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Divergence, Gradients and Eigenvalues

Let psi be an eigenfunction of "div(grad)" + V for a real eigenvalue lambda.

Note: Here Psi is not necessarily in L^2 and V is real-valued.

If j = 2 Im( psi-bar x grad psi) , then show that div(j) = 0

Also, compute div (j) when lambda is in C/R and then give an example with an explicit V,

psi, lambda, j, and div(j).

Thank you and please show all steps. Please see the attached file for the fully formatted problems.


Solution Preview


First take a look at div(j):

div(j) = 2 Im[ (grad psibar)*(grad psi) ] + 2 Im[ psibar*Delta psi ]

The quantity (grad psibar)*(grad psi) = |grad psi|^2 is pure real and so its imaginary part is zero.
Therefore, in any case

div(j) = 2 Im[ psibar*Delta psi ] ...

Solution Summary

Divergence, gradients and eigenvalues are investigated.