Thermal bath spectral density and frequency signs

On $(0,\mathrm{\infty })$ put $w_{-}(\omega )=\mathrm{\mid }\kappa (\omega ){\mathrm{\mid }}^2{\omega }^{2}N(\omega )$ and $w_{+}(\omega )=\mathrm{\mid }\kappa (\omega ){\mathrm{\mid }}^2{\omega }^2[N(\omega )+1]$. These are nonnegative Borel functions with ${\int }_0^{\mathrm{\infty }}(w_{-}+w_{+})=M<\mathrm{\infty }$, so each is integrable. Define $J_{\ast }(x)=A{w}_{-}(-x)$ for $x<0$, $J_{\ast }(x)=A{w}_{+}(x)$ for $x>0$, and $J_{\ast }(0)=0$. The reflection $T:(-\mathrm{\infty },0)\to (0,\mathrm{\infty })$, $T(x)=-x$, is a $C^1$ diffeomorphism with $\mathrm{\mid }{T}^{\mathrm{\prime }}(x)\mathrm{\mid }=1$. Apply [F6] to $w_{-}$ on these open half-lines to obtain ${\int }_{-\mathrm{\infty }}^0{w}_{-}(-x)dx={\int }_0^{\mathrm{\infty }}{w}_{-}(\omega )d\omega$. Thus $J_{\ast }$ is nonnegative Borel and integrable, with integral $AM$. By [F5], $\mu (E)={\int }_E{J}_{\ast }(x)dx$ is a finite measure. For any Borel $E\subseteq \mathbb{R}$, the function $f_E(\omega )=w_{-}(\omega ){\mathbf{1}}_E(-\omega )$ is Borel and bounded in absolute value by $w_{-}$, hence is in $L^1(0,\mathrm{\infty })$. Applying the same reflection to $f_E$ gives $\mu (E)=A{\int }_0^{\mathrm{\infty }}[w_{-}(\omega ){\mathbf{1}}_E(-\omega )+w_{+}(\omega ){\mathbf{1}}_E(\omega )]d\omega$. This is exactly the stated point-mass mixture, now with countable additivity supplied by [F5].

To establish the possible failure of time integrability, fix any $c,\beta ,\gamma >0$ and take the real Borel coupling $\kappa (\omega )={\mathbf{1}}_{(0,1)}(\omega )\mathrm{/}(\omega \sqrt{2N(\omega )+1})$ for $\omega >0$. Its denominator is strictly positive and finite at every positive frequency. Assign $\kappa (0)=0$ if a value at zero is wanted. Its thermal envelope is exactly ${\mathbf{1}}_{(0,1)}$, so $M=1$ and [F1] applies with the supplier's specified regulators and normalization (one may take $d_0=1$ and $\mathcal{E}=\kappa$). Write $C(t)={\mathcal{J}}_{\alpha \alpha }(t)$ for this coupling. Taking real parts in the absolutely convergent frequency integral of [F1] and using [F7] gives $\mathrm{Re}C(t)=A{\int }_0^1\cos (\omega t)d\omega =A\sin t\mathrm{/}t$ for $t\mathrm{\ne }0$, by applying the Lebesgue fundamental theorem to $v(\omega )=\sin (\omega t)\mathrm{/}t$ on $[0,1]$. At $t=0$ the same real part is $A$. Thus $\mathrm{\mid }C(t)\mathrm{\mid }\ge A\mathrm{\mid }\sin t\mathrm{\mid }\mathrm{/}t$ for $t>0$.

