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If I have a self-adjoint operator which operates on twice differentiable functions defined by: $$ Lx(t) = [k(t)x'(t)]' + g(t)x(t) $$ How can I show that $k(t)$ is real-valued given that $k \in C^1[a,b]$ and $g \in C[a,b]$? Not really sure where to begin other than using the self-adjoint condition; that is, $(Lu,v) = (u,Lv)$ where $u,v$ are twice differentiable functions and $$ (u,v) = \int_a^b u(t) \overline{v(t)} \, dt $$ Any ideas to help me out?

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