Qm Controlled Approximations

Let $\mathrm{\ell }>0$ and $E_{\star }>0$ be real reference scales, let $m>0$ and $\mathrm{\hslash }>0$ be real, and define $h:=\frac{\mathrm{\hslash }}{\mathrm{\ell }\sqrt{2m{E}_{\star }}}>0.$ Let $I\subseteq \mathbb{R}$ be a nonempty open interval, let $W\in C^2(I;\mathbb{R})$, let $\lambda \in \mathbb{R}$, and let $u:I\to \mathbb{C}$ be a function. Then:

(a) $u$ is of class $C^2$ on $I$ and satisfies $u^{\mathrm{\prime }\mathrm{\prime }}(\xi )=\frac{W(\xi )-\lambda }{h^2}u(\xi )(\xi \in I)$ if and only if the function $\psi :\mathrm{\ell }I\to \mathbb{C}$, $\psi (x):=u(x\mathrm{/}\mathrm{\ell })$, is of class $C^2$ on the open interval $\mathrm{\ell }I=\{\mathrm{\ell }\xi :\xi \in I\}$ and satisfies $-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}(x)+E_{\star }W(x\mathrm{/}\mathrm{\ell })\psi (x)=E_{\star }\lambda \psi (x)(x\in \mathrm{\ell }I);$

(b) if $a<b$ are points of $I$, then each of the following identities holds under its indicated strict-sign hypothesis:

(i) if $W(\xi )-\lambda >0$ for every $\xi \in [a,b]$, then $\frac{1}{h}{\int }_a^b\sqrt{W(\xi )-\lambda }d\xi =\frac{1}{\mathrm{\hslash }}{\int }_{\mathrm{\ell }a}^{\mathrm{\ell }b}\sqrt{2m(E_{\star }W(x\mathrm{/}\mathrm{\ell })-E_{\star }\lambda )}dx;$

(ii) if $\lambda -W(\xi )>0$ for every $\xi \in [a,b]$, then $\frac{1}{h}{\int }_a^b\sqrt{\lambda -W(\xi )}d\xi =\frac{1}{\mathrm{\hslash }}{\int }_{\mathrm{\ell }a}^{\mathrm{\ell }b}\sqrt{2m(E_{\star }\lambda -E_{\star }W(x\mathrm{/}\mathrm{\ell }))}dx\mathrm{.}$

All square roots here are the nonnegative real square roots of strictly positive numbers. Part (i) is the forbidden-region exponent identity; part (ii) is the allowed-region phase identity.

The identities are statements about the displayed functions and integrals only. The number $h$ is dimensionless because $\mathrm{\hslash }$ is in $\mathrm{J}\mathrm{s}$ and $\mathrm{\ell }\sqrt{2m{E}_{\star }}$ is in $\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-1}=\mathrm{J}\mathrm{s}$; no physical interpretation, no approximation and no error estimate is asserted here.

• Role of the lemma. The lemma is the exact bookkeeping that lets the dimensionless equation $u^{\mathrm{\prime }\mathrm{\prime }}=(W-\lambda )u\mathrm{/}{h}^2$, in which the semiclassical parameter $h$ is explicit, be read as the physical line-model equation with potential $E_{\star }W(\cdot \mathrm{/}\mathrm{\ell })$ at energy $E_{\star }\lambda$. Both the equation and the phase integral transfer exactly; no approximation is involved.
• Small $h$. The regime $h\ll 1$ is the regime $\mathrm{\hslash }\ll \mathrm{\ell }\sqrt{2m{E}_{\star }}$; the lemma itself makes no smallness assumption, and the error theorems that use smallness state it separately.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be an arbitrary complex Hilbert space (Hilbert space) and $H_0=H_0^{\ast }$ a self-adjoint operator on its exact dense domain $D=D(H_0)$ (Symmetric, self-adjoint and essentially self-adjoint operators). Let $E\in \mathbb{R}$ be an isolated eigenvalue with $\mathrm{dist}(E,\sigma (H_0)\setminus \{E\})\ge g>0$, and let its orthogonal spectral projection $P_0=E_{H_0}(\{E\})$ have finite rank $r\ge 1$. The PVM and gap calculus are those of CC spectral calculus on an arbitrary complex Hilbert space by compatible vector-local Cayley subspaces. Let $V=V^{\ast }\in \mathcal{B}(H)$ and $v=\mathrm{\parallel }V\mathrm{\parallel }\ge 0$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). Put $H_{\lambda }=H_0+\lambda V$ on $D$, and define ${\mathrm{\Omega }}_v=\{\begin{array}{ll}\{\lambda \in \mathbb{C}:\mathrm{\mid }\lambda \mathrm{\mid }<g\mathrm{/}(2v)\},& v>0,\\ \mathbb{C},& v=0.\end{array}$ For $\lambda \in {\mathrm{\Omega }}_v$ write $q_{\lambda }=2\mathrm{\mid }\lambda \mathrm{\mid }v\mathrm{/}g<1$, $\mathrm{\Gamma }=\{z:\mathrm{\mid }z-E\mathrm{\mid }=g\mathrm{/}2\}$ oriented counterclockwise, and $\mathrm{\Delta }=\{z:\mathrm{\mid }z-E\mathrm{\mid }<g\mathrm{/}2\}$.

(a) Resolvent and exact domain. $H_{\lambda }$ is closed and densely defined on $D$. Every $z\in \mathrm{\Gamma }$ is a resolvent point, with the convention $R_{\lambda }(z)=(z-H_{\lambda }{)}^{-1}:H\to D$ of Resolvent and spectrum of an unbounded operator, and $R_{\lambda }(z)=R_0(z){\sum }_{k\ge 0}(\lambda V{R}_0(z){)}^k,\mathrm{\parallel }{R}_{\lambda }(z)\mathrm{\parallel }\le \frac{2\mathrm{/}g}{1-q_{\lambda }},\mathrm{\parallel }{R}_{\lambda }(z)-R_0(z)\mathrm{\parallel }\le \frac{2}{g}\frac{q_{\lambda }}{1-q_{\lambda }}\mathrm{.}$ The series converges in operator norm uniformly on $\mathrm{\Gamma }$ and locally uniformly in $\lambda \in {\mathrm{\Omega }}_v$.

(b) Finite-rank holomorphic cluster. The norm contour integral $P_{\lambda }=\frac{1}{2\pi i}{\oint }_{\mathrm{\Gamma }}{R}_{\lambda }(z)dz$ is a bounded idempotent of rank exactly $r$ throughout ${\mathrm{\Omega }}_v$, with $P_{\lambda =0}=P_0$ and $\mathrm{\parallel }{P}_{\lambda }-P_0\mathrm{\parallel }\le q_{\lambda }\mathrm{/}(1-q_{\lambda })$. It is holomorphic in operator norm. Its range $M_{\lambda }=P_{\lambda }H$ lies in $D$ and is invariant under $H_{\lambda }$, and $P_{\lambda }{H}_{\lambda }=H_{\lambda }{P}_{\lambda }$ on $D$. Both $H_{\lambda }{P}_{\lambda }\in \mathcal{B}(H)$ and $P_{\lambda }:H\to (D,\mathrm{\parallel }x\mathrm{\parallel }+\mathrm{\parallel }{H}_{0}x\mathrm{\parallel })$ are holomorphic in their respective operator norms.

(c) Exactly the enclosed spectrum. Let $A_{\lambda }=H_{\lambda }{\mathrm{\mid }}_{M_{\lambda }}:M_{\lambda }\to M_{\lambda }$. Then $\sigma (H_{\lambda })\cap \mathrm{\Delta }=\sigma (A_{\lambda })\subset \mathrm{\Delta }\mathrm{.}$ This is a nonempty finite set of at most $r$ eigenvalues. The restriction of $w-H_{\lambda }$ to $\ker P_{\lambda }$ has a bounded two-sided inverse on its exact domain $D\cap \ker P_{\lambda }$ for every $w\in \mathrm{\Delta }$. Consequently all generalized eigenvectors at an enclosed eigenvalue belong to $M_{\lambda }$; their eigenspaces and generalized eigenspaces are those of this finite-dimensional restriction.

(d) Real coupling and zero perturbation. For real $\lambda$, $H_{\lambda }$ is self-adjoint on $D$. For real $\lambda \in {\mathrm{\Omega }}_v$, $P_{\lambda }=E_{H_{\lambda }}((E-g\mathrm{/}2,E+g\mathrm{/}2))$ is the orthogonal projection on the entire enclosed spectral cluster (Projection valued measure). If $v=0$, then $V=0$, $H_{\lambda }=H_0$ and $P_{\lambda }=P_0$ for every complex coupling, with $A_{\lambda }=E{I}_{M_0}$ and enclosed spectrum $\{E\}$.

No separability or arbitrary-index Axiom of Choice is assumed. The claim supplies a cluster and its finite restriction; for $r>1$ it supplies no ordering or individually holomorphic scalar eigenvalue branches.

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2 and the autonomous-dynamics postulate P4 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) for a declared system, in the state reading of The state space of the framework: preparations, pure rays and density operators, the sharp observable reading of Assigned sharp observables and the outcome law of the framework and the evolution reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution; units are those of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Assigned energy observable. As an additional model assumption of the pair, fix a self-adjoint operator $H:D(H)\subseteq \mathfrak{H}\to \mathfrak{H}$ on the declared state space $\mathfrak{H}$ and declare the pair $(H,\mathrm{J})$ to be the system's sharp energy observable in the sense of Assigned sharp observables and the outcome law of the framework clause (a); this identification of the operator with the energy of the system is an assumption of the model, not a consequence of P1, P2 or P4. Assume $H$ is bounded below: there is a real $c$ with $⟨Hu,u⟩\ge c\mathrm{\parallel }u{\mathrm{\parallel }}^2$ for every $u\in D(H)$. The spectrum $\sigma (H)$ is the set of the cited spectral theorem, and the model bottom is $E_0:=\inf \sigma (H)\in [c,+\mathrm{\infty }),$ read in joules. An assignment of a concrete material system to the pair $(H,\mathfrak{H})$ is a tested identification in the sense of the preparation and apparatus discipline P7 and is not made here.

(c) Form domain and trial value. Let $Q(H):=D((H-cI{)}^{1\mathrm{/}2})$ and $q_H$ be the form domain and the closed semibounded form of Spectral form domain and core of a semibounded operator for the lower bound $c$ of clause (b), with $q_H[u]={\int }_{\mathbb{R}}\lambda d{E}_u(\lambda )(u\in Q(H)),$ $E$ the projection-valued measure of $H$ and $E_u(B)=⟨E(B)u,u⟩$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure). The definition uses the facts, proved by that lemma, that $q_H$ does not depend on the choice of the lower bound $c$, that $D(H)$ is dense in $Q(H)$, and that $q_H[u]=⟨Hu,u⟩$ for $u\in D(H)$. A trial vector of the model is a vector $u\in Q(H)$ with $\mathrm{\parallel }u\mathrm{\parallel }=1$; its trial value is $q_H[u]$, read in joules. For a trial vector that lies in $D(H)$, the trial value is the mean energy of the pure preparation $[u]$ under the observable $(H,\mathrm{J})$ in the moment reading of Mean, second moment and variance of an assigned observable in a preparation; for a general trial vector in $Q(H)\setminus D(H)$ the trial value is the form value and is not asserted to be the mean of any outcome law.

(d) Bottom gap and discrete ladder. If the bottom is an isolated spectral value, declare a gap to be a real $g>0$ with $\sigma (H)\cap (E_0,E_0+g)=\mathrm{\varnothing },$ and put $P:=E(\{E_0\})$ for the bottom spectral projection. The discrete spectrum below the essential spectrum is the part of ${\sigma }_{\mathrm{d}}(H)$ of Discrete and essential spectrum of a self-adjoint operator below $\mathrm{\Lambda }:=\inf {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(H)$, listed with multiplicity in nondecreasing order as $E_1\le E_2\le \cdots$, with the convention $E_n:=\mathrm{\Lambda }$ once the list is exhausted, $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. The definition asserts neither that $E_0$ is an eigenvalue, nor that any eigenvector exists. The ladder convention does give $E_0=E_1$: the nonempty closed spectrum of the bounded-below operator contains its bottom; if the bottom is discrete, it is the first listed eigenvalue, and otherwise it lies in the essential spectrum, so $\mathrm{\Lambda }=E_0$ and the empty below-threshold list is padded by $E_1=\mathrm{\Lambda }$. This equality does not imply that the bottom is attained by an eigenvector.

(e) Scope. Clauses (a)-(d) are a reading of the framework's state, observable and dynamics postulates together with the declared operator $H$; they assert nothing about whether a preparation attains the trial value, about the behaviour of any preparation under time evolution beyond P4, about equilibration, environmental coupling, measurement or apparatus, and about any measured energy. All spectral statements are statements about the declared operator.

(a) Framework data. Adopt P1, P2 and P4 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) for a declared system, with the state, sharp-observable and evolution readings of Assigned sharp observables and the outcome law of the framework and Assigned energy observable, unitary flow and the two pictures of autonomous evolution; units are those of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice) as the choice hypotheses of the cited mathematical suppliers.

(b) The perturbed family. As an additional model assumption of the pair, fix a self-adjoint operator $H_0:D(H_0)\subseteq \mathfrak{H}\to \mathfrak{H}$ on the declared state space, declare $(H_0,\mathrm{J})$ to be the unperturbed sharp energy observable, fix a bounded self-adjoint operator $V\in \mathcal{B}(\mathfrak{H})$ (A bounded linear operator between normed spaces) with $v:=\mathrm{\parallel }V\mathrm{\parallel }<\mathrm{\infty }$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum) and fix a real coupling $\lambda$ (dimensionless: $\lambda V$ is read in joules). Declare the perturbed family $H(\lambda ):=H_0+\lambda V,D(H(\lambda )):=D(H_0),$ to be the model's sharp energy observable at coupling $\lambda$. Since $V$ is $H_0$-bounded with $H_0$-bound $0\le a<1$ (the bound inequality holds with $a=0$ and $b=v$), the Kato-Rellich theorem applies and $H(\lambda )$ is self-adjoint on $D(H_0)$ for every real $\lambda$ (Kato-Rellich theorem, Symmetric, self-adjoint and essentially self-adjoint operators). This self-adjointness is a supplied theorem about the declared operators; the reading of the family as the system's energy is the model assumption.

(c) The gapped label. Fix an isolated eigenvalue $E\in \mathbb{R}$ of $H_0$ of finite multiplicity $r\ge 1$ with gap $\mathrm{dist}(E,\sigma (H_0)\setminus \{E\})\ge g>0,$ and put $P_0:=E_{H_0}(\{E\})$ and $Q_0:=I-P_0$ (Discrete and essential spectrum of a self-adjoint operator, Projection valued measure, Resolvent and spectrum of an unbounded operator). The definitions used by the pair are those of the cited suppliers:

(i) for real $\lambda$ with $2\mathrm{\mid }\lambda \mathrm{\mid }v\mathrm{/}g<1$, the followed spectral cluster is the spectrum of $H(\lambda )$ inside $\mathrm{\mid }z-E\mathrm{\mid }<g\mathrm{/}2$, and $P_{\lambda }:=\frac{1}{2\pi i}{\oint }_{\mathrm{\mid }z-E\mathrm{\mid }=g\mathrm{/}2}(z-H(\lambda ){)}^{-1}dz$ (counterclockwise) is its rank-$r$ orthogonal spectral projection, by Finite-rank gapped spectral cluster under a bounded perturbation. The cluster is a nonempty finite set and is exactly the spectrum of the restriction to $P_{\lambda }\mathfrak{H}$. In the simple case $r=1$ only, write $E_{\lambda }$ for its unique spectral value; this scalar is the followed perturbed label of Contour perturbation projection and is analytic on the valid coupling disk. For $r>1$ no unique scalar label, branch ordering or individual scalar holomorphy is presumed. At $\lambda =0$ the cluster is $\{E\}$ and its projection is $P_0$. If $v=0$, these data are constant for all real couplings, with no radius involving division by $v$.

(ii) the first-order operator is the Hermitian operator $V_E:=\left(P_{0}V{P}_0\right){\mid }_{P_0\mathfrak{H}}$ on the finite-dimensional eigenspace $P_0\mathfrak{H}$, whose spectrum $\mathrm{spec}(V_E)\subseteq [-v,v]$ is the model's first-order shift data (joules per unit coupling);

(iii) the complementary reduced inverse $S:={\iota }_{Q_0}(H_0{\mathrm{\mid }}_{D(H_0)\cap Q_0\mathfrak{H}}-E{)}^{-1}{Q}_0$ is a bounded operator with the displayed exact complementary-domain inverse. For every finite multiplicity $r\ge 1$, including $v=0$, its existence follows by applying part (c) of Finite-rank gapped spectral cluster under a bounded perturbation at coupling $\lambda =0$ and $w=E$: since the cluster projection is then $P_0$, the restriction of $E-H_0$ to $D(H_0)\cap Q_0\mathfrak{H}$ has a bounded two-sided inverse $R_Q(E):Q_0\mathfrak{H}\to D(H_0)\cap Q_0\mathfrak{H}$, and $S=-{\iota }_{Q_0}{R}_Q(E)Q_0$. For $r=1$ the same operator is supplied by Simple-level reduced-resolvent coefficients. Its role is to record that the whole complementary spectral subspace, continuum included, contributes to the second-order data.

(d) Scope. Clauses (a)-(c) cover bounded perturbations of a self-adjoint operator and a fixed isolated finite-multiplicity label only. The definition asserts no time-dependent perturbation, no adiabatic ramping of $\lambda$, no statement that a preparation occupies the branch of $E_{\lambda }$ or of any eigenvalue of $V_E$, no branch-population or relaxation statement, no measurement and no identification of the model with a laboratory system; the labels $E_{\lambda }$ are spectral data of the declared family.

(a) Framework data and line model. Adopt P1, P2, P4 and P5 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), the state and observable readings cited there, and the declared effective one-dimensional line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions with a mass parameter $m>0$, spin label $s=0$, action constant $\mathrm{\hslash }>0$ and units from Dimensions, SI unit conventions and quantity types of the framework (The massive spin-s particle model of the framework: state space, operators, units and frame readings). Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix a real potential $V\in C_c^5(\mathbb{R};\mathbb{R})$ (five times continuously differentiable with compact support); it is real, bounded and measurable, so the line model's assigned energy operator $H_V$ of its clause (c) is defined with domain $D(H_V)=W^{2,2}(\mathbb{R};\mathbb{C})$, acts by $H_{V}f=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f+Vf$, and its stationary equation at energy $E$ reads $-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}+V\psi =E\psi$ for classically $C^2$ solutions.

