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.
Agmon, Bounds on exponential decay of eigenfunctions of Schr¨ odinger operators, in ”Schr¨ odinger Operators” (Como, 1984), 1–38, Lecture Notes in Math
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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.