PT-Symmetric Radial Oscillator
Exact Solution on Shifted Contour
Starting from the radial Schrödinger equation on the complex-shifted contour $r=x-i\varepsilon,; x\in(-\infty,\infty),; \varepsilon>0$,
\[\left[-\frac{d^2}{dr^2}+r^2+\frac{\beta^2-\frac14}{r^2}\right]\psi(r)=E\psi(r),\]we derive the exact bound-state spectrum and wavefunctions.
Near-Origin Analysis
Near $r=0$, the inverse-square term dominates, reducing the equation to
\[\left[-\frac{d^2}{dr^2}+\frac{\beta^2-\frac14}{r^2}\right]\psi(r)\approx 0.\]Assume a power-law behavior,
\[\psi(r)\sim r^s.\]Then
\[\psi’’(r)=s(s-1)r^{s-2},\]and substitution yields
\[-s(s-1)r^{s-2}+\left(\beta^2-\frac14\right)r^{s-2}=0.\]Hence the indicial equation is
\[s(s-1)=\beta^2-\frac14,\]with solutions
\[s=\frac12\pm\beta.\]Therefore, the two admissible local behaviors are
\[\psi(r)\sim r^{\frac12+\beta}, \qquad \psi(r)\sim r^{\frac12-\beta}.\]These two independent behaviors generate the two quasi-parity branches of the spectrum.
Quasi-Parity in PT-Symmetric Quantum Mechanics
In PT-symmetric Quantum Mechanics, quasi-parity $Q=\pm1$ is introduced because the ordinary parity operator $P$ no longer commutes with the Hamiltonian after the complex coordinate shift $|r| = x - i\varepsilon$, even though combined $PT$ symmetry remains preserved.As a consequence:
- The two independent solutions of the radial equation can no longer be classified simply as even or odd under spatial reflection.
- Instead, they are labeled by a generalized parity-like quantum number called quasi-parity.
Role of Quasi-Parity
Quasi-parity distinguishes the two admissible branches of the spectrum through the relation $$ \alpha = -Q\beta $$ thereby unifying both solution families into a single spectral formula. Quasi-parity serves as the natural replacement for ordinary parity in non-Hermitian but PT-symmetric systems.Hermitian Limit
In the Hermitian limit $\varepsilon \to 0$:
- The Hamiltonian regains ordinary parity symmetry.
- Quasi-parity smoothly reduces to standard spatial parity.
Thus, quasi-parity acts as the proper extension of parity within the PT-symmetric framework.
Asymptotic Factorization
For large $|r|$, the harmonic term dominates, so normalizability requires
\[\psi(r)\sim e^{-r^2/2}.\]Combining asymptotic and near-origin behavior, use the ansatz
\[\psi(r)=r^{\rho+\frac12}e^{-r^2/2}F(r^2),\]where
\[\rho=\pm\beta.\]Define
\[z=r^2.\]Then
\[\psi(r)=r^{\rho+1/2}e^{-z/2}F(z).\]Reduction to Laguerre Equation
Substituting the ansatz into the Schrödinger equation gives
\[zF’’(z)+(\rho+1-z)F’(z)+\frac{E-2\rho-2}{4}F(z)=0.\]This is the associated Laguerre differential equation,
\[z y’’+(\alpha+1-z)y’+Ny=0,\]whose polynomial solutions are
\[y(z)=L_N^{(\alpha)}(z), \qquad N=0,1,2,\dots\]provided
\[N=\frac{E-2\rho-2}{4}.\]Solving for the energy,
\[E=4N+2\rho+2.\]Since $\rho=\pm\beta$, the two branches are
\[E=4N+2\beta+2, \qquad E=4N-2\beta+2.\]Quasi-Parity Formulation
Introduce quasi-parity $Q=\pm1$ via
\[\rho=-Q\beta.\]Then the spectrum becomes
\[E_N^{(Q)}=4N+2\rho+2,\]or equivalently,
\[E_N^{(Q)}=4N+2-2Q\beta.\]Exact Eigenfunctions
Using
\[F(z)=L_N^{(\rho)}(z),\]the normalized wavefunctions are
\[\psi(r)=\mathcal N\;r^{\rho+1/2}e^{-r^2/2}L_N^{(\rho)}(r^2),\]with $\rho=-Q\beta$. Thus,
\[\psi(r)=\mathcal N\;r^{-Q\beta+1/2}e^{-r^2/2}L_N^{(-Q\beta)}(r^2).\]Quasi-Parity Sectors
-
For $Q=-1$: \(\rho=\beta, \qquad E=4N+2\beta+2, \qquad \psi(r)\sim r^{1/2+\beta}.\)
-
For $Q=+1$: \(\rho=-\beta, \qquad E=4N-2\beta+2, \qquad \psi(r)\sim r^{1/2-\beta}.\)
The two quasi-parity sectors correspond to the two admissible singular behaviors at the origin on the shifted complex contour.
Final Exact Solution
The PT-symmetric radial oscillator admits the exact spectrum
\[E=4N+2\rho+2, \qquad \rho=-Q\beta, \qquad Q=\pm1, \qquad N=0,1,2,\dots\]with eigenfunctions
\[\psi(r)=\mathcal N\;r^{\rho+1/2}e^{-r^2/2}L_N^{(\rho)}(r^2);\;\;\qquad \mathcal N=\frac{N!}{\Gamma(N+\rho+1)}\]When the PT-regularization is removed by taking $\varepsilon \to 0$, the PT-symmetric solutions continuously reduce to those of ordinary Hermitian quantum mechanics: at $\beta=\tfrac{1}{2}$ the spectrum reproduces the standard linear harmonic oscillator states, and quasi-parity becomes identical to ordinary spatial parity. The two independent solution branches behave near the origin as
\[\psi(r)\sim r^{1/2-\beta}\]and
\[\psi(r)\sim r^{1/2+\beta}\]where the branch proportional to $r^{1/2-\beta}$ is more singular near $r=0$, while the branch proportional to $r^{1/2+\beta}$ is less singular. For $\beta\ge1$, the more singular solution ceases to be square-integrable and must therefore be discarded as unphysical. This shows that PT-symmetric regularization temporarily enlarges the admissible solution space by permitting both branches, but upon returning to ordinary Hermitian quantum mechanics only the normalizable physical states survive, leaving the singular non-normalizable branch excluded.