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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?

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).

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.

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