Confining Potentials on the Real Line

The real line is identified with $\mathbb{R}^{1}$ as in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. Differentiability of a function $\mathbb{R}\to\mathbb{R}$ at a point is that of Derivative at an Interior Point, convexity on $\mathbb{R}=\mathbb{R}^{1}$ is that of that definition with $n=1$, and $|s|$ is the absolute value of $s\in\mathbb{R}$. Let $V:\mathbb{R}\to\mathbb{R}$.

(Confining potential)¶ The function $V$ is a confining potential if it is differentiable at every point of $\mathbb{R}$ with derivative $V'$, the function $V'$ is differentiable at every point of $\mathbb{R}$ with a continuous derivative $V''$, $V$ is convex on $\mathbb{R}$, and the following three conditions hold.

(a) (Superquadratic growth)¶ For every positive $M\in\mathbb{R}$ there is a positive $K\in\mathbb{R}$ with $Mx^{2}\le V(x)$ for every $x\in\mathbb{R}$ with $K\le|x|$.

(b) (Slope bound)¶ There is $C\in\mathbb{R}$ with $|V'(x)|\le C\bigl(1+|V(x)|\bigr)$ for every $x\in\mathbb{R}$.

(c) (Curvature small against the potential)¶ For every positive $\varepsilon\in\mathbb{R}$ there is $C_{\varepsilon}\in\mathbb{R}$ with $V''(x)\le\varepsilon|V(x)|+C_{\varepsilon}$ for every $x\in\mathbb{R}$.