I was wondering if the following holds:

If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the end-points.

If you now add a nice smooth + bounded -potential $V \in C^{\infty}(\mathbb{R})$ to your current potential, so that you end up with the ODE

$$-y''(x) + (q(x)+V(x)) y(x) = \lambda y(x),$$

is it still clear that your differential equation is limit-circle or limit-point at the endpoints?

I mean, somehow I feel that this statement should hold, as it is somehow natural to assume that a nice potential should keep the nice properties of the operator, but I could not find a reference for this.