The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability

In the setting of Second-Order Equations on Euclidean Open Sets, let $\beta,\sigma,\lambda\in\mathbb{R}$ be positive with $\sigma^{2}<\beta$, where $\sigma^{2}=\sigma\sigma$, and let $V:\mathbb{R}\to\mathbb{R}$ be a confining potential, with derivative $V'$ and second derivative $V''$ as in that definition. Natural numbers are regarded as real numbers through the canonical map, which is suppressed from the notation. We write $\exp$ for the exponential function, $\operatorname{tr}$ for the trace, $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$, and, for a natural number $m\ge1$, $\mathcal{L}^{m}$ for Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{m})$.

For each natural number $N\ge2$ let $W_{N}$ be the Weyl chamber in $\mathbb{R}^{N}$, let

which are positive real numbers since $N-1\ge1$, and let $P_{N}:W_{N}\to\mathbb{R}$ be

where $H_{b_{N}}$ is the logarithmic energy of strength $b_{N}$ on $W_{N}$.

1. (Admissible confinement)¶ $V$ is of class $C^{2}$ on $\mathbb{R}=\mathbb{R}^{1}$, with $\partial_{1}V=V'$ and $\partial_{1}\partial_{1}V=V''$; $0\le V''(t)$ for every $t\in\mathbb{R}$; there are $a_{0},b_{0}\in\mathbb{R}$ with $0<a_{0}$ and $a_{0}|t|-b_{0}\le V(t)$ for every $t\in\mathbb{R}$; and $V$ has regular growth, that is, for every positive $\eta\in\mathbb{R}$ there is $C_{\eta}\in\mathbb{R}$ with $V''(t)\le C_{\eta}+\eta\,V'(t)^{2}$ for every $t\in\mathbb{R}$. Consequently $V$ satisfies every hypothesis that Well-Posedness of the Dyson Hamilton-Jacobi Equation in the Weyl Chamber below the Collision Threshold places on its confinement $V_{1}$, and for every $N\ge2$ the function $P_{N}$ is the function $P$ of The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality with strength $b_{N}$ in place of $\beta$ and with $V_{1}=V$.

2. (Penalty and monotone gradient)¶ For every $N\ge2$, $P_{N}$ is a penalty on $W_{N}$, and $0\le\bigl(DP_{N}(x)-DP_{N}(y)\bigr)\cdot(x-y)$ for all $x,y\in W_{N}$. Moreover $0<\kappa_{N}<2b_{N}$.

3. (Dissipation with constants uniform in $N$)¶ Let $\varepsilon=1-\sigma^{2}/\beta$, so that $0<\varepsilon<1$. There is $C_{0}\in\mathbb{R}$ with $0\le C_{0}$ such that for every natural number $N\ge2$ and every $x\in W_{N}$

4. (Lower bound uniform in $N$)¶ There are $c_{1},C_{1}\in\mathbb{R}$ with $0<c_{1}$ and $0\le C_{1}$ such that for every natural number $N\ge2$ and every $x\in W_{N}$

5. (Gibbs integrability)¶ For every natural number $N\ge2$ let $g_{N}:\mathbb{R}^{N}\to\mathbb{R}$ be the function with

Then $0\le g_{N}(x)$ for every $x\in\mathbb{R}^{N}$, $g_{N}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{N})$, and its integral satisfies

that is, $\int_{W_{N}}\bigl(1+|P_{N}(x)|+\lVert DP_{N}(x)\rVert^{2}+\lVert x\rVert^{2}\bigr)e^{-P_{N}(x)/a_{N}}\,dx<\infty$.