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Static Bohr-frequency evaluation of the jump integral

Statement

Assume Countable Choice. Work on $C^{d}$ , $d \geq 1$ , with its standard complex inner product, conjugate-transpose adjoint $†$ , and induced operator norm. Let $H_{S} = H_{S}^{†}$ have an orthonormal eigenbasis $∣ m ⟩$ , $1 \leq m \leq d$ , with real eigenvalues $E_{m}$ . Let $X_{α} = X_{α}^{†}$ and $∥ X_{α} ∥ \leq 1$ , $1 \leq α \leq N$ . Use the normalized $N \times N$ frequency and time kernels $h$ and the positive parameter $γ$ of the normalization lemma. Assume the entrywise Fourier identities $\int_{R} h (u) e^{i ω u} d u = 2 π h (ω)$ at every Bohr frequency $ω = E_{n} - E_{m}$ . For real $s, t$ , define $U (t, s) = e^{- i H_{S} (t - s)}$ by the finite-dimensional matrix-exponential series. For $1 \leq λ \leq N$ , the Schrödinger-picture jump $L_{λ} (t) = \sqrt{γ} \sum_{α = 1}^{N} \int_{R} h_{λ α} (t - s) U (t, s) X_{α} U (t, s)^{†} d s$ exists absolutely, is independent of $t$ , and equals $2 π \sqrt{γ} \sum_{m, n = 1}^{d} \sum_{α = 1}^{N} h_{λ α} (E_{n} - E_{m}) (X_{α})_{m n} ∣ m ⟩ ⟨ n ∣$ , where $(X_{α})_{m n}$ denotes the matrix entry in the given eigenbasis.

Facts & Assumptions

[F1]

The normalized time kernel $h$ has finite zeroth and first mixed-norm moments, and $γ > 0$ . ULE kernel adjoint symmetry and rate identities

[F2]

Assuming Countable Choice, on a finite-dimensional system, measurable Hermitian couplings of operator norm at most one give absolutely convergent jump integrals for this kernel. Convergent jump integrals and Hermitian Lamb operators

[F3]

For matrices on $C^{d}$ with the standard complex inner product, the adjoint is the conjugate transpose, the operator norm is the unit-ball supremum, and $e^{z A} = \sum_{k = 0}^{\infty} z^{k} A^{k} / k!$ converges absolutely in operator norm for every complex $z$ . Finite-dimensional matrix exponential

[F4]

If $A = A^{†}$ on $C^{d}$ and $r \in R$ , then $e^{- i r A}$ is unitary and depends differentiably, hence continuously, on $r$ in operator norm. This is the supplier with $ℏ = 1$ . Unitarity and differentiation of the finite Hermitian matrix exponential

[A1]

Assume Countable Choice for the convergence supplier [F2] and the change-of-variables theorem [F7]. The Axiom of Countable Choice (AC_ω)

[F5]

For a measurable real function $p$ , its positive and negative parts are $p^{+} = \max {p, 0}$ and $p^{-} = \max {- p, 0}$ . For integrable $p$ , both parts have finite integral and $\int p = \int p^{+} - \int p^{-}$ . For a complex integrable $f = p + i q$ , the integral is $\int f = \int p + i \int q$ , and integrability means finite integral of $∣ f ∣$ . Integrable real and complex functions, and their integrals

[F6]

The space $L^{1} (μ)$ is a complex vector space and its Lebesgue integral is complex-linear. Thus finite sums and constant complex factors commute with the integral of integrable functions. The Lebesgue integral is linear on L¹(μ)

[F7]

Assume Countable Choice. For open $U, V \subseteq R^{n}$ , a $C^{1}$ diffeomorphism $T : U \to V$ , and nonnegative Lebesgue measurable $a$ , the change-of-variables formula is $\int_{V} a (y) d y = \int_{U} a (T (x)) ∣ \det D T (x) ∣ d x$ . We use $U = V = R$ and $T_{t} (u) = t - u$ , for which $T_{t}^{- 1} = T_{t}$ and $∣ T_{t}^{'} ∣ = 1$ . A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions

Proof

Given: Countable Choice, the finite-dimensional complex system, Hermitian $H_{S}$ , real times, bounded Hermitian couplings, normalized kernel, and pointwise inversion identities in the Statement. Goal: Prove absolute convergence and evaluate $L_{λ} (t)$ .

