# Boundary localization of transmission eigenfunctions in spherically
stratified media

Yan Jiang  Department of Mathematics, Jilin University, Changchun, Jilin,
China<EMAIL_ADDRESS>,  Hongyu Liu  Department of Mathematics,
City University of Hong Kong, Hong Kong SAR, China<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Jiachuan Zhang  School of Physical and Mathematical
Sciences, Nanjing Tech University, Nanjing, Jiangsu, China.
<EMAIL_ADDRESS>and  Kai Zhang  Department of Mathematics, Jilin
University, Changchun, Jilin, China<EMAIL_ADDRESS>
###### Abstract.

Consider the transmission eigenvalue problem for $u\in H^{1}(\Omega)$ and
$v\in H^{1}(\Omega)$:

$\left\\{\begin{array}[]{ll}\nabla\cdot(\sigma\nabla
u)+k^{2}\mathbf{n}^{2}u=0&\text{in}\ \ \Omega,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \Delta v+k^{2}v=0&\text{in}\ \ \Omega,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \displaystyle u=v,\ \sigma\frac{\partial
u}{\partial\nu}=\frac{\partial v}{\partial\nu}&\text{on}\ \
\partial\Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \end{array}\right.$

where $\Omega$ is a ball in $\mathbb{R}^{N}$, $N=2,3$. If $\sigma$ and
$\mathbf{n}$ are both radially symmetric, namely they are functions of the
radial parameter $r$ only, we show that there exists a sequence of
transmission eigenfunctions $\\{u_{m},v_{m}\\}_{m\in\mathbb{N}}$ associated
with $k_{m}\rightarrow+\infty$ as $m\rightarrow+\infty$ such that the
$L^{2}$-energies of $v_{m}$’s are concentrated around $\partial\Omega$. If
$\sigma$ and $\mathbf{n}$ are both constant, we show the existence of
transmission eigenfunctions $\\{u_{j},v_{j}\\}_{j\in\mathbb{N}}$ such that
both $u_{j}$ and $v_{j}$ are localized around $\partial\Omega$. Our results
extend the recent studies in [15, 16]. Through numerics, we also discuss the
effects of the medium parameters, namely $\sigma$ and $\mathbf{n}$, on the
geometric patterns of the transmission eigenfunctions.

Keywords: Transmission eigenfunctions, spectral geometry, boundary
localization, wave localization

2010 Mathematics Subject Classification: 35P25, 78A46 (primary); 35Q60, 78A05
(secondary).

##  1\. Introduction

In this paper, we study the geometric patterns of transmission eigenfunctions.
To begin with, we briefly discuss the physical origin of the transmission
eigenvalue problem.

Let us consider the time-harmonic wave scattering caused by the interaction of
an incident wave field and an inhomogeneous medium. Let $u^{i}$ denote the
incident field, which is an entire solution to $(\Delta+k^{2})u^{i}=0$ in
$\mathbb{R}^{N}$, $N=2,3$. Here, $k\in\mathbb{R}_{+}$ signifies the
(normalized) angular frequency of the wave propagation. Let
$(\Omega;\sigma,\mathbf{n})$ signify the inhomogeneous medium. Here, $\Omega$
denotes the support of the inhomogeneity of the medium, which is a bounded
Lipschitz domain such that $\mathbb{R}^{N}\backslash\overline{\Omega}$.
$\sigma$ and $\mathbf{n}$ signify the medium parameters. It is assumed that
$\sigma$ and $\mathbf{n}$ are $L^{\infty}$ functions and both are bounded
below by a positive constant. We also let $\sigma=\mathbf{n}\equiv 1$ in
$\mathbb{R}^{N}\backslash\overline{\Omega}$. Let $u$ and $u^{s}:=u-u^{i}$,
respectively, denote the total and scattered wave fields. The wave scattering
is governed by the following Helmholtz system:

