# Quantum Scattering of Spinless Particles in Riemannian Manifolds

Lars Meschede, Benjamin Schwager, Dominik Schulz, Jamal Berakdar  Institut für
Physik, Martin-Luther-Universität Halle-Wittenberg

(August 28, 2024)

###### Abstract

Quantum mechanics is sensitive to the geometry of the underlying space. Here,
we present a framework for quantum scattering of a non-relativistic particle
confined to a two-dimensional space. When the motion manifold hosts localized
curvature modulations, scattering occurs from an emergent geometric potential
and the metric tensor field. Analytical and full numerical simulations
identify the geometric potential as the primary source for low-energy
scattering, while the metric tensor field of the curved space governs high-
energy diffraction. Compared to flat spaces, important differences in the
validity range of perturbation approaches are found and demonstrated by full
numerical simulations using combined finite element and boundary element
methods. As an illustration, we consider a Gaussian-shaped dent leading to
effects known as gravitational lensing. Experimentally, the considered setup
is realizable based on geometrically engineered 2D materials.

##  I Introduction

Quantum dynamics under constrained geometries can be approached with different
methods. One way is to quantize the constrained classical system, as has been
put forward by DeWitt [1, 2], followed by numerous studies (for a review of
the development until 1980, we refer to [3]). Possible ambiguities in this
approach were addressed in Refs. [4, 5]. Another way is to model the
constrained quantum dynamics by embedding the constraint in higher-dimensional
space which results in a confining potential for the particles [6, 7]. This
confining potential approach (CPA) entails the use of local coordinate frames
attached to the manifold. It leads (for sufficiently strong and homogeneous
confinement [8, 9]) to a well-defined decoupling of the motion tangent from
that perpendicular to the constraint. The resulting effective tangent
Schrödinger equation includes several geometry-induced terms that originate
from the embedding procedure [10]. The most prominent one is a scalar
potential known as the geometric potential.

Analytical and numerical studies revealed several physical phenomena
influenced by the geometry of the manifold of motion. For instance, the
geometric potential of bumped surfaces induces bound states relevant to
quantum dots [11, 12]. Also, electronic transport is sensitive to space
geometry [13]. A further finding is the modification of waveguide dispersion
of surface plasmon polaritons on metallic wires by geometry-induced momenta
[14]. The dynamics of strain-driven nanostructures may depend strongly on the
geometric potential, as well [15]. A further study [16] shows how the physics
of the Tomonaga-Luttinger liquids is altered by the curvature of space. The
coupling of constrained charged particles to electromagnetic fields was
addressed in Refs. [17, 18, *Jensen2010] and novel spin-orbit coupling
mechanisms were proposed [20, 21, 22, 23].

Experimentally, Szameit et al. [24] investigated the analog system of a
photonic topological crystal and reported good agreement between the measured
propagation of light with theoretical simulations based on the effective
tangent Schrödinger equation. However, their experiments were reproduced only
by properly incorporating the constraining potential. Onoe et al. [25]
observed an alteration of the Luttinger exponent for curved molecules. Further
proposed experimental setups include semiconductor-heterostructures [26],
electrons floating over the surface of liquid helium droplets [27] or trapped
atomic gases (possibly in microgravity) [28, 29]. Especially binary scattering
processes of ultracold dipolar atomic gases trapped within flat two-
dimensional manifolds have already been studied in some detail, see [30, 31]
and references therein.

For investigating geometry-induced effects and testing the quality of the
constraining potential, Mostafazadeh et al. studied scattering from a
localized curvature modulation in a two-dimensional space [32, 33, 34]. In
this setting, the scattering amplitude is determined by the interplay of the
externally applied potential fields, the geometry-induced potential, and the
motion in the curved region with deviating metric. In addition, a Gaussian
dent was considered analytically using approximation methods that led to
closed expressions for the total scattering cross length (which is the two-
dimensional analog to the total scattering cross section encountered in three-
dimensional studies and should not be confused with the scattering length) and
the optical theorem. Following these ideas, Ref. [35] compared the classical,
quantum mechanical, and semi-classical regimes for a spherical protuberance of
a plane.

With this study we wish to extend these investigations beyond the perturbative
approaches. Numerical computations support our analysis for different
spectrally resolved scattering cross lengths in the elastic channel. The
results evidence that the geometric potential plays the dominant role at low
energies. For small amplitudes, it is well-captured by first-order
perturbation theory, in contrast to the metric tensor field of the curved
space whose influence becomes predominant at high energies. Our results
demonstrate how dent-like structures in space act as two-dimensional quantum
lenses.

The article is organized as follows: In Sec. II we formulate the scattering
problem on asymptotically flat two-dimensional Riemannian manifolds according
to the CPA and recapitulate and compare results obtained from both the
Lippmann-Schwinger equation and the partial wave analysis (PWA). Then, we
study a Gaussian dent in Sec. III and summarize our findings in Sec.IV.
Finally, derivations of expressions and further analysis of the induced
effects are deferred to appendices.

##  II Scattering Theory on a Riemannian Manifold

We consider the quasi-free motion of a nonrelativistic spinless particle whose
dynamics are constrained to a Riemannian two-dimensional manifold
$\left(\mathcal{M},g\right)$ isometrically embedded in the three-dimensional
Euclidean space as $\mathcal{E}:\mathcal{M}\rightarrow\mathbb{R}^{3}$. For
simplicity, we restrict ourselves to cases where $\mathcal{M}$ can be globally
parameterized by

$\displaystyle\begin{aligned} \mathcal{Y}:\mathbb{R}^{2}\supset
Q&\rightarrow\mathcal{M}\subset\mathbb{R}^{3}\ ,\\\
\textbf{q}=(q^{1},q^{2})^{T}&\mapsto\mathcal{Y}(\textbf{q})\ .\end{aligned}$
(1)

The Riemannian metric tensor field shall be the canonically induced one, and
the Levi-Civita connection is assumed to act on the tangent bundle
$T\mathcal{M}$.

As stated, there are different ways to formulate quantum mechanics under
constraints. The CPA [6, 7] is an approach arguably [7, 36, 8, 37] close to
physical reality. It starts by inserting (1) together with a normal
displacement, so that
$\mathcal{X}(\textbf{q},y)=\mathcal{Y}(\textbf{q})+y\,\textbf{n}$, into the
Schrödinger equation and hence pulls it back into the parameter space
$Q\subset\mathbb{R}^{2}$. The manifold is introduced to the dynamics by the
action of a constraining potential $V_{\lambda}$ that shall possess the
following properties in accordance with [36, 37, 38, 10] and references
therein: (i) $V_{\lambda}$ depends only on the normal displacement coordinate
$y$; (ii) $V_{\lambda}$ has a deep minimum on $\mathcal{M}$ (for $y=0$) so
that it can be expanded around $\mathcal{M}$; (iii) $V_{\lambda}$ preserves
the gauge group representing the subgroup of the isometry group. This
construction enables a mechanism of dimensional reduction. We note that
statement (ii) accounts for quantum fluctuations by securing the uncertainty
principle of Heisenberg. The CPA thus allows for a perturbative expansion
around $\mathcal{M}$ to separate the effective tangent motion from the normal
one. The effective tangent Schrödinger equation reads

$\displaystyle\left[-\dfrac{h^{2}}{2m_{0}}\,\nabla_{\mu}\nabla^{\mu}+V_{\mathrm{geo}}\,\hat{I}\right]\chi_{\mathrm{t}}(\textbf{q},t)=\mathrm{i}\hbar\,\chi_{\mathrm{t}}(\textbf{q},t)\
.$ (2)

The Laplace-Beltrami operator (calling $|g|\coloneqq|\mathrm{det}(g)|$) can be
expressed as

$\displaystyle\nabla_{\mu}\nabla^{\mu}=\dfrac{1}{\sqrt{|g|}}\partial_{\mu}\,\left(\sqrt{|g|}\,g^{\mu\nu}\,\partial_{\nu}\right)\
,$ (3)

and the geometry-induced potential field, in dependence of the Gaussian
curvature $K$ and the mean curvature $M$, has the form

$\displaystyle
V_{\mathrm{geo}}=-\dfrac{\hbar^{2}}{2m_{0}}\,\left(M^{2}-K\right)\ .$ (4)

This potential always acts attractive for the case given by (1) [37].

