Analytic susceptibility and dispersion restrictions

The affine differentiation needed here follows directly from difference quotients: if $f$ is real differentiable on an open real domain, or complex differentiable on an open complex domain, and $b,d$ are fixed scalars in that field with $bt+d$ in the domain, then for real $h\mathrm{\ne }0$ and $b\mathrm{\ne }0$, $\frac{f(b(t+h)+d)-f(bt+d)}{h}=b\frac{f(bt+d+bh)-f(bt+d)}{bh}\to b{f}^{\mathrm{\prime }}(bt+d)$ as $h\to 0$; for $b=0$ the function is constant. Apply this to the exponential in [F2] with $b=-i\omega$, $d=0$, and repeat to obtain $(e^{-i\omega t}{)}^{\mathrm{\prime }}=-i\omega e^{-i\omega t}$ and $(e^{-i\omega t}{)}^{\mathrm{\prime }\mathrm{\prime }}=-{\omega }^2{e}^{-i\omega t}$. For a real harmonic history $E(t)=\mathrm{Re}(e{e}^{-i\omega t})$, extend the real coefficients of [A1] complex-linearly and substitute ${\tilde{P}}_{\mathrm{\ell }}(t)=p_{\mathrm{\ell }}{e}^{-i\omega t}$. The complex amplitude equation is $({\omega }_{\mathrm{\ell }}^2-{\omega }^2-i{\gamma }_{\mathrm{\ell }}\omega )p_{\mathrm{\ell }}=a_{\mathrm{\ell }}e$. Taking real parts commutes with real difference quotients and limits, so this gives a real harmonic particular solution of [A1]. Wherever the scalar denominator is nonzero, the amplitude is uniquely $p_{\mathrm{\ell }}=\frac{a_{\mathrm{\ell }}}{{\omega }_{\mathrm{\ell }}^2-{\omega }^2-i{\gamma }_{\mathrm{\ell }}\omega }e$, including when $e=0$. Summing gives the stated harmonic particular-solution multiplier without dividing by the vector amplitude. This does not identify the undamped coefficient with a retarded zero-past response.

The roots of ${\omega }_{\mathrm{\ell }}^2-z^2-i{\gamma }_{\mathrm{\ell }}z=0$ are $z=-i{\gamma }_{\mathrm{\ell }}\mathrm{/}2\pm \sqrt{{\omega }_{\mathrm{\ell }}^2-{\gamma }_{\mathrm{\ell }}^2\mathrm{/}4}$. If $0\le {\gamma }_{\mathrm{\ell }}<2{\omega }_{\mathrm{\ell }}$, the square root is real and both roots have imaginary part $-{\gamma }_{\mathrm{\ell }}\mathrm{/}2\le 0$; at ${\gamma }_{\mathrm{\ell }}=0$ they are the real values $\pm {\omega }_{\mathrm{\ell }}$. If ${\gamma }_{\mathrm{\ell }}=2{\omega }_{\mathrm{\ell }}$, the repeated root is $-i{\omega }_{\mathrm{\ell }}$. If ${\gamma }_{\mathrm{\ell }}>2{\omega }_{\mathrm{\ell }}$, write $\beta =\sqrt{{\gamma }_{\mathrm{\ell }}^2\mathrm{/}4-{\omega }_{\mathrm{\ell }}^2}$; then $0<\beta <{\gamma }_{\mathrm{\ell }}\mathrm{/}2$ and the roots $-i({\gamma }_{\mathrm{\ell }}\mathrm{/}2\mp \beta )$ both have strictly negative imaginary part. Hence no denominator vanishes in the open upper half-plane, proving analyticity there; the undamped real poles are exactly the stated boundary exceptions.

Fix one oscillator, denote its parameters by $a,{\omega }_0,\gamma$, assume $\gamma >0$, and set $g(t)=0$ for $t<0$. For $t\ge 0$, define the retarded kernel by $g(t)=e^{-\gamma t\mathrm{/}2}\sin (\alpha t)\mathrm{/}\alpha$ if $0<\gamma <2{\omega }_0$, $g(t)=t{e}^{-{\omega }_{0}t}$ if $\gamma =2{\omega }_0$, and $g(t)=(e^{-(\gamma \mathrm{/}2-\beta )t}-e^{-(\gamma \mathrm{/}2+\beta )t})\mathrm{/}(2\beta )$ if $\gamma >2{\omega }_0$, where $\alpha =\sqrt{{\omega }_0^2-{\gamma }^2\mathrm{/}4}$ and $\beta =\sqrt{{\gamma }^2\mathrm{/}4-{\omega }_0^2}$ in their respective cases. Differentiating with [F2] and the affine rule proved in step 1.1 gives $g(0)=0$, $g^{\mathrm{\prime }}(0+)=1$ and $g^{\mathrm{\prime }\mathrm{\prime }}+\gamma g^{\mathrm{\prime }}+{\omega }_0^{2}g=0$ for $t>0$. The underdamped $g,g^{\mathrm{\prime }}$ decay at rate $\gamma \mathrm{/}2$; the critical formula and derivative are bounded by $C{e}^{-{\omega }_{0}t\mathrm{/}2}$; the overdamped formula and derivative are bounded by $C{e}^{-(\gamma \mathrm{/}2-\beta )t}$, with $\gamma \mathrm{/}2-\beta >0$. Hence $g,g^{\mathrm{\prime }}\in L^1(0,\mathrm{\infty })$. For each positive decay rate $\lambda$, $t{e}^{-\lambda t}$ is integrable by integration by parts on $[0,T]$ and [F3] as $T\to \mathrm{\infty }$, so also $tg\in L^1(0,\mathrm{\infty })$.

