Floquet Fourier–Galerkin assembly

Assume the Axiom of Choice and Countable Choice. Fix integers $d\ge 1$, $b,N\ge 0$, and real numbers $\mathrm{\hslash }>0$, $\omega >0$, and $\tau >0$. On every coordinate space ${\mathbb{C}}^r$ used below, use the standard inner product $⟨x,y⟩:={\sum }_{a=1}^r{x}_a\overline{y_a}$, linear in the first argument, and induced norm $\mathrm{\parallel }x\mathrm{\parallel }=({\sum }_{a=1}^r\mathrm{\mid }{x}_a{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, as fixed in the finite-dimensional matrix exponential definition; matrix adjoints are conjugate transposes. Let $H_k\in M_d(\mathbb{C})$ for $-b\le k\le b$ satisfy $H_{-k}=H_k^{\ast }$; set $H_k=0$ when $\mathrm{\mid }k\mathrm{\mid }>b$. Thus $H(t)={\sum }_{k=-b}^b{H}_k{e}^{ik\omega t}$ is a finite Fourier polynomial with period $T=2\pi \mathrm{/}\omega$. Put $q=(2N+1)d$ and use this coordinate inner product and norm on ${\mathbb{C}}^q$. Let $S_{\tau }$ be a caller-supplied total classical routine on $\{M\in M_q(\mathbb{C}):M=M^{\ast }\}$ whose declared interface, on input $M$, returns real numbers ${\tilde{\lambda }}_1,\dots ,{\tilde{\lambda }}_q$, an orthonormal basis $x_1,\dots ,x_q$ for this inner product, and residuals $r_j=\mathrm{\parallel }M{x}_j-{\tilde{\lambda }}_j{x}_j\mathrm{\parallel }\le \tau$ for $1\le j\le q$. This algorithm returns that finite record unchanged and does not independently certify its residual claims.

The algorithm takes $(d,b,N,\mathrm{\hslash },\omega ,(H_k{)}_{\mathrm{\mid }k\mathrm{\mid }\le b},\tau ,S_{\tau })$ and returns a matrix $K_N$ and the eigensolver's output record:

1. Order the Fourier indices as $-N,-N+1,\dots ,N$ and allocate a $(2N+1)d$ by $(2N+1)d$ block matrix.

2. For each ordered pair $n,m\in \{-N,\dots ,N\}$, assign the $d$ by $d$ block $(K_N{)}_{nm}:=H_{n-m}+n\mathrm{\hslash }\omega {\delta }_{nm}{I}_d,$ where ${\delta }_{nm}=1$ if $n=m$ and $0$ otherwise.

3. Set $R:=S_{\tau }(K_N)$ by one call, then return $(K_N,R)$.

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Finite-level Sambe operator with Fourier domain

Finite-dimensional matrix exponential

For the special case $b=1$, an input Fourier sector $m$ contributes to output sector $n$ through block $H_{n-m}$; the diagonal also contains $n\mathrm{\hslash }\omega I_d$. The arrows show block locations only and are not used in the proof of the compression theorem.