The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity

In the setting of Second-Order Equations on Euclidean Open Sets, let $N\ge2$ be a natural number, let $\beta\in\mathbb{R}$ be positive, and let $W_{N}$ and $H=H_{\beta}$ be the Weyl chamber and the logarithmic energy of strength $\beta$. For $x\in W_{N}$ and $k,j\in[N]$ let $a_{kj}(x)=(x_{k}-x_{j})^{-1}$ if $k\ne j$, which exists since $x_{k}\ne x_{j}$ by the definition of $W_{N}$, and $a_{kk}(x)=0$; and let

We write $\operatorname{tr}$ for the trace and abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$.

1. (The chamber is open and nonempty)¶ $W_{N}$ is open, and it contains the point whose $k$th coordinate is $N-k+1$ for $k\in[N]$.

2. (Derivatives)¶ $H$ is of class $C^{2}$ on $W_{N}$, and for every $x\in W_{N}$ and $k\in[N]$

3. (Symmetrisation)¶ For every $x\in W_{N}$ and $z\in\mathbb{R}^{N}$,

where $\tfrac{\beta}{2}$ is the product of $\beta$ with the multiplicative inverse of $2=1+1$.

4. (Monotone gradient)¶ For all $x,y\in W_{N}$, $0\le\bigl(DH(x)-DH(y)\bigr)\cdot(x-y)$.

5. (Calogero identity)¶ For every $x\in W_{N}$, $\lVert DH(x)\rVert^{2}=\beta^{2}S(x)$; consequently $\beta\operatorname{tr}\bigl(D^{2}H(x)\bigr)=\lVert DH(x)\rVert^{2}$.

6. (Euler identity)¶ Let $d_{N}=\sum_{k=1}^{N}\sum_{j=1}^{N}\epsilon_{kj}$, where $\epsilon_{kj}=1$ if $k\ne j$ and $\epsilon_{kk}=0$. For every $x\in W_{N}$, $DH(x)\cdot x=-\tfrac{\beta}{2}d_{N}$.