$\nabla\cdot(\sigma\nabla u)+k^{2}\mathbf{n}^{2}u=0\quad\mbox{in}\
\mathbb{R}^{N};\ \
\lim_{r\rightarrow+\infty}r^{(N-1)/2}\left(\partial_{r}u^{s}-\mathrm{i}ku^{s}\right)=0,$
(1.1)

where $\mathrm{i}:=\sqrt{-1}$, $r:=|x|$ for $x\in\mathbb{R}^{N}$, and
$\partial_{r}u:=\hat{x}\cdot\nabla_{x}u$ with
$\hat{x}:=x/|x|\in\mathbb{S}^{N-1}$. The limit in (1.1) is called the
radiation condition and characterizes the outgoing nature of the scattered
wave field $u^{s}$. In the physical setup, when $N=2$, (1.1) describes the
transverse electromagnetic scattering, where $\sigma^{-1}$ and
$\mathbf{n}^{2}$ specify the electric permittivity and magnetic permeability
of the optical medium [27]; whereas when $N=3$, (1.1) describes the acoustic
scattering, where $\sigma^{-1}$ and $\mathbf{n}^{2}$ are respectively the
density and modulus of the acoustic medium [29]. We refer to [29, 32] for the
well-posedness of the scattering problem (1.1) with a unique solution $u\in
H_{loc}^{1}(\mathbb{R}^{N})$. It holds that:

$u(x)=u^{i}(x)+\frac{e^{\mathrm{i}kr}}{r^{(N-1)/2}}u_{\infty}(\hat{x})+\mathcal{O}\left(\frac{1}{r^{(N+1)/2}}\right)\quad\mbox{as}\
\ r\rightarrow+\infty.$ (1.2)

In (1.2), $u_{\infty}(\hat{x})$ is known as the far-field pattern, which
encodes the scattering information caused by the perturbation of the incident
field $u^{i}$ due to the scatterer $(\Omega;\sigma,\mathbf{n})$.

Associated with the scattering problem describe above, a practical inverse
scattering problem of industrial importance is to recover
$(\Omega;\sigma,\mathbf{n})$ by knowledge of $u_{\infty}(\hat{x})$. The
inverse problem can be abstractly recast as the following operator equation:

$\mathcal{F}(\Omega;\sigma,\mathbf{n})=u_{\infty}(\hat{x}),$ (1.3)

where $\mathcal{F}$ is defined via the scattering problem (1.1). For (1.3),
one peculiar case is that $u_{\infty}\equiv 0$. In such a case, the scatterer
$(\Omega;\sigma,\mathbf{n})$ produces no scattering information to the outside
observation, namely it is invisible/transparent with respect to the wave
probing. Noting that $u^{i}\equiv 0$ readily yields $u^{i}=u^{s}$ in
$\mathbb{R}^{N}\backslash\overline{\Omega}$ by Rellich’s Theorem [17], one can
directly derive that $u|_{\Omega}\in H^{1}(\Omega)$ and $v:=u^{i}|_{\Omega}\in
H^{1}(\Omega)$ fulfil that:

$\left\\{\begin{array}[]{ll}\nabla\cdot(\sigma\nabla
u)+k^{2}\mathbf{n}^{2}u=0&\text{in}\ \ \Omega,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \Delta v+k^{2}v=0&\text{in}\ \ \Omega,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \displaystyle u=v,\ \sigma\frac{\partial
u}{\partial\nu}=\frac{\partial v}{\partial\nu}&\text{on}\ \
\partial\Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\end{array}\right.$ (1.4)

where $\nu\in\mathbb{S}^{N-1}$ is the exterior unit normal vector to
$\partial\Omega$. (1.4) is referred to as the transmission eigenvalue problem.
It is clear that $u=v\equiv 0$ are trivial solutions to (1.4). If there exists
a nontrivial pair of solutions $(u,v)$, $k$ is called a the transmission
eigenvalue and $u,v$ are the associated transmission eigenfunctions. Clearly,
according to our discussion above, the transmission eigenfunctions depict the
wave propagation inside the scatterer when invisibility/transparency occurs.

