My personal experience is that the people who are entertained by 
the abstract
geometry of the Sophus Lie's symmetries usually do not feel too
much at ease with any specific physical pragmatism, and vice versa.
Not the present authors. Having reviewed the details of the Liean
construction of the symmetries (and of a sample bunch of the
related particular analytic and exceptional closed solutions) for
the specific parabolic partial differential Schroedinger equation
(mentioned in the title and, for the nonce, non-separable) they
surprise us all by the climax of their story in section 4.
There, they put things in a complete order (from the point of view 
of a quantum physicist at least) and guarantee the correct
(viz, spatially vanishing) asymptotic behaviour of what they called
- as if knowingly - wave functions from the very beginning. A
number of benefits follows [for me, eg, the really amazing
observation represented by eq. (4.16)] bringing a new insight into 
the old problem.

MR2147626
Andriopoulos, K. ; Leach, P. G. L. 
Wavefunctions for the time-dependent linear oscillator 
and Lie point symmetries. 
J. Phys. A 38 (2005), no. 20, 4365--4374.