Saved representations and contraction hysterises of N-dimensional oscillators plus a constant force

Abstract:
I study the contraction of the $N$-dimensional oscillator with a constant force ${\bf f}$,
\[
 H=\frac {{\bf p}^2}{2m} + \frac k 2 {\bf x}^2- {\bf f} \cdot {\bf x},
\]
and show that one obtains a `contraction hysterisis', as the parameters $k$ and $f$ approach zero in different order. I show that the quadrupole moments provide a natural saved realization of the contracted $su(N)$ algebras.

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