The spectral theory of transmission eigenvalue problem has received
considerable interest in the literature. We refer to [11, 18, 28] for survey
and review on the spectral properties of transmission eigenvalues. Recently,
several intrinsic local and global geometric patterns of the transmission
eigenfunctions have been revealed. Roughly and heuristically speaking, the
transmission eigenfunctions tend to (globally) localize/concentrate on
$\partial\Omega$ while (locally) vanish around singular/high-curvature points
on $\partial\Omega$. Here, by localization/concentration, we mean that the
$L^{2}$-energies of the eigenfunctions in $\Omega$ are localized/concentrated
around $\partial\Omega$; and by a singular point, we mean the boundary point
on $\partial\Omega$ at which the normal vector $\nu$ is no longer
differentiable. A singular point can be regarded as having (extrinsic)
curvature being infinity. The local geometric property was first discovered
and investigated in [6, 5], and was further studied in [2, 3, 4, 9, 19, 22,
23, 24] for different geometric and physical setups. We also refer to [2, 4,
7, 8, 9, 10, 12, 13, 14, 22, 23, 30, 33, 34] for related studies in
characterizing non-scattering waves (locally) around corner/singular points.
The global geometric property was first discovered and investigated in [15],
and was further studied in [16, 20, 21] for different geometric and physical
setups. Those geometric patterns are physically interpretable. In fact, in
order to achieve invisibility/transparency, the wave propagates in a “smart”
way which slides over the boundary surface of the scattering object while
avoids the singular/highly-curved places to avoid being trapped. More
intriguingly, the geometric properties have been used to produce super-
resolution imaging schemes for inverse acoustic and electromagnetic scattering
problems [15, 25], artificial mirage [21], and pseudo surface plasmon
resonance [21].

In this paper, we further study the boundary concentration of the transmission
eigenfunctions and extend the related studies in [15, 16] to more general
setups. Specifically, in [15, 16], the theoretical justifications are mainly
concerned with the case that $\sigma\equiv 1$, $\mathbf{n}$ is constant and
$\Omega$ is smooth and convex, though the general case with variable medium
parameter $\mathbf{n}$ and non-convex and non-smooth $\Omega$ is numerically
investigated. We shall include both (being possibly variable) $\sigma$ and
$\mathbf{n}$ into the current study and rigorously justify the boundary
concentration phenomenon for the transmission eigenfunctions. Moreover, we
shall present some novel numerical observations which strengthen the medium
effect on the geometric patterns of the transmission eigenfunctions. More
detailed discussion about the main results shall be given in Section 2, and
the corresponding proofs are provided in Sections 3 and 4.

##  2\. Statement of the main results and discussion

Following [15], we first provide a quantitative description of
surface/boundary localization of a function $\varphi\in L^{2}(\Omega)$. In
what follows, for $\varepsilon\in\mathbb{R}_{+}$, we define

$\mathcal{N}_{\varepsilon}(\partial\Omega):=\\{x\in\Omega;\
\mathrm{dist}(x,\partial\Omega)<\varepsilon\\},$ (2.1)

where $\mathrm{dist}$ signifies the Euclidean distance in $\mathbb{R}^{N}$,
$N=2,3$. Clearly, $\mathcal{N}_{\varepsilon}(\partial\Omega)$ defines an
$\varepsilon$-neighbourhood of $\partial\Omega$.

######  Definition 2.1.

A function $\varphi\in L^{2}(\Omega)$ is said to be _boundary-localized_ (or,
_surface-localized_) if there exists $\varepsilon\ll 1$ such that

$\frac{\|\varphi\|_{L^{2}(\Omega\backslash\mathcal{N}_{\varepsilon}(\partial\Omega))}}{\|\varphi\|_{L^{2}(\Omega)}}\ll
1.$ (2.2)

According to Definition 2.1, the $L^{2}(\Omega)$-energy of the function
$\varphi$ is mainly localized in a small neighbourhood of $\partial\Omega$,
namely $\mathcal{N}_{\varepsilon}(\partial\Omega)$. In what follows, the
asymptotic parameters involved in Definition 2.1 shall become more rigorous.
In the case that $\Omega$ is a ball in $\mathbb{R}^{N}$, $N=2,3$, by scaling
and translation if necessary, we can assume without loss of generality that
$\Omega$ is the unit ball, namely $\Omega:=\\{x\in\mathbb{R}^{N};|x|<1\\}$. In
such a case, we also set $\Omega_{\tau}:=\\{x\in\mathbb{R}^{N};|x|<\tau\\},\
\tau\in(0,1)$ to signify the ball of radius $\tau$.

Our main results can be stated as follows.

######  Theorem 2.1.

Consider the transmission eigenvalue problem (3.1). Let $\Omega$ be the unit
ball in $\mathbb{R}^{N}$, $N=2,3$. Assume that $\mathbf{n}$, $\sigma$ are
functions of the radial parameter $r$ only which fulfil Assumption A in what
follows. Then for any given $\tau\in(0,1)$, there exists a sequence of
eigenfunctions $\\{u_{m},v_{m}\\}_{m\in\mathbb{N}}$ associated to eigenvalues
$k_{m}\rightarrow\infty$ as $m\rightarrow\infty$ such that:

$\lim_{m\rightarrow\infty}\frac{\|v_{m}\|_{L^{2}(\Omega_{\tau})}}{\|v_{m}\|_{L^{2}(\Omega)}}=0.$
(2.3)

According to Theorem 2.1, if one takes $\tau$ to be sufficiently close to 1,
namely $\varepsilon=1-\tau\ll 1$, it is clear that $v_{m}$ in (2.7) with $m$
sufficiently large are all boundary-localized according to Definition 2.1. The
Assumption A in Theorem 2.1 is stated as follows.

Assumption A. Let $\mathbf{n}_{1}$, $\mathbf{n}_{2}$, $M_{1}$, $M_{2}$ be
positive constants, and the radial functions $\mathbf{n}(r)\in C[0,1]$,
$\sigma(r)\in C^{2}[0,1]$ satisfy the following properties:

$\begin{array}[]{ll}(A1)~{}1<\mathbf{n}_{1}<\mathbf{n}(r)<\mathbf{n}_{2},&(A2)~{}1<M_{1}<M_{2},\\\
(A3)~{}\sigma^{\prime}(r)\leq
0,&(A4)~{}M_{1}\leq\frac{\mathbf{n}^{2}}{\sigma(r)}\leq M_{2},\\\
(A5)\sigma^{\prime}(r)^{2}\geq 2\sigma^{\prime\prime}(r)\sigma(r).\end{array}$
(2.4)

It is remarked that in our subsequent analysis, we actually can combine
conditions (A4) and (A5) by requiring a slightly less restrictive condition:

$M_{1}k^{2}\leq\frac{\mathbf{n}^{2}}{\sigma}k^{2}+\frac{\sigma^{\prime
2}}{4\sigma^{2}(r)}-\frac{\sigma^{\prime\prime}(r)}{2\sigma(r)}\leq
M_{2}k^{2}.$ (2.5)

However, the condition (2.5) involves the eigenvalue $k^{2}$, and we split it
into the two conditions (A4) and (A5) in Assumption A. It is directly verified
that when both $\sigma$ and $\mathbf{n}$ are constant, the assumptions in
(2.4) yield that

$\mathbf{n}>1\quad\mbox{and}\quad 0<\sigma\leq\mathbf{n}^{2}.$ (2.6)

Nevertheless, if both $\sigma$ and $\mathbf{n}$ are constant, we can prove a
stronger boundary-localization result.

######  Theorem 2.2.

Consider the same setup as Theorem 2.1 and assume that $\mathbf{n}$, $\sigma$
are both positive constants satisfying $\mathbf{n}^{2}>\sigma$. Then for any
given $\tau\in(0,1)$, there exists a sequence of eigenfunctions
$\\{u_{m},v_{m}\\}_{m\in\mathbb{N}}$ associated to eigenvalues
$k_{m}\rightarrow\infty$ as $m\rightarrow\infty$ such that

$\lim_{m\rightarrow\infty}\frac{\|\varphi_{m}\|_{L^{2}(\Omega_{\tau})}}{\|\varphi_{m}\|_{L^{2}(\Omega)}}=0,\quad\varphi_{m}=u_{m}\
\mbox{or}\ v_{m}.$ (2.7)

That is, if $\sigma$ and $\mathbf{n}$ are both constant, there exists a
sequence of transmission eigenfunctions where both the $u$-parts and $v$-parts
are localized around $\partial\Omega$.

######  Remark 2.1.

In Theorem 2.2, the condition $\mathbf{n}^{2}>\sigma$ is required. We would
like to point that the condition can also be replaced to be
$\mathbf{n}^{2}<\sigma$. In fact, in the latter case, for $(u,v)$ to (1.4), we
can set

$\tilde{k}=k\cdot\frac{\mathbf{n}}{\sigma},\ \
\tilde{\mathbf{n}}=\frac{1}{\mathbf{n}},\ \ \tilde{\sigma}=\frac{1}{\sigma},\
\ \tilde{u}=v,\ \ \tilde{v}=u.$ (2.8)

