I'm considering this simplified Sturm–Liouville problem:
$$ \ddot{y} +q(x)y = -\lambda y $$ $$ \dot y(0) = \dot y(1)= 0 $$ in the general case where $q(x)$ is sign indefinite. I'm interested on what conditions we need to impose on $q(x)$, to guarantee that $\lambda_0>0$ (the smallest eigenvalue of the operator).
A sufficient condition (which one proves via the Rayleigh quotient) is that $q(x)>0,\forall x$. However, I suspect this is not necessary. Similarly, using the Rayleigh quotient (with a constant function as a test function), I can show that a necessary condition is that $\int_0^1 q(x) dx >0$. Does anyone have any insight or can point me in the right direction in the literature?
Edit: Adding Rayleigh quotient for completeness $$ R(z) = \frac{\int_0^1 (\frac{\partial z}{\partial x})^2dx +\int_0^1 q(x) z^2(x)dx }{\int_0^1 z^2 dx} \geq \lambda_0, $$ where $z$ is any test function that satisfies the boundary conditions.