The Inverted Dirac-Moshinsky Oscillator in $(1+1)$ Dimensions

Abstract

We derive and analyze the exact solutions of the inverted Dirac-Moshinsky oscillator (IDMO) in $(1+1)$ dimensions, obtained from the standard model via the substitution $p\to p+im\ddot{I}‰{\hat{I}}^{2}x$. The upper spinor component satisfies a Weber equation with complex spectral parameter $\hat{I}»=(E^2-m^2)/(2m\ddot{I}‰)+i/2$, whose solutions are parabolic cylinder functions $D_{\hat{I}}½(\hat{I}¾)$ with complex order $\hat{I}½=\hat{I}»-1/2$. The physical spectrum is purely continuous ($|E|>m$), with no discrete bound states. Three normalization schemes are developed, and the discrete Gamow resonances at $E_n^{\pm }=\pm \sqrt{m^2+(2n+1)m\ddot{I}‰-im\ddot{I}‰}$ are identified as poles of the resolvent. The negative-energy sector describes antiparticle anti-resonances whose positive imaginary part signals vacuum instability and spontaneous pair production, analogous to the Schwinger effect. The algebraic structure is governed by the principal series of $SU(1,1)$, and the Hamiltonian is $\mathcal{P}\mathcal{T}$-symmetric with unbroken symmetry for $|E|>m$.