The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential

This setting fixes the notation of the Galerkin problems of the Wick-square problem. It introduces no new concept and asserts nothing beyond the facts recorded below, each carried by the reference attached to it.

1. (The problem)¶ We work in the settings of The Wick-Square Problem on the Torus: Standing Notation and Second-Order Equations on Euclidean Open Sets. With a mode as subscript, $q_{k}$ and $a_{k}$ are the Riccati coefficients and the renormalised constants, and $c_{k}$ are the free-field variances. The Euclidean dimension, written $q$ in Second-Order Equations on Euclidean Open Sets §space, is written $m$ below, and the letter $q$ without a subscript is not used; nor are the letters $V$ and $\mu$ of that setting in their meanings there (an open set, a scalar), $V$ being reserved for a Sobolev space and $\mu_{k}$ for the Fourier weights.

2. (Cutoffs and enumerations)¶ The notation of clauses 2 to 4 is introduced for every $N\in\mathbb{N}$ simultaneously. The cube $\Gamma_{N}$ is a nonempty finite set by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite, so by Finite Set it has $m$ elements for a natural number $m$, depending on $N$, with $1\le m$ because $\Gamma_{N}$ is nonempty; we fix such an $m$ and a bijection $\kappa_{N}$ from the initial segment $[m]=\{1,\dots,m\}$ onto $\Gamma_{N}$.

3. (Coordinates)¶ The coordinate map $\iota_{N}:\mathbb{R}^{m}\to H^{-1}$ sends $z=(z_{1},\dots,z_{m})$ to the family with value $z_{i}$ at the mode $\kappa_{N}(i)$, for each $i\in[m]$, and value $0$ at every mode outside $\Gamma_{N}$. This family is $\sum_{i=1}^{m}z_{i}e_{\kappa_{N}(i)}$; it lies in $H^{-1}$ because $H^{-1}$ is a linear subspace of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ (The Wick-Square Problem on the Torus: Standing Notation §state-space) containing the unit families (The Wick-Square Problem on the Torus: Standing Notation §units). The coordinate projection $\mathrm{pr}_{N}:\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})\to\mathbb{R}^{m}$ sends $x$ to $(x(\kappa_{N}(1)),\dots,x(\kappa_{N}(m)))$, and $\Pi_{N}=\iota_{N}\circ\mathrm{pr}_{N}$ sends $x$ to the family equal to $x$ on $\Gamma_{N}$ and to $0$ outside $\Gamma_{N}$, which lies in $H^{-1}$.

4. (The free Galerkin quadratic and the Riccati potential)¶ $\tilde{u}_{N},\Phi_{N}:\mathbb{R}^{m}\to\mathbb{R}$ denote the functions

$$\tilde{u}_{N}(z)=\sum_{i=1}^{m}\bigl(q_{\kappa_{N}(i)}\,z_{i}^{2}+a_{\kappa_{N}(i)}\bigr),\qquad\Phi_{N}(z)=\tfrac12\sum_{i=1}^{m}\bigl(\mu_{\kappa_{N}(i)}+2q_{\kappa_{N}(i)}\bigr)z_{i}^{2}.$$

Both are of class $C^{2}$ on $\mathbb{R}^{m}$, being built from constant and coordinate functions by sums, scalar multiples and products (Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, claims 2 and 3).