Geometric scattering in the presence of line defects

Abstract

A nonrelativistic scalar particle moving on a curved surface undergoes a geometric scattering whose behavior is sensitive to the theoretically ambiguous values of the intrinsic and extrinsic curvature coefficients entering the expression for the quantum Hamiltonian operator. This suggests using the scattering data to settle the ambiguity in the definition of the Hamiltonian. It has recently been shown that the inclusion of point defects on the surface enhances the geometric scattering effects. We perform a detailed study of the geometric scattering phenomenon in the presence of line defects for the case that the particle is confined to move on a Gaussian bump and the defect(s) are modeled by delta function potentials supported on a line or a set of parallel lines normal to the scattering axis. In contrast to a surface having point defects, the scattering phenomenon associated with this system is generically geometric in nature in the sense that for a flat surface, the scattering amplitude vanishes for all scattering angles $\theta$ except $\theta =\theta _0$ and $\pi -\theta _0$, where $\theta _0$ is the angle of incidence. We show that the presence of the line defects amplifies the geometric scattering due to the Gaussian bump. This amplification effect is particularly strong when the center of the bump is placed between two line defects.

Appendices

Appendix A: Derivation of Eq. (35)

The scattering solution (33) of the Schrödinger equation (22) has the form

$$\begin{aligned} \psi _0(\mathbf {x})=\psi _{\mathrm{inc}}(\mathbf {x})+\psi _{\mathrm{scatt}}(\mathbf {x}), \end{aligned}$$

(60)

where $\psi _{\mathrm{inc}}(\mathbf {x})$ and $\psi _{\mathrm{scatt}}(\mathbf {x})$ respectively represent the incident and scattered waves and have the form $\psi _{\mathrm{inc}}(\mathbf {x}){:}{=}\hbox {e}^{i\mathbf {k}\cdot \mathbf {x}}/2\pi$ and

$$\begin{aligned} \psi _{\mathrm{scatt}}(\mathbf {x}){:}{=}\frac{-i}{2\pi } \sum _{m,n=1}^N \hbox {e}^{i k_x a_m}A^{-1}_{mn}\, \hbox {e}^{i(k_x |x-a_n|+k_y y)}. \end{aligned}$$

(61)

In view of (8) and (60),

$$\begin{aligned} \psi _{\mathrm{scatt}}(\mathbf {x})\rightarrow \frac{\hbox {e}^{ik r}}{2\pi \sqrt{r}}\,\mathfrak {f}_0(\mathbf {k}',\mathbf {k})~~~\mathrm{for}~~~r\rightarrow \infty . \end{aligned}$$

(62)

This shows that in order to compute the scattering amplitude, we should determine the large-r behavior of the right-hand side of (61). First, we express this equation as

$$\begin{aligned} \psi _{\mathrm{scatt}}(\mathbf {x})=\frac{\hbox {e}^{ik_y y}}{2 \pi } \sum _{n=1}^N \Big [ \mathfrak {t}^+_{n}\ \Theta (x-a_n)\hbox {e}^{ik_x x} + \mathfrak {t}^-_{n}\ \Theta (a_n-x)\hbox {e}^{-ik_x x}\Big ], \end{aligned}$$

(63)

where

$$\begin{aligned} \mathfrak {t}^\pm _{n}{:}{=}-i\sum _{m=1}^N A^{-1}_{mn} \ \hbox {e}^{i k_x (a_m\mp a_n)}, \quad \Theta (x){:}{=}\left\{ \begin{array}{ll} 0 &{}\quad \mathrm{for}\, x<0,\\ 1 &{}\quad \mathrm{for}\, x\ge 0.\end{array}\right. \end{aligned}$$

(64)

