TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Finite Floquet unitary normal form

Statement

Assume Countable Choice. Let $d \geq 1$ , $T > 0$ , $ℏ > 0$ , $ω = 2 π / T$ , and let $H : R \to M_{d} (C)$ be continuous, $T$ -periodic, and Hermitian at every time. Let $U (t, s)$ be the unitary propagator solving $i ℏ \partial_{t} U (t, s) = H (t) U (t, s)$ and $U (s, s) = I_{d}$ . There are an orthonormal basis $v_{1}, \dots, v_{d}$ and real numbers $ε_{1}, \dots, ε_{d}$ such that $U (T, 0) v_{j} = e^{- i T ε_{j} / ℏ} v_{j} (1 \leq j \leq d) .$ Define the Hermitian matrix $H_{F} v_{j} = ε_{j} v_{j}$ and the quasienergy classes $ε_{j} + ℏ ω Z$ . With $P (t) : = U (t, 0) e^{i t H_{F} / ℏ}$ , $P (t)$ is a $T$ -periodic unitary matrix and $U (t, 0) = P (t) e^{- i t H_{F} / ℏ} (t \in R) .$ The real representative $ε_{j}$ may be changed by any integer multiple of $ℏ ω$ without changing its eigenphase; no preferred real branch is asserted.

Statement references

Norm Dyson series and uniqueness

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

$t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle

$\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$

Finite-dimensional matrix exponential

Facts & Assumptions

Given: Countable Choice, $d \geq 1$ , $T > 0$ , $ℏ > 0$ , and the continuous periodic Hermitian matrix family $H$ .

[A1]

Countable Choice is assumed, as required by the norm-continuous bounded-dynamics supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Norm Dyson series and uniqueness: on each compact time interval a norm-continuous bounded self-adjoint family has a unique unitary propagator with the differential equation, initial identity, composition law, and uniqueness for every initial value.

[F2]

A unitary endomorphism is normal by Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, and the finite-dimensional complex spectral theorem gives it an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F3]

Every point of the real unit circle has the form $(\cos θ, \sin θ)$ for some $θ \in [0, 2 π)$ ( $t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle); Euler's identity gives $e^{i θ} = \cos θ + i \sin θ$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F4]

Two complex exponential values agree exactly when their exponents differ by an element of $2 π i Z$ ( $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

[F5]

The finite matrix exponential of a Hermitian matrix is unitary, differentiable, and obeys the exponential group law (Finite-dimensional matrix exponential, Unitarity and differentiation of the finite Hermitian matrix exponential).

Proof

Given: [A1] and [F1]–[F5].

1.1A1F1given

On any compact interval, $H$ is a norm-continuous bounded self-adjoint family, so [F1] supplies the unique unitary propagator there. On overlapping compact intervals the propagators agree by uniqueness; hence $U (t, s)$ is consistently defined for all real $s, t$ . For each fixed $t$ , the curves $r \mapsto U (r + T, T)$ and $r \mapsto U (r, 0)$ solve the same differential equation because $H (r + T) = H (r)$ , and both equal $I_{d}$ at $r = 0$ . Uniqueness in [F1] gives $U (t + T, T) = U (t, 0)$ .

1.2F1F2F3F5given

Put $M = U (T, 0)$ . By [F1], $M$ is unitary, so [F2] gives an orthonormal eigenbasis $v_{j}$ with eigenvalues $λ_{j}$ of modulus one. Write $λ_{j} = x_{j} + i y_{j}$ ; then $(x_{j}, - y_{j})$ lies on the real unit circle. By [F3], choose $θ_{j} \in [0, 2 π)$ with $x_{j} = \cos θ_{j}$ and $- y_{j} = \sin θ_{j}$ , so $λ_{j} = e^{- i θ_{j}}$ . Set $ε_{j} = ℏ θ_{j} / T$ and define $H_{F} v_{j} = ε_{j} v_{j}$ . Each $ε_{j}$ is real, hence $H_{F}$ is Hermitian, and [F5] gives $e^{- i T H_{F} / ℏ} v_{j} = e^{- i θ_{j}} v_{j} = λ_{j} v_{j}$ . Equality on the basis proves $e^{- i T H_{F} / ℏ} = U (T, 0)$ .

1.3F4givenalgebra

If $ε_{j}^{'}$ gives the same eigenphase as $ε_{j}$ , then $\exp (- i T (ε_{j}^{'} - ε_{j}) / ℏ) = 1$ . By [F4], $T (ε_{j}^{'} - ε_{j}) / ℏ \in 2 π Z$ , so $ε_{j}^{'} - ε_{j} \in ℏ ω Z$ . Conversely adding $n ℏ ω$ multiplies the phase by $e^{- i 2 π n} = 1$ . Thus the class stated is independent of the real lift.

2.1F1F5step 1.1step 1.2algebra∎

Define $P (t) = U (t, 0) e^{i t H_{F} / ℏ}$ . Both factors are unitary by [F1] and [F5], so $P (t)$ is unitary, and the group law in [F5] gives $P (t) e^{- i t H_{F} / ℏ} = U (t, 0)$ . By composition and step 1.1, $U (t + T, 0) = U (t + T, T) U (T, 0) = U (t, 0) e^{- i T H_{F} / ℏ}$ . Therefore $P (t + T) = U (t, 0) e^{- i T H_{F} / ℏ} e^{i (t + T) H_{F} / ℏ} = U (t, 0) e^{i t H_{F} / ℏ} = P (t),$ where the middle cancellation follows from [F5]. This proves the factorization and periodicity.

Finite Floquet unitary normal form

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources