I'm having trouble using NDEigenvalues
to obtain the first few eigenvalues for a differential operator on the circle of radius one-half.
$\qquad Lf(x) = f''(x)+ (-1-\sin(2x))f'(x)$
Considering this on the space $[0,\,\pi]/\sim$, this is a periodic Sturm-Liouville problem. It can be easily made self adjoint and, thus, should have a real spectrum. However, no matter how I play with the model, I still get complex eigenvalues.
NDEigenvalues[{f''[x] - (1 + Sin[2 x])f'[x], f[0] == f[Pi]}, f[x], {x, 0, Pi}, 10]
{0. +0. I, -4.16187 + 1.97304 I, -4.16187 - 1.97304 I, -16.1379 + 3.99737 I, -16.1379 - 3.99737 I, -36.1651 - 5.99943 I, -36.1651 + 5.99943 I, -64.3269 -7.99966 I, -64.3269 + 7.99966 I, -100.866 - 9.99757 I}
What am I doing wrong? Does this differential operator with these boundary conditions in fact not have real eigenvalues or am I using the function incorrectly?
Please advise!