For the scattering setup we consider, the source of the incident way lies at $x=-\infty$. This implies that $k_x>0$ and the incidence angle $\theta _0$ takes values in the interval $(-\frac{\pi }{2},\frac{\pi }{2})$. Because the angular position of the detector is arbitrary, we take the scattering angle $\theta$ to range over the interval $[-\frac{\pi }{2},\frac{3\pi }{2})$. Next, we introduce the notation:

$$\begin{aligned}&\theta ^+{:}{=}\theta ~~~~~~~~~\mathrm{for}~~~\theta \in \left( -\frac{\pi }{2},\frac{\pi }{2}\right) ,\\&\theta ^{-}{:}{=}\pi - \theta ~~~\mathrm{for}~~~\theta \in \left( \frac{\pi }{2},\frac{3\pi }{2}\right) , \end{aligned}$$

and employ the analysis presented in Appendix A of Ref. [28] to establish the identity:

$$\begin{aligned} \hbox {e}^{ik_y y}\hbox {e}^{\pm ik_x x} \rightarrow \sqrt{\frac{2\pi }{k r}} \ \Big [\hbox {e}^{i(kr-\frac{\pi }{4})}\delta (\theta _0-\theta ^{\pm })+\hbox {e}^{-i(kr-\frac{\pi }{4})} \delta (\theta _0-\theta ^{\pm }+ \pi )\Big ]~~\mathrm{as}~~r\rightarrow \infty . \end{aligned}$$

(65)

With the help of this relation and Eqs. (63) and (64), we obtain (62) with $\mathfrak {f}_0(\mathbf {k}',\mathbf {k})$ given by (35).

Appendix B: Formulas for $I_{mn}, J_{mn}$, and $I_{mm'nn'}$

The following are the formulas we have obtained for $I_{mn}, J_{mn}$, and $I_{mm'nn'}$ by performing the integrals in (46)–(48). Here, $\alpha _n{:}{=}a_n/\sigma$, and $\text {Erf}[x]$ and $\text {Erfc}[x]$ are, respectively, the error and complementary error functions.²

$$\begin{aligned} I_{mn}=&\frac{1}{8} \eta \hbox {e}^{-s \mathfrak {K}(-i {\alpha _m}+i \alpha _n+s \mathfrak {K})} \Bigg \{\sqrt{\pi } s \mathfrak {K}\hbox {e}^{(s \mathfrak {K}+i \alpha _n)^2} \\&\Big [-2 i \alpha _n^2 \lambda _2-2 \alpha _n (\lambda _2-2) s \mathfrak {K}+i \left( 8 \lambda _1+\lambda _2-2 \lambda _2 s^2 \mathfrak {K}^2\right) \Big ]\\&+2 \pi \text {Erf}[\alpha _n-i s \mathfrak {K}] \Big [2 \lambda _2+\lambda _2 s^4 \mathfrak {K}^4+\mathfrak {K}^2 (4 \lambda _1 s^2-1)\Big ]\\&-2 \pi \text {Erfc}[\alpha _n] (\mathfrak {K}^2-2 \lambda _2) \hbox {e}^{s \mathfrak {K}(s \mathfrak {K}+2 i \alpha _n)}\\&+2 \pi \Big [2 \lambda _2+\lambda _2 s^4 \mathfrak {K}^4+\mathfrak {K}^2 (4 \lambda _1 s^2-1)\Big ]\Bigg \}+\mathcal {O}(\eta ^2),\\ J_{mn}=&\frac{1}{8} \eta \hbox {e}^{i {\alpha _m} s \mathfrak {K}} \sqrt{\pi } \Bigg \{2 \hbox {e}^{-s \mathfrak {K}(i {\alpha _n}+s \mathfrak {K})} \sqrt{\pi } \left[ \mathfrak {K}^2 \left( -1+4 s^2 \lambda _1\right) +2 \lambda _2+s^4 \mathfrak {K}^4 \lambda _2\right] \text {Erfc}[{\alpha _n}-i s \mathfrak {K}]\\&-2 \hbox {e}^{i {\alpha _n} s \mathfrak {K}} \sqrt{\pi } (\mathfrak {K}^2-2 \lambda _2) \text {Erf}[{\alpha _n}]+\hbox {e}^{-{\alpha _n} ({\alpha _n}-i s \mathfrak {K})}\\&\Big [(-2 \hbox {e}^{{\alpha _n}^2} \sqrt{\pi } (\mathfrak {K}^2-2 \lambda 2)-i s \mathfrak {K}(-4+8 \lambda _1\\&-2 i {\alpha _n} s \mathfrak {K}(-2+\lambda _2)+\lambda _2+2 s^2 \mathfrak {K}^2 \lambda _2-2 {\alpha _n}^2 (4+\lambda _2))\Big ]\Bigg \}+\mathcal {O}(\eta ^2),\\ I_{mm'nn'}=&g_{mm'nn'}+\Theta (m-n)h_{mm'nn'}+\Theta (n-m)k_{mm'nn'}+l_{mm'nn'}, \end{aligned}$$

