  In order to extend the spectral action principle to non-compact spaces, we
propose a framework for spectral triples where the algebra may be non-unital
but the resolvent of the Dirac operator remains compact. We show that an
example is given by the supersymmetric harmonic oscillator which,
interestingly, provides two different Dirac operators. This leads to two
different representations of the volume form in the Hilbert space, and only
their product is the grading operator. The index of the even-to-odd part of
each of these Dirac operators is 1.
  We also compute the spectral action for the corresponding Connes-Lott
two-point model. There is an additional harmonic oscillator potential for the
Higgs field, whereas the Yang-Mills part is unchanged. The total Higgs
potential shows a two-phase structure with smooth transition between them: In
the spontaneously broken phase below a critical radius, all fields are massive,
with the Higgs field mass slightly smaller than the NCG prediction. In the
unbroken phase above the critical radius, gauge fields and fermions are
massless, whereas the Higgs field remains massive.