Let $\phi$ be a Schwartz test function and write $h(t)={\int }_{\mathbb{R}}\phi (x)e^{ixt}dx$. The first integral exists by [F3]. In the Fourier supplier's convention, $\widehat{\phi }(\xi )={\int }_{\mathbb{R}}\phi (x)e^{-2\pi ix\xi }dx$ and $h(t)=\widehat{\phi }(-t\mathrm{/}(2\pi ))$. The absolute-convergence assertion in [F3] implies $\widehat{\phi }\in L^1(\mathbb{R})$. The map $T:\mathbb{R}\to \mathbb{R}$, $T(t)=-t\mathrm{/}(2\pi )$, is a $C^1$ diffeomorphism with $\mathrm{\mid }{T}^{\mathrm{\prime }}\mathrm{\mid }=(2\pi {)}^{-1}$, so [F6] applied to $\mathrm{\mid }\widehat{\phi }\mathrm{\mid }$ gives ${\int }_{\mathbb{R}}\mathrm{\mid }h(t)\mathrm{\mid }dt=2\pi {\int }_{\mathbb{R}}\mathrm{\mid }\widehat{\phi }(\xi )\mathrm{\mid }d\xi <\mathrm{\infty }$. After inserting [F1] into the spectral pairing, the absolute value of its double integrand is bounded by $A\mathrm{\mid }h(t)\mathrm{\mid }[w_{-}(\omega )+w_{+}(\omega )]$. These are product-measurable functions on the sigma-finite Lebesgue spaces $\mathbb{R}$ and $(0,\mathrm{\infty })$. Tonelli in [F4] gives the bound $AM\int \mathrm{\mid }h\mathrm{\mid }<\mathrm{\infty }$ on the product integral. Fubini in [F4] therefore permits interchange of time and positive frequency, and the stated pairing is finite.

Apply [F6] with the same real-line diffeomorphism to $f(\xi )=\widehat{\phi }(\xi )e^{-2\pi i\omega \xi }$ and to $f(\xi )=\widehat{\phi }(\xi )e^{2\pi i\omega \xi }$, each integrable because the exponential has modulus one. Fourier inversion [F3] then gives $(2\pi {)}^{-1}{\int }_{\mathbb{R}}h(t)e^{i\omega t}dt=\phi (-\omega )$ and $(2\pi {)}^{-1}{\int }_{\mathbb{R}}h(t)e^{-i\omega t}dt=\phi (\omega )$. Thus the spectral pairing equals $A{\int }_0^{\mathrm{\infty }}[w_{-}(\omega )\phi (-\omega )+w_{+}(\omega )\phi (\omega )]d\omega$. Apply the half-line reflection in [F6] to $w_{-}(\omega )\phi (-\omega )$, which is integrable since a Schwartz function is bounded, to identify this expression with ${\int }_{\mathbb{R}}{J}_{\ast }(x)\phi (x)dx$. Hence the distributional spectrum is the finite measure constructed in step 1.1. The factor $2\pi$ from inversion cancels the convention's $(2\pi {)}^{-1}$; the factor ${\gamma }^{-1}$ remains from [F1].

Expanding $w_{-}$ and $w_{+}$ in $J_{\ast }$ gives the displayed Heaviside density on the respective open half-lines, with each term evaluated only there. Its value at the Lebesgue-null singleton zero is immaterial; in particular $\mu (\{0\})=0$. This proves the almost-everywhere density assertion while allowing any assigned value at zero.

For each integer $k\ge 1$, [F7] and [F8] give ${\int }_{k\pi }^{(k+1)\pi }\mathrm{\mid }\sin t\mathrm{\mid }dt\ge \mathrm{\mid }{\int }_{k\pi }^{(k+1)\pi }\sin tdt\mathrm{\mid }=\mathrm{\mid }\cos (k\pi )-\cos ((k+1)\pi )\mathrm{\mid }=2$. Since $t\le (k+1)\pi$ on that interval, step 1.2 implies ${\int }_{k\pi }^{(k+1)\pi }\mathrm{\mid }C(t)\mathrm{\mid }dt\ge 2A\mathrm{/}((k+1)\pi )$. If $C$ belonged to $L^1(\mathbb{R})$, then $\mathrm{\mid }C\mathrm{\mid }$ would be measurable with finite integral and [F5] would make $E\mapsto {\int }_E\mathrm{\mid }C(t)\mathrm{\mid }dt$ a finite measure. The intervals can be made disjoint by omitting endpoints, which have zero Lebesgue integral. For every $m\ge 0$, summing over $2^m\le k<2^{m+1}$ gives a lower bound $2^{m}2A\mathrm{/}(2^{m+1}\pi )=A\mathrm{/}\pi$. Finite additivity and monotonicity therefore bound the integral of $\mathrm{\mid }C\mathrm{\mid }$ over the first $L$ such disjoint blocks below by $LA\mathrm{/}\pi$ for every positive integer $L$. Since $A>0$, the full time integral is infinite. This proves the Statement's nonintegrability possibility without altering its distributional spectrum.