potentials are studied: the Scarf-II potential V1(x)=iA1sech(x)tanh(x), which decays exponentially for large ∣x∣; the rational potentials V2(x)=iA2x/(1+x4) and V3(x)=iA3x/(1+∣x∣3), which decay algebraically for large ∣x∣; the step-function potential V4(x)=iA4sgn(x)θ(2.5−∣x∣), which has compact support; the regulated Coulomb potential V5(x)=iA5x/(1+x2), which decays slowly as ∣x∣→∞ and may be viewed as a long-range potential. The real parameters An measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrödinger eigenvalue problems associated with these potentials reveal that the spectra of the corresponding Hamiltonians exhibit universal properties. In general, the eigenvalues are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from 0 to +∞. The complex eigenvalues occur in discrete complex-conjugate pairs and for Vn(x) (1≤n≤4) the number of these pairs is finite and increases as the value of the strength parameter An increases. However, for V5(x) there is an {\it infinite} sequence of discrete eigenvalues with a limit point at the origin. This sequence is complex, but it is similar to the Balmer series for the hydrogen atom because it has inverse-square convergence.
The spectrum of a one-dimensional Hamiltonian with potential V(x)=ix2 for negative x and V(x)=−ix2 for positive x is analyzed. The Schrödinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.
PT Symmetric Hamiltonian Model and Exactly Solvable Potentials
Searching for non-Hermitian (parity-time)PT-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a PT symmetric non-Hermitian Hamiltonian model which is given as H^=ω(b^†b^+21)+α(b^2−(b^†)2) where ω and α are real constants, b^ and b†^ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of PT symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian H is pseudo-Hermitian, we have obtained the Hermitian equivalent of H which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. H is called pseudo-Hermitian, if there exists a Hermitian and invertible operator η satisfying H†=ηHη−1. For the Hermitian Hamiltonian h, one can write h=ρHρ−1 where ρ=η is unitary. Using this ρ we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function ρ is obtained for each potential case.
We exhibit the exact ground state and the corresponding binding energy of the system composed of N (nonidentical) quantum-mechanical particles interacting in one dimension via the potential ${V}_{\mathrm{ij}}({x}_{i}\ensuremath{-}{x}_{j})=g{({x}_{i}\ensuremath{-}{x}_{j})}^{\ensuremath{-}2}+({f}_{i}\ensuremath{-}{f}_{j}){({x}_{i}\ensuremath{-}{x}_{j})}^{\ensuremath{-}1}$. We discuss some special choices of the constants fi and g that yield examples of possible physical interest.