Equation (2) describes the equilibrium states of the particle that is
constrained to $(\mathcal{M},g)$. We proceed in assuming that the scattering
occurs from any localized geometric perturbation of a manifold $\mathcal{M}$
that can asymptotically be identified with the Euclidean plane,
$\mathcal{M}{\simeq}\mathbb{R}^{2}$, for
$\|\mathcal{Y}(\textbf{q})\|_{2}\rightarrow\infty$. As seen from the
asymptotic metric tensor field, called $\eta$ in what follows, the latter is
flat and possesses trivial topology, so we call $\mathcal{M}$ an
asymptotically flat manifold. This condition implies that $\mathcal{M}$
exhibits only localized deviations of the flat metric, and the curved region
can hence be approximately modeled as having compact support. Thus, the
effective Hamilton operator in (2) can be decomposed in the general form
$\hat{H}=\hat{H}^{0}+\hat{H}^{\mathrm{s}}$ with

$\displaystyle\hat{H}^{0}$
$\displaystyle=-\dfrac{\hbar^{2}}{2m_{0}}\,\dfrac{1}{\sqrt{|\eta|}}\partial_{\mu}\,\left(\sqrt{|\eta|}\,\eta^{\mu\nu}\partial_{\nu}\right)+V_{\mathrm{geo}}^{0}\,\hat{I}\
,$ (5) $\displaystyle\hat{H}^{\mathrm{s}}$
$\displaystyle=-\dfrac{\hbar^{2}}{2m_{0}}\,\biggl{[}\left(g^{\mu\nu}-\eta^{\mu\nu}\right)\partial_{\mu}\partial_{\nu}+\left(\dfrac{1}{\sqrt{|g|}}\partial_{\mu}\left(\sqrt{|g|}g^{\mu\nu}\right)-\dfrac{1}{\sqrt{|\eta|}}\partial_{\mu}\left(\sqrt{|\eta|}\,\eta^{\mu\nu}\right)\right)\partial_{\nu}\biggl{]}+\left[V_{\mathrm{geo}}-V_{\mathrm{geo}}^{0}\right]\,\hat{I}\
.$ (6)

The superscripts $0$ and $\mathrm{s}$ denote the asymptotic and scattered
part, respectively. Hence, the origin of the scattering processes are twofold
deviations between the asymptotic and scattering regions: The difference of
the respective geometry-induced potentials acts as a scalar scattering
potential. In addition to that, the disturbance in the metric tensor field
yields additional terms that can be regarded as geometry-induced tensor
potentials (of first and second rank). The contributions of both terms to the
scattering Hamilton operator $\hat{H}^{\mathrm{s}}$ are additive but their
effects on scattering observables cannot be disentangled in general. Due to
the presence of the derivative terms, the scattering processes investigated
here obtain a stronger k-dependence than those found in the familiar situation
of pure scalar potential scattering. It is this influence that we will
identify to be responsible for our observations. $\hat{H}^{0}$ governs the
asymptotically free particle according to the eigenvalue problem

$\displaystyle\hat{H}^{0}\chi_{\mathrm{t}}^{0}(\textbf{q})=E\,\chi_{\mathrm{t}}^{0}(\textbf{q})\
.$ (7)

In the following part, we will study Riemannian manifolds that are globally
parameterized in Monge form with

$\displaystyle\mathcal{Y}(\textbf{q})\coloneqq\left(q^{1},q^{2},f(q^{1},q^{2})\right)^{T}\
,$

referring to standard Cartesian coordinates. That means $Q=\mathbb{R}^{2}$ can
be identified so that $q^{1}=x$, $q^{2}=y$, and $z=f(x,y)$ with $f$ is assumed
to be a smooth, axially symmetric function with vanishing radial derivative at
its center located at a certain $\textbf{q}_{0}$. Furthermore, it is expected
to decay to a constant value so that

$\displaystyle f(\textbf{q})\simeq z_{0}\in\mathbb{R}\ ,\
\left(\|\textbf{q}\|_{2}\rightarrow\infty\right)\ ,$

identifying the resulting $\mathcal{M}$ as asymptotically flat. Then, using
the reparametrization with respect to polar coordinates, fulfilling the
relations $r=\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}$ and
$\mathrm{tan}(\varphi)=\frac{y-y_{0}}{x-x_{0}}$ so that $f=f(r)$, because of
the axial symmetry, we find the Riemannian metric tensor field

$\displaystyle\left(g_{\mu\nu}\right)=\left(\begin{matrix}1+(\partial_{r}f)^{2}&0\\\
0&r^{2}\end{matrix}\right)\ .$ (8)

From this, we deduce the Gaussian and mean curvatures,

$\displaystyle K=\dfrac{F\,\partial_{r}F}{r}\ ,\quad
M=\dfrac{1}{2}\,\left(\dfrac{F}{r}+\partial_{r}F\right)\ ,$ (9)

and, consequently, the geometric potential (4),

$\displaystyle
V_{\mathrm{geo}}=-\dfrac{\hbar^{2}}{8m_{0}}\,\left(\dfrac{F}{r}-\partial_{r}F\right)^{2}$
(10)

with the definition

$\displaystyle F\coloneqq\dfrac{\partial_{r}f}{\sqrt{1+(\partial_{r}f)^{2}}}\
.$ (11)

###  II.1 Effective Tangent Lippmann-Schwinger Equation

For studying scattering processes, we use the Lippmann-Schwinger equation
which reads

$\displaystyle|\chi_{\mathrm{t}}^{\pm}\rangle=|\chi_{\mathrm{t}}^{0}\rangle+\left[(E\pm\mathrm{i}\varepsilon)\,\hat{I}-\hat{H}^{0}\right]^{-1}\hat{H}^{\mathrm{s}}|\chi_{\mathrm{t}}^{\pm}\rangle$
(12)

for the state vectors ($+$ or $-$ correspond to outgoing or incoming wave
boundary conditions). In the representation parameterized by (1)
($\langle\textbf{q}|\equiv\langle\textbf{x}(\textbf{q})|$), we find for the
wave function

$\displaystyle\begin{aligned}
\langle\textbf{q}|\chi_{\mathrm{t}}^{\pm}\rangle&=\langle\textbf{q}|\chi_{\mathrm{t}}^{0}\rangle+\dfrac{2m_{0}}{\hbar^{2}}\,\int
G^{\pm}(\textbf{q},\textbf{q}^{\prime})\,\langle\textbf{q}^{\prime}|\hat{H}^{\mathrm{s}}|\chi_{\mathrm{t}}^{\pm}\rangle\
\mathrm{d}\sigma(\textbf{q}^{\prime})\\\
&=\langle\textbf{q}|\chi_{\mathrm{t}}^{0}\rangle+\langle\textbf{q}|\chi_{\mathrm{t}}^{\mathrm{s}\,\pm}\rangle\
,\end{aligned}$ (13)

where

$\displaystyle
G^{\pm}(\textbf{q},\textbf{q}^{\prime})=\dfrac{\hbar^{2}}{2m_{0}}\,\left\langle\textbf{q}\biggl{|}\left[(E\pm\mathrm{i}\varepsilon)\,\hat{I}-\hat{H}_{0}\right]^{-1}\biggl{|}\textbf{q}^{\prime}\right\rangle$
(14)

is a Green’s function for the asymptotic form of (2), i. e.,
$\hat{H}\simeq\hat{H}^{0}$.

Expression (13) describes the stationary spatial profile of the wave function
due to the scattering event. It consists of a superposition of the incident
wave and the scattered wave labeled by the superscripts $0$ and $\mathrm{s}$,
respectively. We identify

$\displaystyle\langle\textbf{q}|\chi_{\mathrm{t}}^{\mathrm{s}\,\pm}\rangle$
$\displaystyle=\dfrac{2m_{0}}{\hbar^{2}}\,\int
G^{\pm}(\textbf{q},\textbf{q}^{\prime})\,\langle\textbf{q}^{\prime}|\hat{H}^{\mathrm{s}}|\chi_{\mathrm{t}}^{\pm}\rangle\
\mathrm{d}\sigma(\textbf{q}^{\prime})$ $\displaystyle\sim
a(\textbf{k},\textbf{k}^{\mathrm{s}})\ ,$

where $a$ is the scattering amplitude between states characterized by the wave
vectors k and $\textbf{k}^{\mathrm{s}}$. In the case of asymptotically flat
manifolds, the asymptotic form of the effective tangent Schrödinger equation
(2) is the two-dimensional Helmholtz equation. This possesses plane wave
solutions so that
$\langle\textbf{q}|\chi_{\mathrm{t}}^{0}\rangle=\frac{1}{2\pi}\,\mathrm{e}^{\mathrm{i}\textbf{k}\cdot\textbf{q}}$.
From (13), we derive the wave profile incorporating the outgoing waves

$\displaystyle\langle\textbf{q}|\chi_{\mathrm{t}}^{+}\rangle\simeq\dfrac{1}{2\pi}\,\left(\mathrm{e}^{\mathrm{i}\textbf{k}\cdot\textbf{q}}+a(\textbf{k},\textbf{k}^{\mathrm{s}})\,\dfrac{\mathrm{e}^{\mathrm{i}kr}}{\sqrt{r}}\right)\
.$ (15)

The scattering amplitude is found to be

$\displaystyle
a(\textbf{k},\textbf{k}^{\mathrm{s}})=\dfrac{m_{0}\,\mathrm{e}^{-\mathrm{i}\frac{3\pi}{4}}}{\hbar^{2}}\,\sqrt{\dfrac{2\pi}{k}}\,\int\mathrm{e}^{-\mathrm{i}\textbf{k}^{\mathrm{s}}\cdot\textbf{q}^{\prime}}\,\langle\textbf{q}^{\prime}|\hat{H}^{\mathrm{s}}|\chi_{\mathrm{t}}^{+}\rangle\
\mathrm{d}\sigma(\textbf{q}^{\prime})$ (16)

which is obtained by inserting the asymptotic form of the Green’s function
[39] that is proportional to the Hankel function of the first kind [40],

$\displaystyle G(\textbf{q},\textbf{q}^{\prime})$
$\displaystyle=-\dfrac{\mathrm{i}}{4}\,H^{(1)}_{0}(k\,\|\textbf{q}-\textbf{q}^{\prime}\|_{2})$
$\displaystyle\simeq-\dfrac{\mathrm{i}}{4}\,\sqrt{\dfrac{2}{\pi
kr}}\,\mathrm{e}^{\mathrm{i}\,\left(kr+\textbf{k}^{\mathrm{s}}\cdot\textbf{q}^{\prime}-\frac{\pi}{4}\right)}\
,\ \left(r\rightarrow\infty\right)\ ,$

with $k=\|\textbf{k}\|_{2}$, $r=\|\textbf{q}\|_{2}$, and
$\textbf{k}^{\mathrm{s}}=k\,\textbf{e}_{r}$. Setting the incident direction
without loss of generality as the positive $x$-direction,
$\textbf{k}=k\,\textbf{e}_{x}$, we can express the scattering amplitude in
dependence of the scattering angle $\vartheta\in[0,\pi]$ between the incident
and scattered wave vectors k and $\textbf{k}^{\mathrm{s}}$. Thus, the
differential scattering cross length is given by
$\frac{\mathrm{d}\sigma}{\mathrm{d}\vartheta}\coloneqq|a(\vartheta)|^{2}$. The
total scattering cross length and the momentum transfer cross length are

$\displaystyle\sigma_{\mathrm{tot}}=\int\limits_{0}^{2\pi}|a(\vartheta)|^{2}\
\mathrm{d}\vartheta$ (17)

and

$\displaystyle\sigma_{\mathrm{M}}=\int\limits_{0}^{2\pi}\left(1-\mathrm{cos}(\vartheta)\right)\,|a(\vartheta)|^{2}\
\mathrm{d}\vartheta\ .$ (18)

Normalizing the partial cross length to the total one, we obtain a quantity
that can be interpreted as a probability density,

$\displaystyle
w(\vartheta)\coloneqq\dfrac{1}{\sigma_{\mathrm{tot}}}\,\dfrac{\mathrm{d}\sigma}{\mathrm{d}\vartheta}(\vartheta)\
,$ (19)

so that $\mathrm{d}p(\vartheta)=w(\vartheta)\,\mathrm{d}\vartheta$ describes
the probability of an incident wave being scattered into the infinitesimal
angular segment around a certain direction.

By inserting the results of the Monge parametrization (8) – (11), we obtain,
as in Ref. [33], an expression for the scattering amplitude in the first Born
approximation, $a(\textbf{k},\textbf{k}^{\mathrm{s}})\simeq
a^{(1)}(\textbf{k},\textbf{k}^{\mathrm{s}})$ (full details are in the
appendix),

$\displaystyle\begin{aligned}
a^{(1)}(\vartheta)=\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}\,\sqrt{\dfrac{\pi}{2k}}\,\int\limits_{0}^{\infty}r^{\prime}\,&\left[\left(F^{2}k^{2}\,\mathrm{sin}^{2}\left(\nicefrac{{\vartheta}}{{2}}\right)+\dfrac{1}{4}\,\left(\dfrac{F}{r^{\prime}}-\partial_{r^{\prime}}F\right)^{2}\right)\,J_{0}(2\,k\,r^{\prime}\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right))\right.\\\
&\hskip
8.53581pt\left.+\left(\dfrac{F^{2}k}{2\,r^{\prime}\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right)}+k\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right)\,F\,\partial_{r^{\prime}}F\right)\,J_{1}(2\,k\,r^{\prime}\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right))\right]\
\mathrm{d}r^{\prime}\ ,\end{aligned}$ (20)

where $J_{n}$ are Bessel functions of the first kind of order
$n\in\mathbb{N}_{0}$. We want to emphasize that the formula (20) is valid for
general axially symmetric asymptotically flat two-dimensional submanifolds of
Euclidean space $\mathbb{R}^{3}$. Its integrand consists of a sum of four
contributions. One of them (the second one in the first row) is given by the
geometric potential field (10), and the remaining terms depend directly on the
deformation via the quantity $F$. These other contributions confirm that the
non-flat metric acts as a scattering center itself, thus generalizing the
short-range potential scattering process.

To elucidate the influence of these terms on the first Born scattering
amplitude, we compare the familiar situation of short-ranged potential
scattering [41] in flat space where only an external potential may be present
as an additional term in the integrand of (20). There, we would have $F=0$ and
hence no geometric potential, leading to the vanishing $a^{(1)}\rightarrow 0$
as $k\rightarrow\infty$. Typically, the first Born approximation yields a
reliable description of this energy regime. For the scattering upon geometric
structures, however, it behaves differently because the wave has to travel
through a region with nontrivial metric. We can infer that the geometric
potential obtains a role similar to that of an external scattering potential.
Inspecting the spectrally resolved scattering amplitude, one deduces from the
explicit and implicit (in the argument of the Bessel functions) dependencies
on $k$ that the geometric potential dominates the process in the regime of
small incident energies, leading to the known divergent behavior of the total
scattering cross length, $\sigma_{\mathrm{tot}}^{(1)}\sim k^{-1}$, as it is
attractive [42]. In the limit of large energies, its influence becomes
marginal so that the remaining terms determine the scattering process in (20).

###  II.2 Effective Tangent Partial Wave Analysis

The PWA is a standard approach in scattering theory. We refer, for example, to
[41, 43] for flat two-dimensional spaces, i. e. $\mathcal{M}=\mathbb{R}^{2}$,
and to [44] for the three-dimensional analog. Performing it, using the ansatz
(15) for the wave profile, we express the scattering amplitude as (see
appendix)

$\displaystyle
a(\vartheta)=\mathrm{e}^{-\mathrm{i}\frac{\pi}{4}}\sqrt{\dfrac{2\pi}{k}}\,\sum\limits_{m=-\infty}^{+\infty}\left(S_{m}-1\right)\,\mathrm{e}^{\mathrm{i}m\vartheta}\
,$ (21)

where $S_{m}\coloneqq\mathrm{e}^{2\mathrm{i}\delta_{m}}$ is the S-matrix of
the corresponding orbital angular momentum channel in dependence of the
scattering phase shift $\delta_{m}$. With this expression, the cross lengths
(17) and (18) attain the forms

$\displaystyle\begin{aligned}
\sigma_{\mathrm{tot}}&=\sum\limits_{m=-\infty}^{+\infty}\sigma_{m}=\dfrac{1}{k}\,\sum\limits_{m=-\infty}^{+\infty}\left|S_{m}-1\right|^{2}\\\
&=\dfrac{4}{k}\,\sum\limits_{m=-\infty}^{+\infty}\mathrm{sin}^{2}(\delta_{m})\
,\end{aligned}$ (22)

where $\sigma_{m}$ is the partial scattering cross length of the respective
orbital angular momentum channel with $m\in\mathbb{Z}$, and

$\displaystyle\begin{aligned}
\sigma_{\mathrm{M}}&=\dfrac{1}{2k}\,\sum\limits_{m=-\infty}^{+\infty}\left|S_{m+1}-S_{m}\right|^{2}\\\
&=\dfrac{2}{k}\,\sum\limits_{m=-\infty}^{+\infty}\mathrm{sin}^{2}(\delta_{m+1}-\delta_{m})\
.\end{aligned}$ (23)

The relevant data can be derived from the K-matrix as this is the complex
Cayley-transform of the S-matrix,

$\displaystyle
K_{m}=\mathrm{tan}(\delta_{m})=\mathrm{i}\,\dfrac{1-S_{m}}{1+S_{m}}\ .$ (24)

For a given spatial profile of the wave function we use the relation

$\displaystyle
K_{m}=\dfrac{\dfrac{p_{m}(r_{1})}{p_{m}(r_{2})}\,J_{m}(kr_{2})-J_{m}(kr_{1})}{\dfrac{p_{m}(r_{1})}{p_{m}(r_{2})}\,Y_{m}(kr_{2})-Y_{m}(kr_{1})}\
,$ (25)

for computation, where $J_{m}$ and $Y_{m}$ denote the Bessel functions of the
first and second kind, respectively, and

$\displaystyle
p_{m}(r)\coloneqq\left\langle\chi_{\mathrm{t}}(r,\vartheta),\mathrm{e}^{\mathrm{i}m\vartheta}\right\rangle_{L^{2}(\mathbb{S}^{1})}$
(26)

indicates projection of the tangent wave function onto the angular part of the
cylindrical harmonics.

To evaluate these formulas, we solve the scattering problem employing a
combination of a finite element method and a boundary element method (FEM-BEM)
in two dimensions. The two components of our simulations depend on the FEniCSx
[45, 46, 47] library. After obtaining the wave function $\chi_{\mathrm{t}}$,
we compute the K-matrix inserting two arbitrary but fixed values for the
radii. From this, we determine both the scattering phase shift and the
S-matrix, either of which could be used to yield the cross lengths.

##  III Results for the Gaussian Dent

For illustration, we study a Gaussian dent described by

$\displaystyle f(r)\doteq f_{0}\,\mathrm{e}^{-\frac{r^{2}}{2\,\sigma^{2}}}\ ,$
(27)

with $f_{0}\in\mathbb{R}$ describing the peak amplitude and
$\sigma\in\mathbb{R}^{+}$ the standard deviation that act as measures for the
depth and width, respectively. Although this function does not have compact
support, it decays sufficiently fast so that we can cut it beyond an effective
range. Evaluating (11) gives

$\displaystyle
F=-\dfrac{f_{0}\,r\,\mathrm{e}^{-\frac{r^{2}}{2\,\sigma^{2}}}}{\sigma^{2}\,\sqrt{1+\left(\frac{f_{0}\,r}{\sigma^{2}}\right)^{2}\,\mathrm{e}^{-\frac{r^{2}}{\sigma^{2}}}}}\
,$ (28)

hence the geometric potential (4) reads

$\displaystyle
V_{\mathrm{geo}}=-\dfrac{\hbar^{2}}{8m_{0}}\cdot\dfrac{f_{0}^{2}\,r^{4}\,\left(f_{0}^{2}+\sigma^{2}\mathrm{e}^{\frac{r^{2}}{\sigma^{2}}}\right)^{2}}{\left(f_{0}^{2}\,r^{2}+\sigma^{4}\,\mathrm{e}^{\frac{{r}^{2}}{\sigma^{2}}}\right)^{3}}\
,$ (29)

which is axially symmetric, varies radially, and has a ring-shaped structure.
A more detailed discussion of the geometric quantities is given in the
appendix.

Using FEM-BEM, we solved the direct scattering problem as defined by (2)
numerically and computed the scattering data from the result. For comparison,
we repeated the simulation twice, neglecting either the deviation of the
metric or the geometric potential, which allows for estimating the individual
contributions to the geometric scattering process.

###  III.1 Discussion of the Scattering Data for Varying Geometric Parameters

\begin{overpic}[width=433.62pt]{figures/setup_1.png} \put(1.0,98.0){(a)}
\put(1.0,68.7){(d)} \put(1.0,42.0){(g)} \put(1.0,23.0){(j)}
\put(36.3,98.0){(b)} \put(36.3,68.7){(e)} \put(36.3,39.0){(h)}
\put(36.3,20.7){(k)} \put(95.0,98.0){(c)} \put(95.0,68.7){(f)}
\put(95.0,42.0){(i)} \put(95.0,23.0){(l)} \end{overpic} Figure 1: (a-c)
Directional scattering probability amplitude as function of the scattering
angle; (d-f) scattering phase shifts for the first few orbital angular
momentum channels; (g-i) partial scattering cross lengths in dependence of the
incident wavenumber; (j-l) total and momentum transfer scattering cross
lengths in dependence of the incident wavenumber for the direct scattering
problem upon a Gaussian dent. Overall, $\sigma=a_{0}/\sqrt{2}$ and
$f_{0}\in\\{0.5;1;2\\}\,a_{0}$ in the columns from left to right. A
characteristic length of the problem serves as the unit $a_{0}$.

Fig. 1 presents results for the scattering process. The two lower rows of fig.
1 (g)-(l) show the partial, total, and momentum transfer cross lengths
obtained from both the analytical approach using the first Born approximation
and our numerical simulations. The two upper rows in fig. 1 (a)-(f) show the
corresponding energy resolved directional scattering probability density $w$
in dependence of the scattering angle $\vartheta$ and the scattering phases
$\delta_{m}$ for all orbital angular momentum channels. Different amplitudes
$f_{0}\in\\{0.5;1;2\\}\,a_{0}$ of the dent are associated with the columns,
where $\sigma=a_{0}/\sqrt{2}$ is kept constant and a unit of
$a_{0}=10\,\mathrm{nm}$ was chosen. Higher amplitudes determine larger
variation in the metric and hence stronger curvature.

The integrated cross lengths evidence that localized geometric deviations
cause scattering processes. In order to understand them, at first, we can
compare the results for the spectrally resolved total cross length given by
the two approaches (compare in fig. 1 (j)-(l) the purple solid line (full
numerics) with the red dot-dashed line (analytical first Born approximation)).
Thereby, we recognize significant deviations, which are the stronger, the
greater the deformation of space is. Particularly at higher energies (roughly
for $k\gtrsim 1\,a_{0}^{-1}$), the first Born approximation appears to be not
sufficient to describe the simulation. Better agreement is found for low wave
vectors. According to (20) and the discussion of the significance of its
different contributions in the low- and high-energy regimes, we can firstly
conclude that the scattering process upon localized geometric structures is
indeed governed by the scalar geometric potential in the low-energy regime,
stating a familiar situation for which the first Born approximation yields a
reasonable description. Secondly, it expects leading contributions by the
metric tensor field for higher energies but does not capture these properly.
However, the known behavior is reproduced in the limiting case of flat spaces.
Based on the first Born approximation, scattering from localized geometric
structures was proposed as a possibility to study geometry-induced potential
fields [32, 33, 34].

At higher values of $k$, the numerical results reveal an oscillatory behavior
of the total cross length while the momentum transfer cross length (see the
green dashed line in fig. 1 (j)-(l)) converges. Interpreting these as measures
of the lateral size of the target, we infer that the momentum transfer cross
length delivers a systematically lower result than the total one. In addition,
both increase as the degree of deformation of space increases, which means
more scattering. That is, higher amplitudes $f_{0}$ of the dent and hence its
overall shape affects the quality of the scattering cross lengths as size
measures in the observed spectral range because the width parameter $\sigma$
was kept constant in all three cases. This behavior is in contrast to pure
scalar potential scattering and is caused by the tensor potentials.

Considering the scattering probability densities in figs. 1 (d)-(f), we notice
two scattering characteristics: while the pattern is isotropic for low
energies, anisotropic behavior with a pronounced forward peak is visible for
high ones. The partial cross lengths depicted in figs. 1 (g)-(i) confirm that
the low-energy process is governed by the $\mathrm{s}$-channel ($m=0$).
Indeed, this is the only one yielding a nonvanishing component of the total
scattering cross length and a nonzero scattering phase shift in the relevant
regime (see figs. 1 (a)-(c)). From the scattering phase shifts follows that
the number of involved scattering channels grows linearly with the incident
wavenumber $k$ and that channels with opposite quantum numbers of orbital
angular momentum contribute symmetrically (${S_{m}=S_{-m}\ \forall\
m\in\mathbb{Z}}$). Generally, the phases become more sensitive to the incident
wavenumber when the curvature of the target varies stronger.

As a consequence, an interference pattern emerges within the wave profile that
can also be observed in the angular dependence of the scattering probability
densities for higher incident wavenumbers. This effect can be understood from
a wave mechanical point of view (see below). Recalling that the geometric
potential is less critical for the regime of high incident energies, the
target is essentially reduced to a curved region through which the wave
travels freely. Then, there is a family of elementary waves that pass the
structure along continuously varying geodesic trajectories. As these possess
different path lengths, the respective representatives of the family
experience varying phase shifts while traversing the target, and their
recombination behind it leads to the interference effect. Therefore,
scattering of an incoming plane wave upon a Gaussian dent leads to
diffraction, and the substructure visible in the density plots figs. 1 (d)-(f)
shows the diffraction orders. Their number increases with the incident
wavenumber.

###  III.2 Influences of the Metric and the Geometric Potential

From (6) follows that scattering is due to both variation in the metric and
the induced geometric potential. Here, we study these two effects separately.
To this end, we use the parameters $f_{0}=a_{0}$ and $\sigma=a_{0}/\sqrt{2}$.

Fig. 2 shows the results when the geometric potential is not taken into
account. Thus, the projectile travels freely but through a curved region. In
the regime of high energies, the results of all plots match those of the above
full analysis (fig. 1), which is expected as the contribution of the geometric
potential vanishes there. For small energies ($k\lesssim a_{0}^{-1}$), the
disappearance of the attractive geometric potential leads to a vanishing of
all partial and integrated scattering cross lengths. Hence, we confirm that
the geometric potential drastically influences the low-energy scattering and
is responsible for the previously observed divergences of scattering cross
lengths.

Fig. 3 presents the results for scattering in flat space off an external
potential field that has the form of our geometric potential (29). Here, we
observe the typical divergence for small wavenumbers and a scattering cross
length approaching zero for high ones, where the scattering from the potential
is negligible. Thus, the deviation of the metric yields significant
contributions to scattering processes that originate from the different
geodesic pathways of the elementary waves and dominate the high energy regime.
The expected scattered wave profiles need to be fundamentally changed compared
to the flat space scenario in order to produce the observed interference
effect. Scattering processes thus appear sensitive to the structure of the
underlying space.

\begin{overpic}[width=290.5246pt]{figures/setup_2a.png} \put(1.0,70.0){(a)}
\put(95.0,70.0){(b)} \put(1.0,35.0){(c)} \put(95.0,35.0){(d)} \end{overpic}
Figure 2: Directional scattering probability (a), scattering phase shifts (b),
partial (c), and integrated scattering cross lengths for the scattering
problem without the geometric potential $V_{\mathrm{geo}}$.
\begin{overpic}[width=290.5246pt]{figures/setup_2b.png} \put(1.0,70.0){(a)}
\put(95.0,70.0){(b)} \put(1.0,35.0){(c)} \put(95.0,35.0){(d)} \end{overpic}
Figure 3: Directional scattering probability (a), scattering phase shifts (b),
partial (c), and integrated scattering cross lengths for the scattering
problem with the metric of a flat space.

###  III.3 Lensing by Gaussian Dent

Fig. 4 shows the squared absolute value of the amplitude of the stationary
wave profile according to equation (15), illustrating how the Gaussian dent
acts like a lens for the two-dimensional plane wave
$\chi_{\mathrm{t}}^{0}\propto\mathrm{e}^{\mathrm{i}kx}$ incoming from
$x\rightarrow-\infty$. We observe an alteration of the probability density in
the form of a diffraction pattern with a pronounced focus centered around the
forward direction. The effect is found for different incident wavenumbers,
with the point of maximal probability amplitude shifting along the positive
$x$-axis as the energy of the projectile is increased.

Assuming that the wavelength $\lambda=2\pi/k$ is small compared to the
expansion of the dent, which holds for high energies, we can describe this
phenomenon by elementary waves that travel through the curved region,
accumulating a phase shift due to path elongation (comparing the arc lengths
of the (classical) geodesic trajectories they follow). As shown in the
previous section, the influence of the geometric potential is marginal for
large incidence wavenumbers $k$ as
$\mathrm{max}\\{V_{\mathrm{geo}}\\}\ll\frac{\hbar^{2}}{2m_{0}}k^{2}$.
Therefore, the oscillating behavior of the total scattering cross length
(especially seen for large $f_{0}$, compare fig. 1 (l)) is caused by the
change of the metric tensor and can be qualitatively explained by an
approximate one-dimensional model, as follows.

Neglecting the geometric potential and the momentum in $y$-direction, the
approximate differential equation (see 2) reads

$\displaystyle\left[\frac{1}{\sqrt{1+(\partial_{x}f)^{2}}}\partial_{x}\left(\frac{1}{\sqrt{1+(\partial_{x}f)^{2}}}\partial_{x}\right)+k^{2}\,\hat{I}\right]\chi_{\mathrm{t}}=0\
.$ (30)

Therefore, its general solutions are given by

$\displaystyle\chi_{\mathrm{t}}(x)=c_{1}\sin\\!\left(c_{2}+ks(x)\right)\ ,$
(31)

with $c_{1},c_{2}\in\mathbb{C}$ and a phase change proportional to the arc
length $s(x)\coloneqq\int_{x_{0}}^{x}\sqrt{1+(\partial_{x^{\prime}}f)^{2}}\
\mathrm{d}x^{\prime}$ of the curve within $(\mathcal{M},g)$. Hence, the
trajectory through the center of the dent yields the greatest effect. From
these expressions we construct the solution for an incoming plane wave as

$\displaystyle\chi_{\mathrm{t}}(x)=\mathrm{e}^{\mathrm{i}k\left(x_{0}+s(x)\right)}\
,$ (32)

where $s(x)<0$ for $x<x_{0}$ and $s(x)\geq 0$ for $x\geq x_{0}$. We choose
$x_{0}$ such that $x_{0}\leq\mathrm{inf}\\{\mathrm{supp}(f)\\}$ or
sufficiently small if $\mathrm{supp}(f)$ does not have a lower bound. Next, we
can refer to the general ansatz

$\displaystyle\chi_{\mathrm{t}}(x)=\begin{cases}\mathrm{e}^{\mathrm{i}kx}+a^{-}\mathrm{e}^{-\mathrm{i}kx}&:x\rightarrow-\infty\\\
a^{+}\mathrm{e}^{\mathrm{i}kx}&:x\rightarrow+\infty\end{cases}\ ,$ (33)

to compute the one-dimensional analog of the scattering amplitude in the
forward ($a^{+}$) and backward ($a^{-}$) directions. We find

$\displaystyle\begin{aligned} a^{-}&=0\ ,\\\
a^{+}&=\mathrm{e}^{2\mathrm{i}k\left(x_{0}+s(0)\right)}-1\ .\end{aligned}$
(34)

Then, the one-dimensional analog of the total scattering cross length can be
calculated as the discrete sum

$\displaystyle\sigma_{\mathrm{tot}}=|a^{+}|^{2}+|a^{-}|^{2}$
$\displaystyle=\bigg{|}\mathrm{e}^{2\mathrm{i}k\left(x_{0}+s(0)\right)}-1\bigg{|}^{2}$
$\displaystyle=|\mathrm{e}^{\mathrm{i}k\wp}-1|^{2}\ ,$ (35)

identifying the effective path extension as

$\displaystyle\wp=\int_{-\infty}^{\infty}\left(\sqrt{1+(\partial_{\tilde{x}}f)^{2}}-1\right)\
\mathrm{d}\tilde{x}$ (36)

since $x_{0}\notin\mathrm{supp}(f)$ or sufficiently small. The last expression
can be transformed to

$\displaystyle\sigma_{\mathrm{tot}}=2\left[1-\cos\left(k\wp\right)\right]\ ,$
(37)

and the $k$-period of the scattering cross length $k_{\mathrm{p}}$ is given by

$\displaystyle k_{\mathrm{p}}=\frac{2\pi}{\wp}\ .$ (38)

This simplified consideration explains the periodicity of
$\sigma_{\mathrm{tot}}$ in the wavenumber $k$ (see fig. 1 (l)) as an
interference effect between the incoming wave and the scattered wave. The
wavelength obtained from figs. 1(k) and (l) deviates by approximately $6\%$
and $3\%$ from our analytical model.

Furthermore, as the dent possesses a nonvanishing diameter that is given by
the integrated cross length, it acts as an aperture. The geometric structure
resembles a gravitational lens for light waves propagating through curved
space. An alteration of the beam trajectory rationalizes both effects due to
deformations of space (or spacetime) [48, 49]. The difference here is that the
extension of the curved region is comparable to the wavelength of the
projectile so that a wave mechanical picture is justified. Note that, in
contrast to electromagnetic radiation, we are dealing with wave functions
$\chi_{\mathrm{t}}$ representing quantum particles, i.e., only the effects of
the diffraction pattern on the observable $|\chi_{\mathrm{t}}|^{2}$ are
physically meaningful. Regarding the diffraction pattern implies the idea of
engineering nanostructures for particle-optical applications.

Figure 4: Absolute square of the wave function at $k=7.5\,a_{0}^{-1}$,
$f_{0}=a_{0}$, and $\sigma=a_{0}/\sqrt{2}$.

##  IV Conclusions

We presented an analysis of the elastic channel for generalized quantum
potential scattering confined to Riemannian manifolds. In addition to possible
external scattering potentials, there exists an effective geometry-induced
potential field that is always attractive when a two-dimensional subset of the
three-dimensional Euclidean space is considered. However, this field is
sensitive to geometric disturbances because of its nonlinear dependencies on
the geometric parameters. Furthermore, the deviating metric yields
contributions to the scattering amplitude. Analytical results and simulations
(that have been extensively tested for convergence and reproduced accurately
the known limiting cases such as flat space behavior) were presented and
compared for a Gaussian dent within a flat two-dimensional space. The first
Born approximation was found to be reliable only in the lower energy limit,
where the geometric potential field is the dominant target. Particularly at
high energies, the adapted metric tensor field leads to wave interference
resulting in a diffraction pattern. This effect appears similar to a
gravitational lens and is envisioned to be exploited in nanostructure
engineering.

As for experiments that can be studied with our formalism, we mention two-
dimensional materials such as free-standing graphene or TMDC sheets. They
develop ripples because long-range ordering in such systems of reduced
dimension is unstable towards spatial fluctuations [50]. However, as the
effects of metric modifications can be hard to distill in transport
(scattering) experiments, since the wavelength of the transport electrons (the
Fermi velocity is $v_{\mathrm{F}}\approx 3\cdot 10^{6}m/s$) is way smaller
than the spatial variations, more promising examples for granting experimental
access are (probably nanoscopic) tip-induced deformations. Indeed, a recent
experiment indicates the formation of tip-induced bound states [51].

##  Appendix A Derivation of the Scattering Amplitude

The scattering problem is described by the Lippmann-Schwinger equation adapted
to the effective tangent quantum dynamics with respect to a Riemannian
manifold $\left(\mathcal{M},g\right)$. Using cylindrical coordinates, the
metric tensor field and the geometric potential field attain the forms

$\displaystyle\left(g_{\mu\nu}\right)$
$\displaystyle\simeq\left(\eta_{\mu\nu}\right)=\left(\begin{matrix}1&0\\\
0&r^{2}\end{matrix}\right)\ ,\ \left(r\rightarrow\infty\right),$
$\displaystyle\quad V_{\mathrm{geo}}$ $\displaystyle\simeq
V_{\mathrm{geo}}^{0}=0\ ,\ \left(r\rightarrow\infty\right)\ ,$

in the asymptotic region. The first Born approximation of the scattering
amplitude (16) is performed by replacing
$|\chi_{\mathrm{t}}^{+}\rangle\simeq|\chi_{\mathrm{t}}^{0}\rangle$. Therefore,
the evaluation of the resulting integrand for the case of asymptotically flat
manifolds that are parameterized in Monge form, so that (8) – (11) hold, leads
to

$\displaystyle\langle\textbf{q}^{\prime}|\hat{H}^{\mathrm{s}}|\chi_{\mathrm{t}}^{+}\rangle$
$\displaystyle\simeq\langle\textbf{q}^{\prime}|\hat{H}^{\mathrm{s}}|\chi_{\mathrm{t}}^{0}\rangle$
$\displaystyle=\dfrac{1}{2\pi}\,\left[-\dfrac{\hbar^{2}}{2m_{0}}\,\left(F^{2}\,\left(\dfrac{\textbf{k}\cdot\textbf{q}^{\prime}}{r^{\prime}}\right)^{2}-\mathrm{i}\,\left(\dfrac{F^{2}}{r^{\prime}}+F\,\partial_{r^{\prime}}F\right)\,\dfrac{\textbf{k}\cdot\textbf{q}^{\prime}}{r^{\prime}}\right)+V_{\mathrm{geo}}\right]\,\mathrm{e}^{\mathrm{i}\textbf{k}\cdot\textbf{q}^{\prime}}$
$\displaystyle=-\dfrac{\hbar^{2}}{4\pi
m_{0}}\,\left[F^{2}\,\left(\dfrac{\textbf{k}\cdot\textbf{q}^{\prime}}{r^{\prime}}\right)^{2}-\mathrm{i}\,\left(\dfrac{F^{2}}{r^{\prime}}+F\,\partial_{r^{\prime}}F\right)\,\dfrac{\textbf{k}\cdot\textbf{q}^{\prime}}{r^{\prime}}+\dfrac{1}{4}\,\left(\dfrac{F}{r^{\prime}}-\partial_{r^{\prime}}F\right)^{2}\right]\,\mathrm{e}^{\mathrm{i}\textbf{k}\cdot\textbf{q}^{\prime}}\
.$

Inserting this expression into (16) yields

$\displaystyle a(\textbf{k},\textbf{k}^{\mathrm{s}})$ $\displaystyle\simeq
a^{(1)}(\textbf{k},\textbf{k}^{\mathrm{s}})$
$\displaystyle=-\dfrac{\mathrm{e}^{-\mathrm{i}\frac{3\pi}{4}}}{\sqrt{8\pi
k}}\,\int\mathrm{e}^{\mathrm{i}\,\Delta\textbf{k}\cdot\textbf{q}^{\prime}}\,\left[F^{2}\,\left(\dfrac{\textbf{k}\cdot\textbf{q}^{\prime}}{r^{\prime}}\right)^{2}-\mathrm{i}\,\left(\dfrac{F^{2}}{r^{\prime}}+F\,\partial_{r^{\prime}}F\right)\,\dfrac{\textbf{k}\cdot\textbf{q}^{\prime}}{r^{\prime}}+\dfrac{1}{4}\,\left(\dfrac{F}{r^{\prime}}-\partial_{r^{\prime}}F\right)^{2}\right]\
\mathrm{d}\sigma(\textbf{q}^{\prime})\ ,$

where $\Delta\textbf{k}\coloneqq\textbf{k}-\textbf{k}^{\mathrm{s}}$ was
introduced. Using it to define an internal $\tilde{x}$-axis and denoting the
angle between the wave vectors as $\vartheta$, basic trigonometric
considerations under incorporation of the conservation of momentum yield

$\displaystyle\|\Delta\textbf{k}\|_{2}$
$\displaystyle=2\,k\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right)=2\,k_{\tilde{x}}$
$\displaystyle\Rightarrow\
k_{\tilde{y}}^{2}-k_{\tilde{x}}^{2}=k^{2}\,\mathrm{cos}(\vartheta)\ .$

The above integral can be computed in polar coordinates using the Bessel
integral of the first kind, and applying trigonometric relations, the result
is expressed as function $\vartheta$ as

$\displaystyle\begin{aligned}
a^{(1)}(\vartheta)=\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}\,\sqrt{\dfrac{\pi}{2k}}\,\int\limits_{0}^{\infty}r^{\prime}\,&\left[\left(F^{2}k^{2}\,\mathrm{sin}^{2}\left(\nicefrac{{\vartheta}}{{2}}\right)+\dfrac{1}{4}\,\left(\dfrac{F}{r^{\prime}}-\partial_{r^{\prime}}F\right)^{2}\right)\,J_{0}(\rho^{\prime})\right.\\\
&\hskip
8.53581pt\left.+\left(\dfrac{F^{2}k}{2\,r^{\prime}\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right)}+k\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right)\,F\,\partial_{r^{\prime}}F\right)\,J_{1}(\rho^{\prime})\right]\
\mathrm{d}r^{\prime}\ .\end{aligned}$ (39)

Therein,
$\rho^{\prime}\coloneqq\|\Delta\textbf{k}\|_{2}\,r^{\prime}=2\,k\,r^{\prime}\,\mathrm{sin}\left(\nicefrac{{\vartheta}}{{2}}\right)$
and $J_{n}$ denotes the Bessel function of the first kind of order
$n\in\mathbb{N}_{0}$. This expression gives the scattering amplitude in the
fist Born approximation. In the main text, it is compared with the total
scattering cross length obtained from numerical simulation.

We note that the optical theorem [52]

$\displaystyle\sigma_{\mathrm{tot}}=2\,\sqrt{\dfrac{\pi}{k}}\,\bigl{(}\Im\\{a(0)\\}-\Re\\{a(0)\\}\bigl{)}$
(40)

yields $\sigma_{\mathrm{tot}}^{(1)}\simeq 0$ when combined with the first Born
approximation of the scattering amplitude (39), because using the series
expansion of the latter for small arguments we find that the forward
scattering (for $\textbf{k}\parallel\textbf{k}^{\mathrm{s}}\ \Leftrightarrow\
\vartheta=0$) is given by

$\displaystyle\begin{aligned}
a^{(1)}(0)&=\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}\,\sqrt{\dfrac{\pi}{32\,k}}\,\int\limits_{0}^{\infty}r^{\prime}\,\left[\left(\dfrac{F}{r^{\prime}}-\partial_{r^{\prime}}F\right)^{2}+2\,k^{2}\,F^{2}\right]\
\mathrm{d}r^{\prime}=\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}\,\sqrt{\dfrac{\pi}{2k}}\,\int\limits_{0}^{\infty}r^{\prime}\,\left[-\dfrac{2m}{\hbar^{2}}\,V_{\mathrm{geo}}+\dfrac{k^{2}\,F^{2}}{2}\right]\
\mathrm{d}r^{\prime}\ .\end{aligned}$ (41)

This circumstance is not surprising as the relation is proven to be valid for
the complete solution of the Lippman-Schwinger equation only and, obviously,
breaks down during the Born approximation for axially symmetric asymptotically
flat Riemannian manifolds.

##  Appendix B Derivation of the Expressions for PWA

The usage of PWA to solve short-range potential scattering problems is well-
documented in the literature. Here, we perform it for the sake of completeness
and show its validity for scattering in asymptotically flat manifolds, i. e.
for scattering upon spatially localized geometric perturbations.

For axially symmetric manifolds parameterized by the radius $r$ from the
symmetry center and the azimuthal angle $\vartheta$, the time-independent
effective tangent Schrödinger equation, as derived from (2), is cast as

$\displaystyle\left[\dfrac{v}{r}\,\partial_{r}\left(r\,v\,\partial_{r}\right)+\dfrac{1}{r^{2}}\,\partial_{\vartheta}^{2}+\left(k^{2}-U\right)\,\hat{I}\right]\chi_{\mathrm{t}}(r,\vartheta)=0$
(42)

where we used $E=\frac{\hbar^{2}k^{2}}{2m_{0}}$. The reduced scattering
potential is $U(r)\coloneqq\frac{2m_{0}}{\hbar^{2}}\,V(r)$, and
$v(r)\coloneqq\frac{1}{\sqrt{1+(\partial_{r}f)^{2}(r)}}$. As the Hamilton
operator $\hat{H}$ is invariant under rotations by the azimuthal angle
$\vartheta$, $\hat{H}$ and the operator of axial orbital angular momentum
$\hat{l}$ possess simultaneous eigenstates and the separation ansatz

$\displaystyle\chi_{\mathrm{t}}^{0}(r,\vartheta)=\frac{v_{m}(r)}{\sqrt{r}}\,\mathrm{e}^{\mathrm{i}m\vartheta}\
,\ m\in\mathbb{Z}$ (43)

is appropriate. Thus, we obtain

$\displaystyle\left[\left(v\,\partial_{r}\right)^{2}+\left(k^{2}-\dfrac{m^{2}}{r^{2}}+\dfrac{v^{2}}{4r^{2}}-U_{\mathrm{eff}}\right)\,\hat{I}\right]u_{m}(r)=0$
(44)

with the effective reduced potential

$\displaystyle
U_{\mathrm{eff}}=U+\dfrac{1}{2}\,\dfrac{v\,\partial_{r}v}{r}=U-\dfrac{1}{2r}\,\dfrac{\partial_{r}f\,\partial_{r}^{2}f}{\sqrt{1+(\partial_{r}f)^{2}}}\
.$ (45)

In general, the scattering potential field is given as the sum of the
geometric and external potential fields, $V\doteq
V_{\mathrm{geo}}+V_{\mathrm{ext}}$. For our parametrization, which gives (8) –
(11), the case of free motion yields an effective reduced potential reading

$\displaystyle U_{\mathrm{eff}}=$
$\displaystyle\,-\dfrac{1}{2r}\,\dfrac{\partial_{r}f\,\partial_{r}^{2}f}{\sqrt{1+(\partial_{r}f)^{2}}}-\dfrac{1}{4r^{2}}\,\dfrac{(\partial_{r}f)^{2}}{1+(\partial_{r}f)^{2}}$
$\displaystyle\,+\dfrac{1}{2r}\,\dfrac{\partial_{r}f\,\partial_{r}^{2}f}{\left(1+(\partial_{r}f)^{2}\right)^{2}}-\dfrac{1}{4}\,\dfrac{(\partial_{r}^{2}f)^{2}}{\left(1+(\partial_{r}f)^{2}\right)^{3}}\
,$ (46)

explicitly. Setting $v(r)\doteq 1+w(r)$, we can express (44) as

$\displaystyle\Biggl{\\{}\left[\partial_{r}^{2}+\left(k^{2}r^{2}-m^{2}+\dfrac{1}{4}\right)\,\hat{I}\right]+\left[\left(w^{2}+2w\right)r^{2}\,\partial_{r}^{2}+\left(w+1\right)\partial_{r}w\,r^{2}\,\partial_{r}+\left(\dfrac{w^{2}}{4}+\dfrac{w}{2}-r^{2}U_{\mathrm{eff}}\right)\,\hat{I}\right]\Biggl{\\}}\,u_{m}(r)=0\
.$ (47)

In contrast, in the asymptotic region equation (42) reduces to

$\displaystyle\left[\partial_{r}^{2}+\dfrac{1}{r}\,\partial_{r}+\dfrac{1}{r^{2}}\,\partial_{\vartheta}^{2}+\left(k^{2}-U^{0}\right)\,\hat{I}\right]\,\chi^{0}_{\mathrm{t}}(r,\vartheta)=0\
,$ (48)

and inserting an ansatz similar to (43) yields

$\displaystyle\left[\partial_{r}^{2}+\left(\dfrac{1}{4r^{2}}+k^{2}-U^{0}(r)-\dfrac{m^{2}}{r^{2}}\right)\,\hat{I}\right]\,u^{0}_{m}(r)=0$
(49)

as a remaining differential equation for the scaled radial part. Comparing
(49) with (47), we see that

$\displaystyle\left[\left(w^{2}+2w\right)r^{2}\,\partial_{r}^{2}+\left(w+1\right)\partial_{r}w\,r^{2}\,\partial_{r}+\left(\dfrac{w^{2}}{4}+\dfrac{w}{2}-r^{2}U_{\mathrm{eff}}\right)\,\hat{I}\right]u_{m}(r)\overset{!}{\simeq}-U^{0}\,u_{m}^{0}(r)=0\
,\ (r\rightarrow\infty)$

must hold in the limit of large distances so that the solution of the
differential equation converges appropriately to its asymptotic solution, i.
e. $u_{m}(r)\simeq u_{m}^{0}(r)$, $r\rightarrow\infty$. As all functions
$u_{m}$ and their derivatives have to be bounded due to the normalization
condition of the wave function, we can extract the conditions

$\displaystyle w\in o(r^{-2})\ ,\qquad U_{\mathrm{eff}}\in o(r^{-2})\ .$

Henceforth, the vanishing $\partial_{r}f\rightarrow 0$ $(r\rightarrow\infty)$
is implied, because by truncating an expansion we find

$\displaystyle w(r)$ $\displaystyle=\dfrac{1}{\sqrt{1+(\partial_{r}f)^{2}}}-1$
$\displaystyle\simeq(\partial_{r}f)^{2}\ ,\ (r\rightarrow\infty)\ ,$

so that the following holds

$\displaystyle w\in o(r^{-2})\ $ $\displaystyle\Leftrightarrow\
\partial_{r}f\in o(r^{-1})$ $\displaystyle\Leftrightarrow\ f\in
o(r^{-\alpha})+c\ ,\ \alpha\in\mathbb{R}:\alpha>0\ .$ (50)

From (46) we see that the condition concerning the effective reduced potential
is automatically fulfilled when (50) holds. Thus, the latter is a necessary
and sufficient condition for applying the PWA. It states, that the manifold
has to be a plane containing a dent that decays at least with a power law.

While (42) describes a scattering problem within curved two-dimensional space,
(48) enables the consideration of scattering upon the entire curved region. As
we deal with the same ansatz that was used by Lapidus [41] for considerations
of potential scattering in flat two-dimensional spaces, the same results are
valid for our situation of scattering upon curved regions. From the
requirement of regularity at the center, i. e. for $r\rightarrow 0$, and the
short range characteristics of $U$, we deduce the asymptotic behavior of the
solutions

$\displaystyle\begin{aligned} u_{m}(r)&\overset{r\rightarrow
0}{\propto}r^{m+\frac{1}{2}}\ ,\\\
u_{m}(r)&\overset{r\rightarrow\infty}{\propto}s_{m}(kr)-\mathrm{tan}(\delta_{m})\,c_{m}(kr)\
,\end{aligned}$ (51)

where $\Theta_{m}(kr)\coloneqq kr-\frac{\pi}{2}\left(m+\frac{1}{2}\right)$
([40, 41, 44])

$\displaystyle s_{m}(kr)$
$\displaystyle\propto\sqrt{kr}\,J_{m}(kr)\overset{r\rightarrow\infty}{\propto}\sqrt{\dfrac{2}{\pi}}\,\mathrm{cos}(\Theta_{m}(kr))\
,$ $\displaystyle c_{m}(kr)$
$\displaystyle\propto\sqrt{kr}\,Y_{m}(kr)\overset{r\rightarrow\infty}{\propto}\sqrt{\dfrac{2}{\pi}}\,\mathrm{sin}(\Theta_{m}(kr))$

are the regular and singular solutions in dependence of the Bessel functions
of the first and second kind, respectively, and $\delta_{m}$ is the scattering
phase shift of the corresponding orbital angular momentum channel. Choosing
$N_{m}=2\,\mathrm{cos}(\delta_{m})\,\mathrm{e}^{\mathrm{i}\delta_{m}}$ as the
respective normalization constant within (51) gives the equivalent expression

$\displaystyle u_{m}(r)$
$\displaystyle\overset{r\rightarrow\infty}{\propto}\sqrt{kr}\,\left(S_{m}\,H_{m}^{(1)}(kr)+H_{m}^{(2)}(kr)\right)$
$\displaystyle\overset{r\rightarrow\infty}{\propto}\sqrt{\dfrac{2}{\pi}}\,\left(\mathrm{e}^{-\mathrm{i}\Theta_{m}(kr)}+S_{m}\,\mathrm{e}^{\mathrm{i}\Theta_{m}(kr)}\right)\
,$

that includes the Hankel functions of first and second kind and the S-matrix
$S_{m}\coloneqq\mathrm{e}^{2\mathrm{i}\delta_{m}}$. Adapting the reasoning of
Burke [44], we find

$\displaystyle
K_{m}=\mathrm{tan}(\delta_{m})=-\dfrac{\pi}{2}\,\dfrac{1}{\sqrt{k}}\,\int\limits_{0}^{\infty}\sqrt{r}\,U(r)\,u_{m}(r)\,J_{m}(kr)\
\mathrm{d}r$ (52)

as an explicit expression for the $K$-matrix.

As for the superposition ansatz (15), we represent the incoming plane wave
according to the Jacobi-Anger expansion [40]

$\displaystyle\mathrm{e}^{\mathrm{i}kr\,\mathrm{cos}(\vartheta)}=\sum\limits_{m=-\infty}^{+\infty}\mathrm{i}^{m}\,J_{m}(kr)\,\mathrm{e}^{\mathrm{i}m\vartheta}\
,$

and, similarly, expand the entire wave formally with respect to the same set
of basis functions,

$\displaystyle\chi_{\mathrm{t}}(r,\vartheta)=\sum\limits_{m=-\infty}^{+\infty}B_{m}(k)\,\dfrac{u_{m}(kr)}{\sqrt{r}}\,\mathrm{e}^{\mathrm{i}m\vartheta}\,,\
B_{m}\in\mathbb{C}\ \forall\ m\in\mathbb{Z}\,.$

Inserting these terms in (15) and considering the asymptotic forms, we find

$\displaystyle\sum\limits_{m=-\infty}^{+\infty}\dfrac{B_{m}(k)}{2\,\mathrm{cos}(\delta_{m})}\,\sqrt{\dfrac{2}{\pi
r}}\,\left(\mathrm{e}^{\mathrm{i}\,(\Theta_{m}+\delta_{m})}+\mathrm{e}^{-\mathrm{i}\,(\Theta_{m}+\delta_{m})}\right)\,\mathrm{e}^{\mathrm{i}m\vartheta}=\dfrac{1}{2\pi}\,\left(\sum\limits_{m=-\infty}^{+\infty}\dfrac{\mathrm{i}^{m}}{2}\,\sqrt{\dfrac{2}{\pi
kr}}\,\left(\mathrm{e}^{\mathrm{i}\Theta_{m}}+\mathrm{e}^{-\mathrm{i}\Theta_{m}}\right)\,\mathrm{e}^{\mathrm{i}m\vartheta}+a(\vartheta)\,\dfrac{\mathrm{e}^{\mathrm{i}kr}}{\sqrt{r}}\right)\
,$

From this we deduce

$\displaystyle
B_{m}(k)=\mathrm{i}^{m}\,\dfrac{\mathrm{cos}(\delta_{m})}{2\pi\,\sqrt{k}}\,\mathrm{e}^{\mathrm{i}\delta_{m}}$

by a comparison of coefficients for the expansions according to orthogonal
basis functions and find the identity

$\displaystyle
a(\vartheta)=\dfrac{\mathrm{e}^{-\mathrm{i}\frac{\pi}{4}}}{\sqrt{2\pi
k}}\,\sum\limits_{m=-\infty}^{+\infty}\left(S_{m}-1\right)\,\mathrm{e}^{\mathrm{i}m\vartheta}\
.$ (53)

This is the relation we state in the main text. Inserting it into the
definitions of the scattering cross lengths, (17) and (18), yields the
expressions (22) and (23), respectively. Furthermore, we are able to express
the orbital angular momentum of the scattered wave with respect to the
$z$-axis as

$\displaystyle
L^{\mathrm{s}}=\dfrac{\hbar}{\sigma_{\mathrm{tot}}}\,\dfrac{1}{k}\,\sum\limits_{m=-\infty}^{+\infty}m\,|S_{m}-1|^{2}\
,$ (54)

but as $S_{m}=S_{-m}\ \forall\ m\in\mathbb{Z}$ for an axially symmetric
scattering potential we expect vanishing orbital angular momentum transfer in
our case. Finally, if the solution $\chi_{\mathrm{t}}$ is known upon
$\mathrm{supp}(U)$, we can find an alternative expression of the the K-matrix
(52) by equating the ratios between radial projections of the wave function
and its values according to (51) taken at two distinct but sufficiently large
radii, and isolating the tangent of the scattering phase shift. The result is
given by the formula (25) in the main text.

##  Appendix C Discussion of the Geometric Potential

\begin{overpic}[width=290.5246pt]{figures/plot_Ugeo_curvatures.png}
\put(1.0,40.0){(a)} \put(96.0,40.0){(b)} \end{overpic} Figure 5: (a) Density
plot of the reduced geometric potential $U_{\mathrm{geo}}$ within the
parameter space and (b) course of the Gaussian and mean curvatures $K$ and $M$
for a Gaussian dent with parameters $f_{0}=1\,a_{0}$ and
$\sigma=\frac{1}{\sqrt{2}}\,a_{0}$, with $a_{0}$ describing an arbitrary
length unit. The geometric potential consists of two contributions dependent
on respectively one type of curvature, and their difference yields an axially
symmetric ring-shaped structure with radial dependence.

Considering the Gaussian dent as given by (27), the relevant quantities of
differential geometry can be evaluated explicitly. The covariant metric tensor
field (8) has the matrix representation

$\displaystyle\left(g_{\mu\nu}\right)=\left(\begin{matrix}\left[1+\left(\frac{f_{0}\,r}{\sigma^{2}}\right)^{2}\,\mathrm{e}^{-\frac{r^{2}}{\sigma^{2}}}\right]&0\\\
0&r^{2}\end{matrix}\right)$

in view of the natural local basis. With the expression $F$ given in (28) we
find the Gaussian and mean curvatures according to (9),

$\displaystyle K$
$\displaystyle=\dfrac{f_{0}^{2}\,\sigma^{2}\,\left(\sigma-r\right)\,\left(\sigma+r\right)\,\mathrm{e}^{\frac{r^{2}}{\sigma^{2}}}}{\left(f_{0}^{2}\,r^{2}+\sigma^{4}\,\mathrm{e}^{\frac{r^{2}}{\sigma^{2}}}\right)^{2}}\
,$ $\displaystyle M$
$\displaystyle=\dfrac{f_{0}\,\left(-f_{0}^{2}\,r^{2}+\sigma^{2}\,\left(r^{2}-2\sigma^{2}\right)\,\mathrm{e}^{\frac{r^{2}}{\sigma^{2}}}\right)\,\mathrm{e}^{-\frac{r^{2}}{2\sigma^{2}}}}{2\,\sigma^{2}\,\left(f_{0}^{2}\,r^{2}+\sigma^{4}\,\mathrm{e}^{\frac{r^{2}}{\sigma^{2}}}\right)\,\sqrt{1+\left(\frac{f_{0}\,r}{\sigma^{2}}\right)^{2}\,\mathrm{e}^{-\frac{r^{2}}{\sigma^{2}}}}}\
,$

and derived the geometric potential (29) from them. The result for the reduced
potential
$U_{\mathrm{geo}}\coloneqq\frac{2\,m_{0}}{\hbar^{2}}\,V_{\mathrm{geo}}$ is
shown in fig. 5 (a) and exhibits an axially symmetric, ring-shaped structure
as all the above mentioned quantities are functions of the radial variable
$r$. The potential vanishes at the center (i. e. $r\rightarrow 0$) where
$K(0)=\frac{f_{0}^{2}}{\sigma^{4}}$ and $M(0)=-\frac{f_{0}}{\sigma^{2}}$. As
can be seen from the equations, the geometric potential is nonlinearly
dependent on both the amplitude $f_{0}$ and the width measure $\sigma$. Hence,
narrowing the width or increasing the amplitude results in larger curvatures.
However, the plots in fig. 5 (b) indicate that the structure of the geometric
potential can not be identified from or associated with either of the
curvatures alone. For the Gaussian dent the two types of curvature yield quite
similar contributions to the geometric potential so that their difference has
a significantly lower range and a different shape. Especially, the potential
vanishes where the curvatures reach their extremes. Nevertheless, we tested
that a variation of the dent in the mentioned way leads to a stronger
geometric potential, while its ring-shaped structure stays invariant.

If we assume $a_{0}=2\,\mathrm{nm}$ which seems experimentally feasible in
current 2D materials, we find amplitude values of $\approx
2.38\,\mathrm{meV}$. Therefore, the geometry-induced potential is significant
for low projectile energies only. The varying metric tensor accompanying its
occurrence still has an effect that depends on the ratio between the
characteristic size of the geometric structure and the wavelength of the
projectile.

Due to symmetry, it is expected that any influence on an incident plane wave
will be mirror symmetric with respect to the propagation direction defined by
k. Therefore, only linear momentum but no angular momentum transfer from the
projectile to the target should occur.

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