Manuscript Hawk02-D@1 · Hawk02-D.tex · proposition · e6e4b72f

Proposition: Spectral transform and amplitudes

Exact amplitudes and quantitative collection bounds · lines 614–690 · proof included or linked

Hawk02-D@1#prop:spectral, Hawk02-D@1#eq:amplitudes, Hawk02-D@1#eq:survival

The sine transform

(S e_{j}) (θ) = \frac{2}{π} sin ((j + 1) θ), 0 < θ < π,

is unitary from $ℓ^{2} (N_{0})$ to $L^{2} ((0, π), d θ)$ and diagonalizes the one-particle Hamiltonian:

S (ω_{0} I + c A) S^{- 1} = ω_{0} + 2 c cos θ .

The operator has purely absolutely continuous spectrum. Its source spectral density is

ρ_{0} (E) = \frac{4 c ^{2} - ( E - ω _{0} ) ^{2}}{2 π c ^{2}} 1_{{∣ E - ω_{0} ∣ < 2 c}} .

For $z = 2 c t$ ,

u_{j} (t) = e^{- i ω_{0} t} (- i)^{j} [J_{j} (z) + J_{j + 2} (z)] .

In particular,

q (t) := ∣ u_{0} (t) ∣^{2} = \frac{J _{1} ( 2 c t )}{c t}^{2}, q (0) = 1.

Proof. The sine functions form an orthonormal basis of the target $L^{2}$ space. The identity

2 cos θ sin ((j + 1) θ) = sin (j θ) + sin ((j + 2) θ)

proves diagonalization, including the boundary condition at $j = 0$ .

The multiplication function has no constant level set of positive measure and is strictly monotone on $(0, π)$ . Its spectral measures are absolutely continuous after change of variables. For $e_{0}$ , changing variables $E = ω_{0} + 2 c cos θ$ gives the displayed density.

The amplitude is

u_{j} (t) = \frac{2}{π} e^{- i ω_{0} t} \int_{0}^{π} sin θ sin ((j + 1) θ) e^{- i z c o s θ} d θ .

Use

2 sin θ sin ((j + 1) θ) = cos (j θ) - cos ((j + 2) θ)

and the Fourier integral for the integer-order Bessel functions. This gives (eq:amplitudes). The recurrence $J_{0} (z) + J_{2} (z) = 2 J_{1} (z) / z$ gives (eq:survival), with its continuous value at zero.

Displayed formulas in this passage

Hawk02-D@1#eq:amplitudes · line 638
$u_{j} (t) = e^{- i ω_{0} t} (- i)^{j} [J_{j} (z) + J_{j + 2} (z)] .$
Hawk02-D@1#eq:survival · line 646
$q (t) := ∣ u_{0} (t) ∣^{2} = \frac{J _{1} ( 2 c t )}{c t}^{2}, q (0) = 1.$
unlabeled · line 617
$(S e_{j}) (θ) = \frac{2}{π} sin ((j + 1) θ), 0 < θ < π,$
unlabeled · line 625
$S (ω_{0} I + c A) S^{- 1} = ω_{0} + 2 c cos θ .$
unlabeled · line 632
$ρ_{0} (E) = \frac{4 c ^{2} - ( E - ω _{0} ) ^{2}}{2 π c ^{2}} 1_{{∣ E - ω_{0} ∣ < 2 c}} .$
unlabeled · line 658
$2 cos θ sin ((j + 1) θ) = sin (j θ) + sin ((j + 2) θ)$
unlabeled · line 672
$u_{j} (t) = \frac{2}{π} e^{- i ω_{0} t} \int_{0}^{π} sin θ sin ((j + 1) θ) e^{- i z c o s θ} d θ .$
unlabeled · line 681
$2 sin θ sin ((j + 1) θ) = cos (j θ) - cos ((j + 2) θ)$

Proposition: Spectral transform and amplitudes

Displayed formulas in this passage

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