  We investigate the spectral and eigenstate properties of the quantum
superexponential oscillator. Our focus is on the quantum signatures of the
recently observed transition of the energy dependent period of the
corresponding classical superexponential oscillator. We show that the ground
state exhibits a remarkable metamorphosis of decentering, asymmetrical
squeezing and the development of a tail. Analyzing the central moments up to
high order a characteristic transition from exponentially decaying moments to
increasing moments is unraveled. A corresponding spectral analysis shows that,
surprisingly, to a good approximation the spectrum is equidistant. A closer
look, however, reveals a spectral scaling behaviour below the transition point
which is replaced by irregular oscillations above the transition energy.
Excited bound states are analyzed up to the continuum threshold. We discuss
future perspectives and possible experimental realizations of the
superexponential oscillator.
