Ground state has always constant sign? Question: What hypothesis on the potential $V$ are required such that the ground state $\phi_0$ has constant sign?
Consider the Schrödinger operator in 1 dimension with potential $V$:
$$\mathcal{H}=-\epsilon^2 \frac{d^2}{dx^2}+V(x)$$
Assuming that $\sigma_{\rm disc}(\mathcal{H})$ is non empty, I am interested in the sign of the ground state $\phi_0$.
$$\mathcal{H}\phi_0 = \lambda_0 \phi_0$$
and 
$$\lambda_0 = \min \left\{\int_\mathbb{R} \phi'(x)^2 + V(x) \phi(x)^2 \, dx, {\lVert\phi\Vert}_{L^2(\mathbb{R})} =1 \right\} $$


*

*If $V\geq 0$ by the elliptic comparison principle it is clear for  me that $\phi_0\geq0$ and in particular has constant sign.

*If $V$ changes sign, I have (numerically at least) the impression that $\phi_0$ changes sign also. For example $V(x)=x^4+U$ and different values of $\epsilon$ and $U=-4$. Is this rigorously true? Furthermore, I have  the impression that the classical result asserting that $\lambda_0 \to \min{V}$ as $\epsilon\to0 $ is not true in this case? 



 A: If $V(x)\to+\infty$ as $x\to\pm\infty$ then the spectrum is discrete, $\lambda_n\to+\infty$, and
the ground state (the eigenfunction corresponding to the smallest eigenvalue)
does not change sign. Moreover, eigenfunction corresponding to $\lambda_n$
in the sequence $\lambda_0<\lambda_1<\ldots$ has exactly $n$ changes of sign.
This follows from Sturm's theory.
See, for example F. Berezin, M. Shubin, The Schrodinger equation, Kluwer, 1991,
Chap. 2, Theorem 3.5.
Remark. $\epsilon$ plays no role in this question, just divide on it. Adding a constant to the potential also plays no role: adding a constant to $V$ just shifts the spectrum, and does not change the eigenstates.
A: No assumptions whatsoever are required other than existence of a ground state. In other words, if the spectrum is bounded below and $\min \sigma(H)$ is an isolated point of the spectrum, then $\phi_0$ has no zeros.
More generally, you can detect where you are in the spectrum by counting zeros. This goes by the name oscillation theory (the wikipedia article is quite insipid, but it has references if you want to know more).
