WKB approximation on an allowed interval of the declared regime model: explicit Gronwall error bound away from turning points

[Both coefficient relations and their matrix equation] By [F1], $M\in C^1$ is invertible and $(u,u^{\mathrm{\prime }}{)}^{\mathsf{T}}\in C^1$, so $c=M^{-1}(u,u^{\mathrm{\prime }}{)}^{\mathsf{T}}$ is $C^1$ and satisfies $Mc=(u,u^{\mathrm{\prime }}{)}^{\mathsf{T}}$. Differentiating the first component of this identity and comparing it with the second gives $w_{+}{c}_{+}^{\mathrm{\prime }}+w_{-}{c}_{-}^{\mathrm{\prime }}=0$. Differentiating the second component, using $u^{\mathrm{\prime }\mathrm{\prime }}=-(p^2\mathrm{/}{h}^2)u$ and $w_{\pm }^{\mathrm{\prime }\mathrm{\prime }}+(p^2\mathrm{/}{h}^2)w_{\pm }=q{w}_{\pm }$, gives $w_{+}^{\mathrm{\prime }}{c}_{+}^{\mathrm{\prime }}+w_{-}^{\mathrm{\prime }}{c}_{-}^{\mathrm{\prime }}=-qu$. Thus $M{c}^{\mathrm{\prime }}=(0,-qu{)}^{\mathsf{T}}$ and, since the Wronskian $W_0=-2i\mathrm{/}h$, $c^{\mathrm{\prime }}=-qu{M}^{-1}(0,1{)}^{\mathsf{T}}=\frac{q}{W_0}\left(\begin{array}{cc}{w}_{+}{w}_{-}& w_{-}^2\\ -w_{+}^2& -w_{+}{w}_{-}\end{array}\right)c=:Bc\mathrm{.}$ Each entry of $B$ has modulus $h\mathrm{\mid }q\mathrm{\mid }\mathrm{/}(2p)$ because $\mathrm{\mid }{w}_{+}\mathrm{\mid }=\mathrm{\mid }{w}_{-}\mathrm{\mid }=p^{-1\mathrm{/}2}$ and $\mathrm{\mid }{W}_0\mathrm{\mid }=2\mathrm{/}h$; the induced ${\mathrm{\ell }}^1$ matrix norm is the largest column sum, hence $\mathrm{\parallel }B{\mathrm{\parallel }}_1=h\mathrm{\mid }q\mathrm{\mid }\mathrm{/}p$.

[Growth bound] Put $v(s):=\mathrm{\parallel }B(s){\mathrm{\parallel }}_1=h\mathrm{\mid }q(s)\mathrm{\mid }\mathrm{/}p(s)$ and $U(x):=\mathrm{\parallel }c(a){\mathrm{\parallel }}_1+{\int }_a^{x}v(s)\mathrm{\parallel }c(s){\mathrm{\parallel }}_{1}ds$. By [F2] and step 1.1, $\mathrm{\parallel }c(x){\mathrm{\parallel }}_1\le U(x)$ and $U^{\mathrm{\prime }}(x)=v(x)\mathrm{\parallel }c(x){\mathrm{\parallel }}_1\le v(x)U(x)$. Therefore $\frac{d}{dx}(e^{-{\int }_a^{x}v}U(x))\le 0$, so $\mathrm{\parallel }c(x){\mathrm{\parallel }}_1\le U(x)\le \mathrm{\parallel }c(a){\mathrm{\parallel }}_1{e}^{{\int }_a^{x}v}$, which is clause (a).

[Deviation bound by integration] Again by [F2] and step 1.1, $\mathrm{\parallel }c(x)-c(a){\mathrm{\parallel }}_1\le {\int }_a^{x}v(s)\mathrm{\parallel }c(s){\mathrm{\parallel }}_{1}ds\le \mathrm{\parallel }c(a){\mathrm{\parallel }}_1{\int }_a^{x}v(s)e^{{\int }_a^{s}v(t)dt}ds\mathrm{.}$ The last integral is $e^{{\int }_a^{x}v}-1$ by the fundamental theorem of calculus applied to $s\mapsto e^{{\int }_a^{s}v}$; this proves clause (b), including the case $c(a)=0$. This derives the deviation estimate from the growth estimate without subtracting two growth bounds or treating $c-c(a)$ as a homogeneous solution.

[Pointwise wavefunction error] The coefficient representation in [F1] gives $u(x)-\mathrm{\Psi }(x)=w_{+}(x)(c_{+}(x)-c_{+}(a))+w_{-}(x)(c_{-}(x)-c_{-}(a))\mathrm{.}$ Since $\mathrm{\mid }{w}_{\pm }(x)\mathrm{\mid }=p(x{)}^{-1\mathrm{/}2}$, its absolute value is at most $p(x{)}^{-1\mathrm{/}2}\mathrm{\parallel }c(x)-c(a){\mathrm{\parallel }}_1$; substituting step 3.1 gives exactly clause (c).

[Scope and degeneracy] The hypotheses used are (W1) and $W\in C^2$ on a neighbourhood of $[a,b]$: in particular $\lambda -W\ge \delta >0$, so $p\ge \sqrt{\delta }$ and no turning point occurs on the interval. The displayed estimate is exact at each fixed $h$; its error is small only when the displayed factor $\exp (h{\int }_a^x\mathrm{\mid }q\mathrm{\mid }\mathrm{/}p)-1$ is small, which is an additional condition on the model data. The line-model units and physical interval $[\mathrm{\ell }a,\mathrm{\ell }b]$ are those of The declared WKB regime model: nondimensionalised line equation, turning points, phase integrals and the declared barrier transmission reading and Nondimensionalisation of the one-dimensional stationary equation and of the WKB exponent. This proves clause (d), without asserting a connection across a turning point or a measured quantity.