Abstract
A simple and accurate numerical technique for finding eigenvalues, node structure, and expectation values of -symmetric potentials is devised. The approach involves expanding the solution to the Schrödinger equation in series involving powers of both the coordinate and the energy. The technique is designed to allow one to impose boundary conditions in -symmetric pairs of Stokes sectors. The method is illustrated by using many examples of -symmetric potentials in both the unbroken- and broken--symmetric regions.
Highlights
Energy quantization is a consequence of demanding that ψ(z) decay exponentially in a -symmetric pair of Stokes sectors
To determine the spectrum associated with a symmetric pair of sectors it suffices to determine the real energies for which c is real
Plots of the wave functions indicate that the cut off λ = 5 is a good approximation for the first few eigenstates; a plot of ψ3(x) is given in figure 3 and the expectation values ázmñn in the N = 3 are listed in table 2
Summary
The area of research known as -symmetric quantum theory began with the discovery that the complex -symmetric Schrödinger equation. For any energy E, real or complex, there is a solution that decays exponentially in any given wedge. The key point of this paper is that if both E and c are real, the solution will decay in the image of the sector This is the crucial step in the numerical procedure because it makes explicit use of the symmetry of the potential. To determine the spectrum associated with a symmetric pair of sectors it suffices to determine the real energies for which c is real This can be implemented graphically by plotting Im c as a function of E. As the double power series are expansions in iz and E, we expect that the truncation is less accurate for higher energies.
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