1.1givenF3F4

Put $u = t - s$ . Since $H_{S}^{k} ∣ m ⟩ = E_{m}^{k} ∣ m ⟩$ , apply the norm-convergent series in [F3] to each eigenvector to obtain $U (t, s) ∣ m ⟩ = e^{- i E_{m} u} ∣ m ⟩$ . As $E_{m}, u$ are real, its adjoint is diagonal with entries $e^{i E_{m} u}$ . By [F4], $U (t, s)$ is unitary for every real $s, t$ .

1.2A1F5F7algebra

Fix real $t$ and put $T_{t} (u) = t - u$ . For any complex $f = p + i q \in L^{1} (R)$ , apply [F7] to $∣ f ∣$ to obtain $\int ∣ f \circ T_{t} ∣ = \int ∣ f ∣ < \infty$ . Apply [F7] separately to the four nonnegative measurable functions $p^{+}, p^{-}, q^{+}, q^{-}$ in [F5]. Their integrals are finite, and composition with $T_{t}$ commutes with taking positive and negative parts. Subtract the two real-part identities and the two imaginary-part identities, then combine them by [F5]. This proves $\int_{R} f (s) d s = \int_{R} f (t - u) d u$ for complex integrable $f$ . In particular, reflection introduces no sign in a Lebesgue integral because the Jacobian is used in absolute value.

2.1givenA1F1F2F3F4step 1.1

For each fixed real $t$ , put $Y_{α}^{(t)} (s) = U (t, s) X_{α} U (t, s)^{†}$ . By [F4] its matrix entries are continuous in $s$ , hence measurable. Conjugate-transpose multiplication gives $(Y_{α}^{(t)} (s))^{†} = Y_{α}^{(t)} (s)$ . Unitarity gives $∥ U v ∥ = ∥ v ∥ = ∥ U^{†} v ∥$ for every vector $v$ ; consequently $∥ U X_{α} U^{†} v ∥ \leq ∥ X_{α} ∥ ∥ v ∥$ , so the unit-ball supremum in [F3] gives $∥ Y_{α}^{(t)} (s) ∥ \leq 1$ . Together with [A1], these meet the hypotheses needed to apply [F2], with the kernel from [F1], proving absolute convergence of each $L_{λ} (t)$ .

3.1F1F5F6step 1.1step 2.1step 1.2algebra

Step 1.1 gives $(Y_{α}^{(t)} (s))_{m n} = (X_{α})_{m n} e^{i (E_{n} - E_{m}) (t - s)}$ . The entrywise integrands are integrable by step 2.1. Apply step 1.2 to each such entry and use [F6] to extract the constant $(X_{α})_{m n}$ and commute the finite matrix and coupling sums with their entrywise integrals. The functions $u \mapsto h_{λ α} (u) e^{i (E_{n} - E_{m}) u}$ are integrable: their modulus is $∣ h_{λ α} (u) ∣$ , bounded by the mixed norm whose integral is finite in [F1]. Thus the coefficient multiplying $∣ m ⟩ ⟨ n ∣$ is $\sqrt{γ} \sum_{α = 1}^{N} (X_{α})_{m n} \int_{R} h_{λ α} (u) e^{i (E_{n} - E_{m}) u} d u$ .

4.1givenstep 3.1∎

Apply the stated entrywise Fourier identity at each real $E_{n} - E_{m}$ to the coefficient in step 3.1. This gives exactly the finite sum in the Statement, containing no $t$ , and proves time independence as well as the claimed formula. The Fourier identities remain explicit hypotheses; no pointwise inversion conclusion is inferred from integrability alone.

Depends on

Used by · 1

ULE jump construction under independent bath hypotheses

Dependency tree · 2 levels

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