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Allowed-region WKB coefficient equation

Statement

Let $a < b$ be real numbers, let $p \in C^{2} ([a, b])$ with $p (x) > 0$ for all $x \in [a, b]$ , let $h > 0$ , put $θ (x) : = \int_{a}^{x} p (s) d s$ and $w_{\pm} (x) : = p (x)^{- 1 / 2} e^{\pm i θ (x) / h}, q (x) : = \frac{3 p^{'} (x)^{2}}{4 p (x)^{2}} - \frac{p^{''} (x)}{2 p (x)} .$ Then:

(a) $w_{\pm} \in C^{2} ([a, b])$ and $w_{\pm}^{''} + \frac{p^{2}}{h^{2}} w_{\pm} = q w_{\pm}$ pointwise on $[a, b]$ .

(b) The Wronskian $W : = w_{+} w_{-}^{'} - w_{+}^{'} w_{-}$ equals $- 2 i / h$ on $[a, b]$ .

(c) Let $y \in C^{2} ([a, b])$ solve $y^{''} + \frac{p^{2}}{h^{2}} y = 0$ , let $M (x) : = (\begin{array}{cc} w_{+} (x) & w_{-} (x) \\ w_{+}^{'} (x) & w_{-}^{'} (x) \end{array})$ and $c (x) : = (c_{+} (x), c_{-} (x))^{T} : = M (x)^{- 1} (y (x), y^{'} (x))^{T}$ (well defined by (b)). Then $c \in C^{1} ([a, b]; C^{2})$ and $c^{'} = B (x) c$ with $B (x) = \frac{q (x)}{W} (\begin{array}{cc} w_{+} (x) w_{-} (x) & w_{-} (x)^{2} \\ - w_{+} (x)^{2} & - w_{+} (x) w_{-} (x) \end{array}),$ and with $∥ \cdot ∥_{1}$ the $ℓ^{1}$ norm on $C^{2}$ one has $∥ B (x) ∥_{1} = \frac{h ∣ q (x) ∣}{p (x)}$ for every $x \in [a, b]$ .

In particular $y = w_{+} c_{+} + w_{-} c_{-}$ , and the coefficient system is a nonsingular linear ODE with explicitly bounded coefficients. No approximation of $y$ and no physical interpretation is asserted here.

Facts & Assumptions

Given: the interval $[a, b]$ , the function $p \in C^{2} ([a, b])$ with $p > 0$ , the number $h > 0$ , the phase $θ$ , the functions $w_{\pm}$ , $q$ , an exact solution $y \in C^{2}$ of $y^{''} + (p^{2} / h^{2}) y = 0$ , and $M$ , $c$ as in the statement.

[F1]

Differentiation rules: sums, products, quotients and compositions of $C^{2}$ functions are $C^{2}$ and obey the product and chain rules; $p^{\pm 1 / 2}$ , $p^{- 1}$ are $C^{2}$ because $p > 0$ on the compact interval (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

[F2]

Integrals of continuous $C^{2}$ -valued curves are taken componentwise as Riemann integrals. Uniform continuity on $[a, b]$ makes prescribed grid sums Cauchy, so completeness of $C^{2}$ gives their limit. The difference quotient of $G (x) = \int_{a}^{x} g (s) d s$ differs from $g (x)$ by at most $\sup_{∣ s - x ∣ \leq ∣ h ∣} ∥ g (s) - g (x) ∥$ , which tends to zero. Thus $G^{'} = g$ , with one-sided endpoint derivatives. A $C^{1}$ curve equals its initial value plus the integral of its derivative, since their difference has zero derivative componentwise. No choice principle is used.

[F3]

Cramer's rule in dimension $2$ : since $\det M = W \neq 0$ by (b), $M (x)^{- 1} = \frac{1}{W} (\begin{array}{cc} w_{-}^{'} & - w_{-} \\ - w_{+}^{'} & w_{+} \end{array})$ and $c = M^{- 1} (y, y^{'})^{T}$ is well defined (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, algebra).

