The Schrödinger equation with complex time

Question

Let $(M,g)$ be a manifold with Laplacian $\Delta$ and $V$ a potential. I define the Schrödinger equation with complex time:

$$\frac{\partial \psi}{\partial z}= - \Delta (\psi)+ V(\psi)$$

with $\psi (z,x), (z,x)\in {\bf C}.M$. $\psi$ is holomorphic in $z$.

Can we find solutions of the Schrödinger equation with complex time?

Comments

Instead of complex time, we often use a complex energy describing decay of the eigenstate, i.e., its transformation into other states (variations of the occupation numbers).

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With purely imaginary time, the schringer quation becomes parabolic instrad of hyperbolic, which makes it unphysical - no longer able to account for the oscillations leading to the observed spectra.

Imaginary time is however a mathematical trick to do calculations in lattice gauge theories. But the results must be analytically continued to real time to make physical sense.