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Would the appropriate formula be $\psi_{t}=i\left(\psi_{zz}z^2 + \psi_{z}z - z \psi\right)$? Then, wave functions that solve Bessel's differential equation would serve as steady-state solutions correct? Is there any corresponding PDE that would allow me to solve for arbitrary values of time. This problem is closely related to the Toda Lattice and inverse scattering transform.