where

$$\begin{aligned} g_{mm'nn'}&{:}{=}\left\{ \begin{array}{lll} q_{mm'nn'}&{}\text {for} &{} m=n,\\ s_{mm'nn'}&{}\text {for} &{} m > n,\\ t_{mm'nn'}&{}\text {for}&{} m<n, \\ \end{array}\right. \\ q_{mm'nn'}&{:}{=}\frac{1}{4} \sqrt{\pi } \eta s \mathfrak {K}[\hbox {e}^{i s \mathfrak {K}(\alpha _m'+{\alpha _n'})}] \hbox {e}^{-\alpha _m^2-i s \mathfrak {K}(\alpha _m-{\alpha _n})-{\alpha _n}^2} \\&\Bigg [\hbox {e}^{\alpha _m^2} \left( \sqrt{\pi } \hbox {e}^{{\alpha _n}^2} s \mathfrak {K}(\text {Erfc}[{\alpha _n})-2]-2 i {\alpha _n}^2+2 {\alpha _n} s \mathfrak {K}+i\right) \\&-i \hbox {e}^{{\alpha _n}^2+2 i s \mathfrak {K}(\alpha _m-{\alpha _n})} \left( -i s \mathfrak {K}\left( \sqrt{\pi } \hbox {e}^{\alpha _m^2} \text {Erfc}[\alpha _m]+2 \alpha _m\right) +2 \alpha _m^2-1\right) \Bigg ]+\mathcal {O}(\eta ^2),\\ s_{mm'nn'}&{:}{=}\frac{1}{4} \sqrt{\pi } \eta s \mathfrak {K}[\hbox {e}^{i s \mathfrak {K}(\alpha _m'+{\alpha _n'})}]\hbox {e}^{-\alpha _m^2-s \mathfrak {K}(s \mathfrak {K}+i ({\alpha _m}+{\alpha _n}))-{\alpha _n}^2} \\&{\Bigg [\hbox {e}^{{\alpha _n}^2}} s \mathfrak {K}\Bigg (\sqrt{\pi } \hbox {e}^{{\alpha _m}^2} \Big (\text {Erfc}[{\alpha _m}-i s \mathfrak {K}]+\text {Erf}[{\alpha _n}-i s \mathfrak {K}]\\&+(\text {Erfc}[{\alpha _n}]-2) \hbox {e}^{s \mathfrak {K}(s \mathfrak {K}+2 i {\alpha _n})}-1\Big )-(\sqrt{\pi } \hbox {e}^{{\alpha _m}^2} \text {Erfc}[{\alpha _m}]+2 {\alpha _m}) \hbox {e}^{s \mathfrak {K}(s \mathfrak {K}+2 i {\alpha _m})}\Bigg )\\&+2 (-2 i {\alpha _n}^2+{\alpha _n} s \mathfrak {K}+i) \hbox {e}^{{\alpha _m}^2+s \mathfrak {K}(s \mathfrak {K}+2 i {\alpha _n})}\Bigg ]+\mathcal {O}(\eta ^2),\\ t_{mm'nn'}&{:}{=}\frac{1}{4} \sqrt{\pi } \eta s \mathfrak {K}\hbox {e}^{i s \mathfrak {K}(\alpha _m'+{\alpha _n'})} \hbox {e}^{-{\alpha _m}^2-s \mathfrak {K}(s \mathfrak {K}+i ({\alpha _m}+{\alpha _n}))-{\alpha _n}^2}\\&\Bigg [s \mathfrak {K}\left( \sqrt{\pi } \hbox {e}^{{\alpha _m}^2} (\text {Erfc}[{\alpha _m}]-2)+2 {\alpha _m}\right) \hbox {e}^{{\alpha _n}^2+s \mathfrak {K}(s \mathfrak {K}+2 i {\alpha _n})}\\&+\hbox {e}^{{\alpha _m} ({\alpha _m}+2 i s \mathfrak {K})} \Bigg (\sqrt{\pi } s \mathfrak {K}\hbox {e}^{{\alpha _n} ({\alpha _n}+2 i s \mathfrak {K})} (\text {Erf}[{\alpha _m}+i s \mathfrak {K}]-\text {Erf}[{\alpha _n}+i s \mathfrak {K}])\\&-\hbox {e}^{s^2 \mathfrak {K}^2} \Big (\sqrt{\pi } \hbox {e}^{{\alpha _n}^2} s \mathfrak {K}\text {Erfc}[{\alpha _n}]+4 i {\alpha _n}^2+2 {\alpha _n} s \mathfrak {K}-2 i\Big )\Bigg )\Bigg ]+\mathcal {O}(\eta ^2),\\ h_{mm'nn'}&{:}{=}\frac{1}{16} \eta \hbox {e}^{i s \mathfrak {K}(\alpha _m'+{\alpha _n'})}\Bigg \{12 \pi \eta (\text {Erf}[{\alpha _n}]-\text {Erf}[{\alpha _m}]) \left( 2 \lambda _2+\left( s^2-1\right) \mathfrak {K}^2\right) \hbox {e}^{i s \mathfrak {K}({\alpha _m}-{\alpha _n})}\\&+4 \pi \eta \hbox {e}^{-i s \mathfrak {K}({\alpha _m}+{\alpha _n}-i s \mathfrak {K})} \text {Erf}[{\alpha _m}-i s \mathfrak {K}] \\&\left( 4 \lambda _1+6 \lambda _2+3 \lambda _2 s^4 \mathfrak {K}^4+\mathfrak {K}^2 \left( (4 \lambda _1+3) s^2-3\right) \right) \\&-4\pi \eta \hbox {e}^{-s \mathfrak {K}(s \mathfrak {K}+i ({\alpha _m}+{\alpha _n}))} \text {Erf}[{\alpha _n}-i s \mathfrak {K}] \\&\left( 4 \lambda _1+6 \lambda _2+3 \lambda _2 s^4 \mathfrak {K}^4+\mathfrak {K}^2 \left( (4 \lambda _1+3) s^2-3\right) \right) \\&+ \sqrt{\pi } \hbox {e}^{-{\alpha _m}^2-{\alpha _n}^2} \Bigg [2 \hbox {e}^{-i s \mathfrak {K}({\alpha _m}+{\alpha _n})} \\&\Bigg (i \hbox {e}^{{\alpha _m}^2+2 i {\alpha _n} s \mathfrak {K}} \Big (6 {\alpha _n}^2 (\lambda _2+8) s \mathfrak {K}+i {\alpha _n} \left( 8 \lambda _1+9 \lambda _2+6 \lambda _2 s^2 \mathfrak {K}^2\right) \\&-s \mathfrak {K}\left( 8 \lambda _1+3 \lambda _2+6 \lambda _2 s^2 \mathfrak {K}^2\right) \Big )\\&+\hbox {e}^{2 i {\alpha _m} s \mathfrak {K}} \Big (3 \hbox {e}^{{\alpha _m}^2} {\alpha _n} (8 \lambda _1+3 \lambda _2)+\hbox {e}^{{\alpha _n}^2} \big (3 \lambda _2 s \mathfrak {K}(-2 i {\alpha _m}^2\\&+2 {\alpha _m} s \mathfrak {K}+2 i s^2 \mathfrak {K}^2+i)-16 {\alpha _m} \lambda _1+8 i \lambda _1 s \mathfrak {K}\big )\Big )\Bigg )\\&-24 i \hbox {e}^{{\alpha _m}^2} {\alpha _n}^3 \lambda _2 \sin (s \mathfrak {K}({\alpha _m}-{\alpha _n}))\Bigg ]\Bigg \}+\mathcal {O}(\eta ^2),\\ k_{mm'nn'}&{:}{=}\frac{1}{8} \sqrt{\pi } \eta \hbox {e}^{i s \mathfrak {K}(\alpha _m'+{\alpha _n'})} \Bigg \{\hbox {e}^{-{\alpha _m}^2-{\alpha _n}^2-i ({\alpha _m}-{\alpha _n}) s \mathfrak {K}} \Big (-{\alpha _m} \hbox {e}^{{\alpha _n}^2} \left( -8 \lambda _1+\left( -3+2 {\alpha _m}^2\right) \lambda _2\right) \\&+{\alpha _n} \hbox {e}^{{\alpha _m}^2} \left( -8 \lambda _1+\left( -3+2 {\alpha _n}^2\right) \lambda _2\right) \Big )\\&+ \hbox {e}^{-{\alpha _m}^2-{\alpha _n}^2-i ({\alpha _m}+{\alpha _n}) s \mathfrak {K}} \Big [\hbox {e}^{{\alpha _m} ({\alpha _m}+2 i s \mathfrak {K})} \Big (8 {\alpha _n} \lambda _1-2 {\alpha _n}^3 \lambda _2\\&+{\alpha _n} \left( 3+2 s^2 \mathfrak {K}^2\right) \lambda _2+2 i {\alpha _n}^2 s \mathfrak {K}(8+\lambda _2)-i s \mathfrak {K}\left( 8 \lambda _1+\lambda _2+2 s^2 \mathfrak {K}^2 \lambda _2\right) \Big )\\&+\hbox {e}^{{\alpha _n} ({\alpha _n}+2 i s \mathfrak {K})} \Big (2 {\alpha _m}^3 \lambda _2-2 i {\alpha _m}^2 s \mathfrak {K}\lambda _2+i s \mathfrak {K}\left( 8 \lambda _1+\lambda _2+2 s^2 \mathfrak {K}^2 \lambda _2\right) \\&-{\alpha _m} \left( 8 \lambda _1+\left( 3+2 s^2 \mathfrak {K}^2\right) \lambda _2\right) \Big )\Big ]\\&+2 \hbox {e}^{-i ({\alpha _m}-{\alpha _n}) s \mathfrak {K}} \sqrt{\pi } \left( \left( s^2-1\right) \mathfrak {K}^2+2 \lambda _2\right) \text {Erf}[{\alpha _m}]\\&-2 \hbox {e}^{-i ({\alpha _m}-{\alpha _n}) s \mathfrak {K}} \sqrt{\pi } \left( \left( s^2-1\right) \mathfrak {K}^2+2 \lambda _2\right) \text {Erf}[{\alpha _n}]\\&-2 \hbox {e}^{i s \mathfrak {K}({\alpha _m}+{\alpha _n}+i s \mathfrak {K})} \sqrt{\pi } \left( x^2 \left( -1+s^2 (1+4 \lambda _1)\right) +2 \lambda _2+s^4 \mathfrak {K}^4 \lambda _2\right) \text {Erf}[{\alpha _m}+i s \mathfrak {K}]\\&+ 2 \hbox {e}^{i s \mathfrak {K}({\alpha _m}+{\alpha _n}+i s \mathfrak {K})} \sqrt{\pi } \left( \mathfrak {K}^2 \left( -1+s^2 (1+4 \lambda _1)\right) +2 \lambda _2+s^4 \mathfrak {K}^4 \lambda _2\right) \text {Erf}[{\alpha _n}+i s \mathfrak {K}]\Bigg \}\\&\quad +\mathcal {O}(\eta ^2),\\ l_{mm'nn'}&{:}{=}\frac{1}{8}\eta \hbox {e}^{i s \mathfrak {K}(\alpha _m'+{\alpha _n'})} \hbox {e}^{-{\alpha _n}^2+i s \mathfrak {K}({\alpha _n}-{\alpha _m})} \\&\Bigg [2 \pi \hbox {e}^{{\alpha _n}^2} \left( 2 \lambda _2+\left( s^2-1\right) \mathfrak {K}^2\right) \left( \text {Erf}[{\alpha _n}]+\text {Erfc}[{\alpha _n}] \hbox {e}^{2 i s \mathfrak {K}({\alpha _m}-{\alpha _n})}\right) \\&+\sqrt{\pi } \Bigg ({\alpha _n} \left( \left( 2 {\alpha _n}^2-3\right) \lambda _2-8 \lambda _1\right) \hbox {e}^{2 i s \mathfrak {K}({\alpha _m}-{\alpha _n})}\\&+2 \sqrt{\pi } \hbox {e}^{{\alpha _n}^2} \left( 2 \lambda _2+\left( s^2-1\right) \mathfrak {K}^2\right) \\&-2 {\alpha _n}^3 \lambda _2+8 {\alpha _n} \lambda _1+3 {\alpha _n} \lambda _2\Bigg )\Bigg ]+\mathcal {O}(\eta ^2). \end{aligned}$$