When a potential grows at infinity, uniqueness for the parabolic Schrödinger equation holds under a weaker integral condition on the solution, with the improvement set by the decay rate of positive stationary solutions.
Gaffney, The conservation property of the heat equation on Riemannian manifolds , Comm
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Relaxed uniqueness conditions for the parabolic Schrodinger equation on Riemannian manifolds
When a potential grows at infinity, uniqueness for the parabolic Schrödinger equation holds under a weaker integral condition on the solution, with the improvement set by the decay rate of positive stationary solutions.