Airy connection across a single simple turning point of the declared regime model, with envelope error control

Adopt the declared WKB regime model of The declared WKB regime model: nondimensionalised line equation, turning points, phase integrals and the declared barrier transmission reading with its data $V,m,\mathrm{\hslash },\mathrm{\ell },E_{\star },W,\lambda ,h$, and assume hypothesis (W3) at a turning point ${\xi }_0$ with $W^{\mathrm{\prime }}({\xi }_0)>0$: on an interval $[{\xi }_0-r_0,{\xi }_0+s_0]$ with $r_0,s_0>0$ the function $W-\lambda$ has the single zero ${\xi }_0$, is strictly negative on $[{\xi }_0-r_0,{\xi }_0)$ (allowed side) and strictly positive on $({\xi }_0,{\xi }_0+s_0]$ (forbidden side). Write, for the two sides, $\theta (\xi ):=\frac{1}{h}{\int }_{\xi }^{{\xi }_0}\sqrt{\lambda -W({\xi }^{\mathrm{\prime }})}d{\xi }^{\mathrm{\prime }}(\xi <{\xi }_0),\mathrm{\Sigma }(\xi ):=\frac{1}{h}{\int }_{{\xi }_0}^{\xi }\sqrt{W({\xi }^{\mathrm{\prime }})-\lambda }d{\xi }^{\mathrm{\prime }}(\xi >{\xi }_0),$the nonnegative dimensionless phase and exponent measured from the turning point into the allowed and forbidden sides. Let$\zeta (\xi ):=\mathrm{sign}(\xi -{\xi }_0)[\frac{3}{2}\mid {\int }_{{\xi }_0}^{\xi }\sqrt{\mathrm{\mid }W({\xi }^{\mathrm{\prime }})-\lambda \mathrm{\mid }}d{\xi }^{\mathrm{\prime }}\mid {]}^{2\mathrm{/}3}$be the Liouville coordinate. Write $R$ for the continuous remainder in the transformed equation $w^{\mathrm{\prime }\mathrm{\prime }}=[\zeta \mathrm{/}{h}^2+R(\zeta )]w$ and put ${\eta }_{-}:=\zeta ({\xi }_0-r_0)$, ${\eta }_{+}:=\zeta ({\xi }_0+s_0)$. Define$\epsilon (h):=C_{\mathrm{k}\mathrm{e}\mathrm{r}}h{\int }_{{\eta }_{-}}^{{\eta }_{+}}\frac{\mathrm{\mid }R(v)\mathrm{\mid }}{\sqrt{\mathrm{\mid }v\mathrm{\mid }+h^{2\mathrm{/}3}}}dv,$where $C_{\mathrm{k}\mathrm{e}\mathrm{r}}$ is the kernel constant of Stable weighted Airy Volterra connection; boundedness of $R$ on the fixed transformed interval gives $\epsilon (h)\le C_{1}h$ for some $C_1>0$ depending only on the fixed data. There are constants $0<C_0<\mathrm{\infty }$ and $0<C_{\mathrm{e}\mathrm{r}\mathrm{r}}<\mathrm{\infty }$, depending only on $W,\lambda ,{\xi }_0,r_0,s_0$ (not on $h$), and $h_0>0$, such that for every $0<h<h_0$: (a) Decaying connection. There is an exact solution $u_{\mathrm{d}\mathrm{e}\mathrm{c}}$ of the model equation $u^{\mathrm{\prime }\mathrm{\prime }}=(W-\lambda )u\mathrm{/}{h}^2$ on $[{\xi }_0-r_0,{\xi }_0+s_0]$ and a real constant $C\mathrm{\ne }0$ withu_{\mathrm{dec}}(\xi)=C\,(W(\xi)-\lambda)^{-1/4} e^{-\Sigma(\xi)}\bigl(1+\delta_{\mathrm{f}}(\xi)\bigr)\qquad(\xi>\xi_{0}),$$$$u_{\mathrm{dec}}(\xi)=2C\,(\lambda-W(\xi))^{-1/4} \Bigl(\sin\bigl(\theta(\xi)+\tfrac{\pi}{4}\bigr)+\delta_{\mathrm{a}}(\xi)\Bigr) \qquad(\xi<\xi_{0}), where the absolute errors obey, for $\xi \mathrm{\ne }{\xi }_0$ away from the turning point,$\mathrm{\mid }{\delta }_{\mathrm{f}}(\xi )\mathrm{\mid }\le C_0(\epsilon (h)+h\mathrm{/}\mathrm{\mid }\zeta (\xi ){\mathrm{\mid }}^{3\mathrm{/}2}),\mid {\delta }_{\mathrm{a}}(\xi )\mid \le C_0(\epsilon (h)+h\mathrm{/}\mathrm{\mid }\zeta (\xi ){\mathrm{\mid }}^{3\mathrm{/}2}),$(b) Growing connection. There is a second exact solution $u_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}}$, linearly independent of $u_{\mathrm{d}\mathrm{e}\mathrm{c}}$, withu_{\mathrm{grow}}(\xi)=C'\,(W(\xi)-\lambda)^{-1/4} e^{+\Sigma(\xi)}\bigl(1+\delta_{\mathrm{f,grow}}(\xi)\bigr)\qquad(\xi>\xi_{0}),$$$$u_{\mathrm{grow}}(\xi)=C'\,(\lambda-W(\xi))^{-1/4} \Bigl(\cos\bigl(\theta(\xi)+\tfrac{\pi}{4}\bigr)+\delta_{\mathrm{a,grow}}(\xi)\Bigr) \qquad(\xi<\xi_{0}), for a real constant $C^{\mathrm{\prime }}\mathrm{\ne }0$, with $\mathrm{\mid }{\delta }_{\mathrm{f},\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}}(\xi )\mathrm{\mid },\mathrm{\mid }{\delta }_{\mathrm{a},\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}}(\xi )\mathrm{\mid }\le C_0(\epsilon (h)+h\mathrm{/}\mathrm{\mid }\zeta (\xi ){\mathrm{\mid }}^{3\mathrm{/}2})$ away from ${\xi }_0$; (c) Derivative and envelope control. Let $A_h(\zeta ):=A(\zeta \mathrm{/}{h}^{2\mathrm{/}3})$ and $B_h(\zeta ):=B(\zeta \mathrm{/}{h}^{2\mathrm{/}3})$ be the Airy functions in Airy envelopes and asymptotic coefficients, let $a(\xi ):=({\zeta }^{\mathrm{\prime }}(\xi ){)}^{-1\mathrm{/}2}$, and put$$ f_h(\zeta):=(1+|\zeta|/h^{2/3})^{-1/4},\qquad \phi_h(\zeta):=\frac{2(\zeta_+)^{3/2}}{3h}.

e_h^-(\zeta):=f_h(\zeta)e^{-\phi_h(\zeta)},\quad d_h^-(\zeta):=h^{-2/3}f_h(\zeta)^{-1}e^{-\phi_h(\zeta)},\quad e_h^+(\zeta):=f_h(\zeta)e^{\phi_h(\zeta)},\quad d_h^+(\zeta):=h^{-2/3}f_h(\zeta)^{-1}e^{\phi_h(\zeta)}. For each transformed coordinate $\eta\in\zeta([\xi_0-r_0,\xi_0+s_0])$, define $w_{\mathrm{dec}}(\eta):=u_{\mathrm{dec}}(\zeta^{-1}(\eta))/a(\zeta^{-1}(\eta))$ and $w_{\mathrm{grow}}(\eta):=u_{\mathrm{grow}}(\zeta^{-1}(\eta))/a(\zeta^{-1}(\eta))$. For the solutions chosen as in the proof, the following transformed derivative error bounds hold on the full transformed interval, with an error constant $C_{\mathrm{err}}$ depending only on the fixed data and not on $h$: $ |w_{\mathrm{dec}}'-A_h'|\le C_{\mathrm{err}} d_h^-(e^{\varepsilon(h)}-1),\qquad |w_{\mathrm{grow}}'-B_h'|\le C_{\mathrm{err}} d_h^+(e^{\varepsilon(h)}-1). $ Here $\varepsilon(h):=\varepsilon_h$ is the control parameter of [F3] and satisfies $\varepsilon(h)\le C_1h$; the corresponding value errors are bounded by $C_{\mathrm{err}}e_h^-(e^{\varepsilon(h)}-1)$ and $C_{\mathrm{err}}e_h^+(e^{\varepsilon(h)}-1)$. Here primes on $w_{\mathrm{dec}},w_{\mathrm{grow}},A_h,B_h$ denote $\eta$-derivatives, and primes on $a,\zeta$ denote $\xi$-derivatives. Consequently, |u_{\mathrm{dec}}'(\xi)-a'(\xi)A_h(\zeta(\xi))-a(\xi)\zeta'(\xi)A_h'(\zeta(\xi))|\le C_{\mathrm{err}}\bigl(|a'(\xi)|e_h^-(\zeta(\xi))+|a(\xi)|\zeta'(\xi)d_h^-(\zeta(\xi))\bigr)(e^{\varepsilon(h)}-1), $$$$ |u_{\mathrm{grow}}'(\xi)-a'(\xi)B_h(\zeta(\xi))-a(\xi)\zeta'(\xi)B_h'(\zeta(\xi))|\le C_{\mathrm{err}}\bigl(|a'(\xi)|e_h^+(\zeta(\xi))+|a(\xi)|\zeta'(\xi)d_h^+(\zeta(\xi))\bigr)(e^{\varepsilon(h)}-1).For $\rho(\xi):=W(\xi)-\lambda>0$ on the forbidden side and $p(\xi):=\sqrt{\lambda-W(\xi)}>0$ on the allowed side, the scaled slope forms on every compact section bounded away from $\xi_0$ areh\rho(\xi)^{-1/4}u_{\mathrm{dec}}'(\xi)=C e^{-\Sigma(\xi)}\bigl(-1+\delta_{\mathrm{f,dec}}^{(1)}(\xi)\bigr),\qquad h\rho(\xi)^{-1/4}u_{\mathrm{grow}}'(\xi)=C'e^{\Sigma(\xi)}\bigl(1+\delta_{\mathrm{f,grow}}^{(1)}(\xi)\bigr),$$hp(\xi)^{-1/2}u_{\mathrm{dec}}'(\xi)=2C\bigl(-\cos(\theta(\xi)+\tfrac{\pi}{4})+\delta_{\mathrm{a,dec}}^{(1)}(\xi)\bigr),\qquad hp(\xi)^{-1/2}u_{\mathrm{grow}}'(\xi)=C'\bigl(\sin(\theta(\xi)+\tfrac{\pi}{4})+\delta_{\mathrm{a,grow}}^{(1)}(\xi)\bigr).$$ Each new slope error is absolute and satisfies $\mathrm{\mid }{\delta }_{\mathrm{f},\mathrm{d}\mathrm{e}\mathrm{c}}^{(1)}\mathrm{\mid },\mathrm{\mid }{\delta }_{\mathrm{f},\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}}^{(1)}\mathrm{\mid },\mathrm{\mid }{\delta }_{\mathrm{a},\mathrm{d}\mathrm{e}\mathrm{c}}^{(1)}\mathrm{\mid },\mathrm{\mid }{\delta }_{\mathrm{a},\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{w}}^{(1)}\mathrm{\mid }\le C_0(\epsilon (h)+h\mathrm{/}\mathrm{\mid }\zeta (\xi ){\mathrm{\mid }}^{3\mathrm{/}2})$; in particular no relative error at a sine or cosine zero is asserted.

(d) Content of the connection. The factors $2$ in the decaying allowed-side form of (a) and $1$ in the growing allowed-side form of (b) are the amplitude relations of the exact Airy pair: the forbidden-side data determine the constants $C$ and $C^{\mathrm{\prime }}$, and the allowed-side oscillations are the sine and cosine forms with phase $\pi \mathrm{/}4$ measured from the turning point into the allowed region. The statement excludes rectangular walls, coalescing or multiple turning points, multi-turn resonances and barrier-top uniformity; it asserts no transmission or reflection coefficient (that is a separate statement requiring hypothesis (W2)), no wave packet, no detector and no measured value; the errors are absolute and blow up as the turning point is approached, as displayed by the $h\mathrm{/}\mathrm{\mid }\zeta {\mathrm{\mid }}^{3\mathrm{/}2}$ term.

• Orientation. The phase $\theta$ and exponent $\mathrm{\Sigma }$ are both measured from the turning point outward into the respective region; with this convention the decaying solution's allowed-side form is the sine with $+\pi \mathrm{/}4$, and the growing solution's is the cosine, exactly as in the Airy asymptotics of [F2]. Replacing the signed phase $\theta$ by $-\theta$ interchanges the leading forms because $\sin (-\theta +\pi \mathrm{/}4)=\cos (\theta +\pi \mathrm{/}4)$ and $\cos (-\theta +\pi \mathrm{/}4)=\sin (\theta +\pi \mathrm{/}4)$. A coordinate reflection with the outward nonnegative phase convention retains the same sine and cosine assignments.
• Use on this page. The barrier-transmission theorem of this page uses the decaying connection on both sides of a separated barrier; the error control of clause (c) is what makes the two matching errors comparable.