  The nonlinear Schroedinger model is a prototypical dispersive wave equation
that features finite time blowup, either for supercritical exponents (for fixed
dimension) or for supercritical dimensions (for fixed nonlinearity exponent).
Upon identifying the self-similar solutions in the so-called "co-exploding
frame", a dynamical systems analysis of their stability is natural, yet is
complicated by the mixed Hamiltonian-dissipative character of the relevant
frame. In the present work, we study the spectral picture of the relevant
linearized problem. We examine the point spectrum of 3 eigenvalue pairs
associated with translation, $U(1)$ and conformal invariances, as well as the
continuous spectrum. We find that two eigenvalues become positive, yet are
attributed to symmetries and are thus not associated with instabilities. In
addition to a vanishing eigenvalue, 3 more are found to be negative and real,
while the continuous spectrum is nearly vertical and on the left-half
(spectral) plane. Finally, the subtle effects of the boundaries are also
assessed and their role in the observed weak eigenvalue oscillations is
clarified.
