$$h''+ \sigma V^{-1}V_{\theta} h' + [(\lambda_1+ V_{\theta, \theta }) V^{-1}+ (\sigma^{-1}+1)k^2]h=0 $$ admits a solution where $h(0)= h(l)$ and $h'(0)= h'(l).$ Here V and k are periodic functions of \theta and V is a smooth bounded from below by a positive constant. \sigma>0.
Sanjiban Santra
No, I have not yet done numerical experiments. Unfortunately $V$ and $K$ are not constants. This seems to be a question which arises in geometry.
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