Proof

1.1F1algebra

[Differentiating the WKB waves] By [F1], $w_{\pm}^{'} = (\pm i p / h - p^{'} / (2 p)) w_{\pm}$ and a second differentiation gives $w_{\pm}^{''} = [(\pm i p / h - p^{'} / (2 p))^{2} \pm i p^{'} / h - \frac{d}{d x} (p^{'} / (2 p))] w_{\pm}$ . Expanding, $(\pm i p / h)^{2} = - p^{2} / h^{2}$ , the cross term in the square is $\mp i p^{'} / h$ , which cancels the explicit $\pm i p^{'} / h$ , and the remaining real terms are $3 p^{' 2} / (4 p^{2}) - p^{''} / (2 p) = q$ ; hence $w_{\pm}^{''} + \frac{p^{2}}{h^{2}} w_{\pm} = q w_{\pm}$ .

2.1step 1.1F1algebra

[The Wronskian] $W = w_{+} w_{-}^{'} - w_{+}^{'} w_{-} = w_{+} w_{-} [(- i p / h - p^{'} / (2 p)) - (i p / h - p^{'} / (2 p))] = - 2 i p w_{+} w_{-} / h$ , and $w_{+} w_{-} = p^{- 1}$ ; hence $W = - 2 i / h$ .

3.1step 1.1step 2.1F1F3algebra

[The coefficient equation] By [F1] and [F3], $M \in C^{1}$ has nonzero determinant and $(y, y^{'})^{T} \in C^{1}$ , so $c = M^{- 1} (y, y^{'})^{T}$ is $C^{1}$ . Its defining identity gives both $y = w_{+} c_{+} + w_{-} c_{-}$ and $y^{'} = w_{+}^{'} c_{+} + w_{-}^{'} c_{-}$ . Differentiating the first identity gives $y^{'} = w_{+}^{'} c_{+} + w_{-}^{'} c_{-} + w_{+} c_{+}^{'} + w_{-} c_{-}^{'}$ ; comparison with the second identity therefore gives $w_{+} c_{+}^{'} + w_{-} c_{-}^{'} = 0$ . Differentiating the second identity and inserting (a) and the exact equation for $y$ yields $0 = y^{''} + \frac{p^{2}}{h^{2}} y = q (w_{+} c_{+} + w_{-} c_{-}) + w_{+}^{'} c_{+}^{'} + w_{-}^{'} c_{-}^{'}$ . These are respectively the first and second rows of $M c^{'} = - q y (0, 1)^{T}$ . Hence $c^{'} = - q y M^{- 1} (0, 1)^{T}$ , and $y = (w_{+}, w_{-}) c$ , so $c^{'} = B c$ with $B : = - q M^{- 1} (\begin{array}{cc} 0 & 0 \\ w_{+} & w_{-} \end{array}) = \frac{q}{W} (\begin{array}{cc} w_{+} w_{-} & w_{-}^{2} \\ - w_{+}^{2} & - w_{+} w_{-} \end{array})$ .

4.1step 2.1step 3.1F1F2algebra∎

[Regularity and coefficient bound] The right side $B c$ is continuous because $q$ , $p$ and $c$ are, so by [F2] the pair $c$ solves the integral equation $c (x) = c (a) + \int_{a}^{x} B (s) c (s) d s$ and is $C^{1}$ with $c^{'} = B c$ ; moreover the entries of $B$ are $q W^{- 1}$ times $w_{+} w_{-}$ , $w_{-}^{2}$ , $- w_{+}^{2}$ , $- w_{+} w_{-}$ , all of modulus $p^{- 1}$ , so $∣ B_{i j} ∣ = \frac{∣ q ∣}{∣ W ∣ p} = \frac{h ∣ q ∣}{2 p}$ . Therefore $∥ B ∥_{1} = \max {∣ B_{11} ∣ + ∣ B_{21} ∣, ∣ B_{12} ∣ + ∣ B_{22} ∣} = \frac{h ∣ q ∣}{p}$ .

Allowed-region WKB coefficient equation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources