# A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation

## Abstract

**:**

## 1. Introduction

## 2. Statement and Re-Statement of the Problem

#### 2.1. Associated IVP and Statement of The Problem

**Proposition**

**1.**

#### 2.2. Change of the Independent Variable and Re-Statement of the Problem

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Calculations in the Normal Form

#### 3.1. Exact and ${\left(JWKB\right)}_{1}$ Solutions of the ${\left(BDE\right)}_{1}$ in the Normal Form

- (i)
- The numerical value of ${d}_{2}$ (=constant) chosen in (18) should not correspond to the classical turning points of the associated normal form differential equation (where ${f(c,\rho )|}_{\rho ={d}_{2}}=0$ in the TISE in Equation (17a)) at which the ${\left(JWKB\right)}_{1}$ method typically fails.
- (ii)
- Similarly, since ${\left(JWKB\right)}_{1}$ fails also in the CIR, ${d}_{2}$ should not be chosen in the CIR, either.
- (iii)
- Numerical values of either ${\alpha}_{2}\left(c\right)$ or ${\beta}_{2}\left(c\right)$ chosen in (18) should not diverge to infinity for all c in the domain of $f(c,\rho )$.
- (iv)

#### 3.2. Asymptotic Matching of the ${\left(BDE\right)}_{1}$ in the Normal Form

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

- Modification of the exact general solution in (22) (which will be used to compare with the modified ${\left(JWKB\right)}_{1}$ solutions):$${y}^{(m.)}(c,\rho )=\left\{\begin{array}{c}y,-\infty <\rho \le c\phantom{\rule{3.33333pt}{0ex}}\left(CAR\right)\hfill \\ {c}_{1}\left(c\right){J}_{1}({e}^{\frac{c-\rho}{2}}),c\le \rho <\infty \phantom{\rule{3.33333pt}{0ex}}\left(CIR\right)\hfill \end{array}\right..$$Note that both exact and ${\left(JWKB\right)}_{1}$ solutions in (22) and (31b) of (30) (or (33) of (20)) have the common form of exponentially-increasing terms in the CIR, $c<\rho <\infty $, where a cancellation according to (41a), (41b) and (42) is required for both exact and ${\left(JWKB\right)}_{1}$ solutions (see also (47a) below for a comparison).
- Modification of the ${\left(JWKB\right)}_{1}$ general solution in (30):$${\tilde{y}}^{(m.)}(c,\rho )=\left\{\begin{array}{c}{\tilde{y}}_{L}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}-\infty <\rho \le c\phantom{\rule{3.33333pt}{0ex}}\left(CAR\right)\\ {\tilde{y}}_{R}^{(m.)}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}c\le \rho <\infty \phantom{\rule{3.33333pt}{0ex}}\left(CIR\right)\end{array}\right.,$$$${\tilde{y}}_{R}^{(m.)}(c,\rho )=\frac{A\left(c\right)}{2\sqrt{\kappa (c,\rho )}}cos\left[\alpha \left(c\right)-\pi /4\right]Exp\left[-\zeta (c,\rho )\right].$$
- Modification of the ${\left(JWKB\right)}_{1}$ general solution in the other form (see Equation (20) via (33)):$${\tilde{y}}^{(m.)}(c,\rho )={\tilde{c}}_{1}\left(c\right){\tilde{y}}_{1}(c,\rho )+{\tilde{c}}_{2}\left(c\right){\tilde{y}}_{2}^{(m.)}(c,\rho ),$$$${\tilde{y}}_{2}^{(m.)}\phantom{\rule{4.pt}{0ex}}(c,\rho )=\left\{\begin{array}{c}\frac{1}{\sqrt{k(c,\rho )}}sin[-\eta (c,\rho )+\pi /4]\phantom{\rule{3.33333pt}{0ex}},-\infty <\rho \le c\phantom{\rule{3.33333pt}{0ex}}\left(CAR\right)\hfill \\ 0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{4.pt}{0ex}},\phantom{\rule{4.pt}{0ex}}c\le \rho <\infty \phantom{\rule{3.33333pt}{0ex}}\left(CIR\right)\hfill \end{array}\right..$$

## 4. Calculations in the Standard Form

#### 4.1. Exact and ${\left(JWKB\right)}_{1}$ Solutions of the ${\left(BDE\right)}_{1}$ in the Standard Form

- $S{D}_{1}:0<x<<0.554858$ (appropriate for the ${\left(JWKB\right)}_{1}$) and
- $S{D}_{2}:0.554858<x<1$ (inappropriate for the ${\left(JWKB\right)}_{1}$)

#### 4.2. Asymptotic Matching of the ${\left(BDE\right)}_{1}$ in the Standard Form

**Remark**

**2.**

## 5. A Physical Application: Exponential Potential Decorated Bound State Problem

## 6. Conclusions

## Conflicts of Interest

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**Figure 1.**Graph of f and g functions for some c values (solid red curve: $c=0$; dotted green: $c=1$; dashed blue: $c=2$).

**Figure 2.**Results of the normal form ${\left(JWKB\right)}_{1}$ solutions: 3D graphs of: (

**a**) ${y}_{EX}(c,\rho )$; (

**b**) $-{y}_{EX}(c,\rho )$; (

**c**) ${y}_{JWKB}(c,\rho )$; (

**d**) $-{y}_{JWKB}(c,\rho )$; (

**e**) $\Delta y(c,\rho )={y}_{EX}(c,\rho )-{y}_{JWKB}(c,\rho )$; (

**f**) $-\Delta y(c,\rho )=-[{y}_{EX}(c,\rho )-{y}_{JWKB}(c,\rho )]$.

**Figure 3.**Graphs of ${\tilde{S}}_{i,j=1}$ (left column) and ${\tilde{S}}_{i,j=2}$ (right column) (where$\phantom{\rule{3.33333pt}{0ex}}i=0,1,2$) in the normal form in $(c,\rho )$ for some specific c values (solid-red curves: $for$ $i=0$; dotted-green curves: $for\phantom{\rule{3.33333pt}{0ex}}i=1$; and dashed-blue curves: $for\phantom{\rule{3.33333pt}{0ex}}i=2$).

**Figure 4.**3D error graphs of asymptotically-modified normal form ${\left(JWKB\right)}_{1}$ general solutions: (

**a**) $\Delta {y}^{m.}(c,\rho )={y}_{EX}^{m.}(c,\rho )-{y}_{JWKB}^{m.}(c,\rho )$; (

**b**) $-\Delta {y}^{m.}(c,\rho )=-[{y}_{EX}^{m.}(c,\rho )-{y}_{JWKB}^{m.}(c,\rho )]$.

**Figure 5.**Results of the standard form ${\left(JWKB\right)}_{1}$ solutions: 3D graphs of: (

**a**) ${y}_{EX}(c,x)$; (

**b**) ${y}_{JWKB}(c,x)$; (

**c**) $\Delta y(c,x)={y}_{EX}(c,x)-{y}_{JWKB}(c,x)$; (

**d**) $-\Delta y(c,x)=-[{y}_{EX}(c,x)-{y}_{JWKB}(c,x)]$.

**Figure 6.**Graphs of ${\tilde{\tilde{S}}}_{i,j=1}$ (left column) and ${\tilde{\tilde{S}}}_{i,j=2}$ (right column) (where$\phantom{\rule{3.33333pt}{0ex}}i=0,1,2$) in the standard form in $(c,x)$ for a specific c value, $c=1$, (solid-red curves: $for$ $i=0$; dotted-green curves: $for\phantom{\rule{3.33333pt}{0ex}}i=1$; and dashed-blue curves: $for\phantom{\rule{3.33333pt}{0ex}}i=2$).

**Figure 7.**Enhancement in the standard form ${\left(JWKB\right)}_{1}$ solution by our asymptotic modifications in the $S{D}_{1}$ for $c=1$.

**Figure 8.**Graph of f and g functions of the exponential potential decorated bound state problem for some c values (solid-red curves: $c=0$; dotted-green curves: $c=1$; dashed-blue curves: $c=2$). First row: normal form representation corresponding to the Time Independent Schrodinger’s Equation (TISE); second row: standard form representation corresponding to the ${\left(BDE\right)}_{1}$.

**Figure 9.**Graphs of ${\tilde{S}}_{i,j=1}$ (left column) and ${\tilde{S}}_{i,j=2}$ (right column) (where $\phantom{\rule{3.33333pt}{0ex}}i=0,1,2$) of the exponential potential decorated bound state problem (which corresponds to the normal form representation via the TISE with the related exponential potential) under study in the normal form representation in $(c,\rho )$ for some specific c values (solid-red curves: $for$ $i=0$; dotted-green curves: $for\phantom{\rule{3.33333pt}{0ex}}i=1$; and dashed-blue curves: $for\phantom{\rule{3.33333pt}{0ex}}i=2$).

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**MDPI and ACS Style**

Deniz, C.
A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation. *Math. Comput. Appl.* **2016**, *21*, 41.
https://doi.org/10.3390/mca21040041

**AMA Style**

Deniz C.
A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation. *Mathematical and Computational Applications*. 2016; 21(4):41.
https://doi.org/10.3390/mca21040041

**Chicago/Turabian Style**

Deniz, Coşkun.
2016. "A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation" *Mathematical and Computational Applications* 21, no. 4: 41.
https://doi.org/10.3390/mca21040041