For real $\omega >0$, put $A_{\mathrm{\ell }}={\omega }_{\mathrm{\ell }}^2-{\omega }^2$ and $B_{\mathrm{\ell }}={\gamma }_{\mathrm{\ell }}\omega$. Away from an undamped pole, $A_{\mathrm{\ell }}-i{B}_{\mathrm{\ell }}\mathrm{\ne }0$ and $\frac{a_{\mathrm{\ell }}}{A_{\mathrm{\ell }}-i{B}_{\mathrm{\ell }}}=\frac{a_{\mathrm{\ell }}(A_{\mathrm{\ell }}+i{B}_{\mathrm{\ell }})}{A_{\mathrm{\ell }}^2+B_{\mathrm{\ell }}^2}$, so $\mathrm{Im}(a_{\mathrm{\ell }}\mathrm{/}(A_{\mathrm{\ell }}-i{B}_{\mathrm{\ell }}))=a_{\mathrm{\ell }}{\gamma }_{\mathrm{\ell }}\omega \mathrm{/}(A_{\mathrm{\ell }}^2+{\gamma }_{\mathrm{\ell }}^2{\omega }^2)\ge 0$. Since ${\epsilon }_{\mathrm{\infty }}$ is real, summing these expressions proves the displayed nonnegative imaginary part of $\epsilon$. At ${\gamma }_{\mathrm{\ell }}=0$ and $\omega ={\omega }_{\mathrm{\ell }}$ the denominator vanishes, so this pointwise computation and pointwise formula do not apply.

For each real vector history $E\in C^2(\mathbb{R};{\mathbb{R}}^3)$ with $E,E^{\mathrm{\prime }},E^{\mathrm{\prime }\mathrm{\prime }}$ bounded, set $P(t)=a{\int }_0^{\mathrm{\infty }}g(s)E(t-s)ds$. Apply [F4] componentwise: the derivative integrands are measurable and bounded, uniformly in $t$, by $\mathrm{\parallel }{E}^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }g\mathrm{\mid }$ and $\mathrm{\parallel }{E}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }g\mathrm{\mid }$, which are integrable by step 2.1. Thus $P_{tt}+\gamma P_t+{\omega }_0^{2}P=a{\int }_0^{\mathrm{\infty }}[g(s)E^{\mathrm{\prime }\mathrm{\prime }}(t-s)+\gamma g(s)E^{\mathrm{\prime }}(t-s)+{\omega }_0^{2}g(s)E(t-s)]ds$.

Using the affine rule of step 1.1 for the shifted histories, integrating by parts twice on $[0,T]$ and taking $T\to \mathrm{\infty }$ is valid because $g,g^{\mathrm{\prime }}$ decay by step 2.1, $g^{\mathrm{\prime }\mathrm{\prime }}=-\gamma g^{\mathrm{\prime }}-{\omega }_0^{2}g\in L^1$, and $E,E^{\mathrm{\prime }}$ are bounded. The boundary values $g(0)=0$, $g^{\mathrm{\prime }}(0+)=1$ give ${\int }_0^{\mathrm{\infty }}g(s)E^{\mathrm{\prime }\mathrm{\prime }}(t-s)ds=E(t)+{\int }_0^{\mathrm{\infty }}{g}^{\mathrm{\prime }\mathrm{\prime }}(s)E(t-s)ds$ and ${\int }_0^{\mathrm{\infty }}g(s)E^{\mathrm{\prime }}(t-s)ds={\int }_0^{\mathrm{\infty }}{g}^{\mathrm{\prime }}(s)E(t-s)ds$. Substitution in step 3.1 cancels the integral by $g^{\mathrm{\prime }\mathrm{\prime }}+\gamma g^{\mathrm{\prime }}+{\omega }_0^{2}g=0$, leaving $P_{tt}+\gamma P_t+{\omega }_0^{2}P=aE$.

The normalized kernel $K=(a\mathrm{/}{\epsilon }_0)g$ is real, causal and locally absolutely continuous; step 2.1 gives $K,K^{\mathrm{\prime }}\in L^1$ and $sK\in L^1$. For $\mathrm{Im}z>0$, step 1.1 differentiates $e^{izs}$, and exponential decay makes the boundary terms vanish in integration by parts, so ${\int }_0^{\mathrm{\infty }}{g}^{\mathrm{\prime }}(s)e^{izs}ds=-izG(z)$ and ${\int }_0^{\mathrm{\infty }}{g}^{\mathrm{\prime }\mathrm{\prime }}(s)e^{izs}ds=-1-z^{2}G(z)$, where $G(z)={\int }_0^{\mathrm{\infty }}g(s)e^{izs}ds$. The Green equation gives $({\omega }_0^2-z^2-i\gamma z)G(z)=1$, hence the normalized transfer is $(a\mathrm{/}{\epsilon }_0)\mathrm{/}({\omega }_0^2-z^2-i\gamma z)$. Summing the finite kernels gives ${\chi }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}={\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}\mathrm{/}{\epsilon }_0$; with [F1], this kernel satisfies every hypothesis of [L1], which yields causality, upper-half-plane analyticity and the unsubtracted Hilbert relations for the strictly damped branch. Multiplication by ${\epsilon }_0$ gives the same relations for $\epsilon -{\epsilon }_{\mathrm{\infty }}={\chi }_{\mathrm{a}\mathrm{b}\mathrm{s}}$.

Steps 1.1–5.1 establish the rational susceptibility, its upper-half-plane analyticity, the nonnegative loss for every real positive frequency where the pointwise response exists, and the unsubtracted dispersion relation exactly in the strictly damped integrable-kernel branch. The undamped poles and their pointwise exclusions remain as stated; no assertion about a universal group-velocity bound follows from these calculations.