Finite-level Sambe operator with Fourier domain

Given: AC and Countable Choice, the finite-level periodic Hermitian family $H$, $T>0$, $\omega =2\pi \mathrm{/}T$, and $\mathrm{\hslash }>0$.

AC is the hypothesis of Kato–Rellich and the periodic derivative theorem; Countable Choice is also a hypothesis of the periodic derivative theorem and the finite Floquet normal form (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

At length $T$ and twist $\theta =0$, The twisted-periodic momentum operator on an interval states that $-iD$ is self-adjoint on $\{u\in H^1(0,T):u^{\ast }(0)=u^{\ast }(T)\}$ and that $T^{-1\mathrm{/}2}{e}^{in\omega t}$ is a complete orthonormal eigenfamily with eigenvalues $n\omega$. In that supplier, $u^{\ast }$ denotes the unique absolutely continuous representative of the $H^1$ class, so $u^{\ast }(0)$ and $u^{\ast }(T)$ are its endpoint traces; its Statement identifies their trace convention as clause (b) of the interval trace result lem-nrqm-intervals-integration-by-parts. The weak derivative defining $H^1$ is that of One-dimensional Sobolev spaces on intervals. The unbounded adjoint is the adjoint defined by representability on the dense operator domain (Adjoint of a densely defined operator).

Kato-Rellich theorem states that a self-adjoint operator plus a symmetric perturbation of relative bound less than one is self-adjoint on the original operator domain, under AC. A bounded symmetric perturbation has relative bound zero (Relative boundedness with respect to an operator, A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). A continuous matrix-valued function on $[0,T]$ has bounded operator norm, since this interval is compact (For a nonempty subset of ${\mathbb{R}}^n$ with $n\ge 1$, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).

Finite Floquet unitary normal form supplies an orthonormal eigenbasis $v_j$ for $H_F$, the periodic unitary $P(t)$, the propagator equation, and the factorization $U(t,0)=P(t)e^{-it{H}_F\mathrm{/}\mathrm{\hslash }}$. Its controlled-dynamics dependency makes $U(t,0)$ $C^1$, while Finite-dimensional matrix exponential and Unitarity and differentiation of the finite Hermitian matrix exponential make the matrix-exponential factor $C^1$; hence $P$ is $C^1$. The identity $H(t)P(t)-i\mathrm{\hslash }{P}^{\mathrm{\prime }}(t)=P(t)H_F$ is derived by differentiating this factorization in step 1.3, not assumed from the normal-form Statement.

The ordinary product rule applies componentwise to finite matrix and vector curves (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$). A $C^1$ vector-valued function has its classical derivative as its weak derivative (Classical derivatives agree with weak derivatives).

The componentwise periodic $H^1$ domain is the finite direct sum of the $\theta =0$ domain from [F1]; each component is represented by its unique absolutely continuous representative, and $f(0)=f(T)$ means equality of the corresponding endpoint traces. The derivative $f^{\mathrm{\prime }}$ in $K$ is the componentwise weak derivative from One-dimensional Sobolev spaces on intervals.