(b) Reference scales and nondimensionalisation. As an additional model assumption of the pair fix a reference length $\mathrm{\ell }>0$ (metres) and a reference energy $E_{\star }>0$ (joules), and put $W(\xi ):=\frac{V(\mathrm{\ell }\xi )}{E_{\star }},\lambda :=\frac{E}{E_{\star }},h:=\frac{\mathrm{\hslash }}{\mathrm{\ell }\sqrt{2m{E}_{\star }}}\mathrm{.}$ Then $W\in C_c^5(\mathbb{R};\mathbb{R})$ is dimensionless with support scaled by $\mathrm{\ell }$, and $h>0$ is dimensionless. By Nondimensionalisation of the one-dimensional stationary equation and of the WKB exponent the physical stationary equation of clause (a) at energy $E$ is equivalent, under $x=\mathrm{\ell }\xi$, $\psi (\mathrm{\ell }\xi )=u(\xi )$, to the dimensionless WKB equation $u^{\mathrm{\prime }\mathrm{\prime }}(\xi )=\frac{W(\xi )-\lambda }{h^2}u(\xi ),\xi \in I,$ on every interval $I$. The exact transfer of the square-root action integrals over a subinterval $[x_a,x_b]=\mathrm{\ell }[a,b]$ is, when the indicated strict sign holds throughout $[a,b]$, $\frac{1}{\mathrm{\hslash }}{\int }_{x_a}^{x_b}\sqrt{2m(V(x)-E)}dx=\frac{1}{h}{\int }_a^b\sqrt{W(\xi )-\lambda }d\xi \text{if }W-\lambda >0,$ $\frac{1}{\mathrm{\hslash }}{\int }_{x_a}^{x_b}\sqrt{2m(E-V(x))}dx=\frac{1}{h}{\int }_a^b\sqrt{\lambda -W(\xi )}d\xi \text{if }\lambda -W>0;$ these are respectively clauses (b)(i) and (b)(ii) of Nondimensionalisation of the one-dimensional stationary equation and of the WKB exponent. The pair calls $h$ the model's semiclassical parameter and calls $h\ll 1$ the semiclassical regime; the physical content of the regime is $\mathrm{\hslash }\ll \mathrm{\ell }\sqrt{2m{E}_{\star }}$, and every statement of the page's theorems is an exact statement about the family indexed by $h$, with the error bounds stated at each $h$. For the mathematical small-$h$ limit, hold the dimensionless profile $W$ and energy $\lambda$ fixed and vary the parameter in $u^{\mathrm{\prime }\mathrm{\prime }}=(W-\lambda )u\mathrm{/}{h}^2$. A corresponding physical family with fixed $m,E_{\star },\mathrm{\hslash }$ is obtained by taking ${\mathrm{\ell }}_h=\mathrm{\hslash }\mathrm{/}(h\sqrt{2m{E}_{\star }})$ and $V_h(x)=E_{\star }W(x\mathrm{/}{\mathrm{\ell }}_h)$, with $E=E_{\star }\lambda$; the physical width then changes, while the natural action constant does not. A single originally chosen $V,\mathrm{\ell },E_{\star }$ supplies one member of this family.

(c) Turning points, regions and phase. For the data $(W,\lambda )$ put $\tau :=\{\xi \in \mathbb{R}:W(\xi )=\lambda \}$ for the set of turning points. A point ${\xi }_0\in \tau$ is simple when $W^{\mathrm{\prime }}({\xi }_0)\mathrm{\ne }0$. An open interval $J$ is an allowed interval when $W(\xi )-\lambda <0$ on $J$ (the dimensionless kinetic energy $-(W-\lambda )$ is positive there), a forbidden interval when $W(\xi )-\lambda >0$ on $J$. On an allowed interval the model uses the positive scaled momentum $p(\xi ):=\sqrt{\lambda -W(\xi )}>0$ with local wavenumber $k_h(\xi ):=p(\xi )\mathrm{/}h$ in the coordinate $\xi$ and WKB phase ${\theta }_h(\xi ):=h^{-1}{\int }_{{\xi }_{\mathrm{r}\mathrm{e}\mathrm{f}}}^{\xi }p(s)ds$ with a declared base point; on a forbidden interval the model uses $\kappa (\xi ):=\sqrt{W(\xi )-\lambda }>0$. The dimensionless forbidden action of a forbidden interval $[{\xi }_a,{\xi }_b]$ is $S:={\int }_{{\xi }_a}^{{\xi }_b}\sqrt{W(\xi )-\lambda }d\xi =\frac{h}{\mathrm{\hslash }}{\int }_{x_a}^{x_b}\sqrt{2m(V(x)-E)}dx,$ The WKB exponent is $S\mathrm{/}h$, the physical action in units of $\mathrm{\hslash }$. For a barrier whose endpoints are simple turning points, the strict-sign hypothesis of clause (b)(i) fails at those endpoints; apply that clause on $[{\xi }_a+\delta ,{\xi }_b-\delta ]$ and let $\delta \downarrow 0$. Both integrands extend continuously with value zero at the endpoints, so the omitted endpoint integrals tend to zero. The three declared regime hypotheses used by the page are: