# Reference Request: First Eigenvalue of Elliptic Operator

Let $$S^2$$ be the 2-sphere equipped with some Riemannian metric. Let $$\Delta$$ denote the Laplace-Beltrami operator. Let $$f\geq 0$$ be a smooth, non-negative function such that $$f(x)>0$$ at some $$x\in S^2$$. Consider the linear second order differential operator $$L=-\Delta+f$$.

I am interested in the eigenvalues of this operator. Specifically, I would like to show that for all the eigenvalues $$\lambda$$ we have $$\lambda \geq \lambda_0 >0$$.

Assuming $$u$$ is an eigenfunction with eigenvalue $$\lambda$$, assuming appropriate regularity, and integrating by parts, we have

$$\lambda ||u||_{L^2}=\int_{S^2} uLu=\int_{S^2}\left( |\nabla u|^2 + fu^2 \right) \geq 0$$

showing $$\lambda \geq 0$$. I think I can use Theorem 8.38 in Gilbarg-Trudinger which says that the minimum eigenvalue is simple and has a positive eigenfunction. In our case this then shows the first eigenvalue is positive. However, this book deals with domains in $$\mathbb{R}^n$$ rather than Riemannian manifolds, but I assume that all the results can be translated to the manifold setting. I'd just like confirmation that this is indeed the case.

Many thanks!

Yes, it is the case. I think there are two different ways to approach this problem. I think the second one is more direct if you already know the behavior of the operator $$-\Delta$$ (without the potential $$f\,$$). Let's start by applying some spectral analysis (this is why I prefer the second approach).
First of all, note that $$-\Delta+f$$ is essentially self-adjoint in $$C_0^\infty(S^2)$$ and self-adjoint in $$H^2(S^2)$$. Thus, its spectrum is non-empty and it is contained in $$\mathbb{R}$$. Moreover, there is a basic result in Spectral Theory saying that: Let $$T$$ be a self-adjoint operator. Then, the spectrum of $$T$$ is contained in $$[M,\infty)$$ if and only if $$T$$ is semi-bounded from below by $$\langle Tu,u\rangle \geq M\Vert u\Vert^2, \qquad \forall u\in D(T).$$ Thus, we can derive a first piece of information following your procedure. In fact, by integration by parts, it is easy to prove that spectrum of $$-\Delta+f$$ is contained in $$[0,\infty)$$.
Now, let's try to follow a different approach. First of all, let's recall Poincare's inequality (valid for $$S^2$$) $$\int_M \vert \nabla g\vert^2 \geq \lambda_1\int_M \vert g\vert^2,$$ where $$\lambda_1$$ is the first eigenvalue of $$-\Delta$$. Thus, following you procedure again, integrating by parts and applying Poincare's inequality, we see that the first eigenvalue $$\lambda_1^\star$$ of $$-\Delta+f$$ satisfies $$\lambda_1^\star\geq \lambda_1$$ (due to the positivity of $$f\,$$). Thus, if you already know that $$\lambda_1>0$$, we conclude the property we are looking for.