It is directly verified that

$\left\\{\begin{array}[]{ll}\nabla\cdot(\tilde{\sigma}\nabla\tilde{u})+\tilde{k}^{2}\tilde{\mathbf{n}}^{2}\tilde{u}=0&\text{in}\
\ \Omega,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\Delta\tilde{v}+\tilde{k}^{2}\tilde{v}=0&\text{in}\ \ \Omega,\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \displaystyle\tilde{u}=\tilde{v},\
\tilde{\sigma}\frac{\partial\tilde{u}}{\partial\nu}=\frac{\partial\tilde{v}}{\partial\nu}&\text{on}\
\ \partial\Omega.\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \end{array}\right.$
(2.9)

Since $\mathbf{n}^{2}<\sigma$, one clearly has
$\tilde{\mathbf{n}}^{2}>\tilde{\sigma}$. Hence, Theorem 2.2 applied to (2.9)
readily yields the existence of a sequence of boundary-localized transmission
eigenfunctions $\\{\tilde{u}_{m},\tilde{v}_{m}\\}_{m\in\mathbb{N}}$ associated
with $\tilde{k}_{m}\rightarrow\infty$ as $m\rightarrow\infty$. Then by using
the relations in (2.8), we have the existence of a sequence of boundary-
localized transmission eigenfunctions $\\{{u}_{m},{v}_{m}\\}_{m\in\mathbb{N}}$
associated with ${k}_{m}\rightarrow\infty$ as $m\rightarrow\infty$. Finally,
we would like to point out that if one takes $\sigma\equiv 1$, the result in
Theorem 2.2 recovers those in [15, 20].

So far, we have mainly considered the radially symmetric cases. In particular,
in Theorem 2.1, we can only show the boundary-localization of the
$v_{m}$-part, though we believe that there exist infinitely many transmission
eigenfunctions such that both $u$\- and $v$-parts are boundary-localized. In
[15], extensive numerical examples show that the surface/boundary-localization
is a generic phenomenon occurring for transmission eigenfunctions, even
associated with variable medium parameters and general domains. In particular,
we note that for the case that $\Omega$ is divided into two connected
subdomains, $\Omega=\Omega^{(1)}\cup\Omega^{(2)}$ with
$\Omega^{(1)}\Subset\Omega$ and
$\Omega^{(2)}=\Omega\backslash\overline{\Omega^{(1)}}$, if the medium
parameters $(\sigma_{1},\mathbf{n}_{1})$ in $\Omega^{(1)}$ and
$(\sigma_{2},\mathbf{n}_{2})$ in $\Omega^{(2)}$ are both constant, then there
exist boundary-localized transmission eigenmodes generically even when
$(\sigma_{1},\mathbf{n}_{1})\neq(\sigma_{2},\mathbf{n}_{2})$ (which
corresponds to variable medium parameters). It is remarked that in the
numerical examples in [15], it is always assumed that
$\sigma_{1}=\sigma_{2}\equiv 1$. Nevertheless, we would like to confirm that
the same numerical conclusion holds when $\sigma_{1}$ and $\sigma_{2}$ are
other constants; see Fig. 1 for typical illustration and comparison. It can be
seen that the boundary-localization phenomenon is still every evident though
less sharper than the case with $\sigma\equiv 1$.

Figure 1. First row: $|u|$ (left) and $|v|$ (right) with $k^{2}=402.989$ and
$(\sigma,\mathbf{n}^{2})=(1,1)\chi_{\Omega_{0.8}}+(2,3)\chi_{\Omega_{1}\backslash\Omega_{0.8}}$.
Second row: $|u|$ (left) and $|v|$ (right) with $k^{2}=402.665$ and
$(\sigma,\mathbf{n}^{2})=(1,0.1)\chi_{\Omega_{0.8}}+(1,10)\chi_{\Omega_{1}\backslash\Omega_{0.8}}$.

Next, we present two more numerical examples which were not considered in
[15], and show that variable medium parameters can make the geometric patterns
of the transmission eigenfunctions more intriguing; see Fig. 2, where
$\Omega^{(1)}$ and $\Omega^{(2)}$ respectively signify the left-subdomain and
right-subdomain of the domain $\Omega$.

Figure 2. First row: $|u|$ (left) and $|v|$ (right) with $k^{2}=395.307$ and
$(\sigma,\mathbf{n}^{2})=(0.1,1)\chi_{\Omega^{(1)}}+(10,1)\chi_{\Omega^{(2)}}$.
Second row: $|u|$ (left) and $|v|$ (right) with $k^{2}=410.024$ and
$(\sigma,\mathbf{n}^{2})=(1,1.1)\chi_{\Omega^{(1)}}+(1,10)\chi_{\Omega^{(2)}}$.

It can be observed that in addition to the boundary localization, it may
happen that the transmission eigenfunctions are localized around the material
interface or even at two “exceptional” points. The numerical observations in
Fig. 2 partly corroborates the necessity of introducing Assumption A in our
current study. On the other hand, they are highly interesting spectral
phenomena that are worth further investigation in our forthcoming work.

##  3\. Proof of Theorem 2.1

In this section, we present the proof of Theorem 2.1. That is, the
transmission eigenvalue problem is given by:

$\left\\{\begin{array}[]{ll}\nabla\cdot(\sigma(r)\nabla
u)+k^{2}\mathbf{n}^{2}(r)u=0&\text{in}\ \ \Omega,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \Delta v+k^{2}v=0&\text{in}\ \ \Omega,\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ \displaystyle u=v,\ \sigma(1)\frac{\partial u}{\partial
r}=\frac{\partial v}{\partial r}&\text{on}\ \ \partial\Omega.\vskip 6.0pt plus
2.0pt minus 2.0pt\\\ \end{array}\right.$ (3.1)

Throughout the rest of this section, we assume that $\Omega$ is the unit ball
in $\mathbb{R}^{N}$, $N=2,3$. We shall divide our analysis into two parts,
respectively, for the two and three dimensions.

###  3.1. Two-dimensional case

In two dimensions, we let $x=(r\cos\theta,r\sin\theta)\in\mathbb{R}^{2}$,
$r=|x|$, denote the polar coordinate. In the sequel, $J_{m}$,
$m\in\mathbb{N}\cup\\{0\\}$, signifies the $m$-th order Bessel function [17,
31].

First, we know that the solutions to (3.1) have the following Fourier
expansions [17, 31]:

$u(x)=\sum_{m=0}^{\infty}\alpha_{m}\phi_{m}(r;k)e^{\mathrm{i}m\theta},\quad
v(x)=\sum_{m=0}^{\infty}\beta_{m}J_{m}(kr)e^{\mathrm{i}m\theta},$ (3.2)

where $\alpha_{m}$, $\beta_{m}\in\mathbb{C}$. Set

$u_{m}(x)=\alpha_{m}\phi_{m}(r;k)e^{\mathrm{i}m\theta},\quad
v_{m}(x)=\beta_{m}J_{m}(kr)e^{\mathrm{i}m\theta},$ (3.3)

then we have

$\sigma\phi_{m}^{\prime\prime}+\left(\sigma^{\prime}+\frac{\sigma}{r}\right)\phi_{m}^{\prime}+\left(k^{2}\mathbf{n}^{2}-\frac{m^{2}}{r^{2}}\sigma\right)\phi_{m}=0,$
(3.4)

where the differentiations are with respect to the variable $r$. Furthermore,
we assume that

$\psi_{m}(r;k)=(r\sigma(r))^{\frac{1}{2}}\phi_{m}(r;k),$

and then

$\psi_{m}^{{}^{\prime\prime}}+\left\\{\frac{k^{2}\mathbf{n}^{2}}{\sigma}-\frac{m^{2}}{r^{2}}+\frac{\sigma^{2}+r^{2}\sigma^{\prime
2}+2r\sigma\sigma^{\prime}}{4r^{2}\sigma^{2}}-\frac{2\sigma^{\prime}+r\sigma^{\prime\prime}}{2r\sigma}\right\\}\psi_{m}=0,$

which implies

$\psi_{m}^{{}^{\prime\prime}}+\left\\{\left(\frac{k^{2}\mathbf{n}^{2}}{\sigma}+\frac{\sigma^{\prime
2}}{4\sigma^{2}}-\frac{\sigma^{\prime\prime}}{2\sigma}\right)-\frac{\sigma^{\prime}}{2\sigma
r}-\left(m^{2}-\frac{1}{4}\right)\frac{1}{r^{2}}\right\\}\psi_{m}=0.$ (3.5)

For the subsequent use, we let $j_{m,s}$ and $j^{\prime}_{m,s}$ denote the
$s$-th positive root of $J_{m}(t)$ and $J^{\prime}_{m}(t)$, respectively, that
are arranged according to the magnitudes [31].

######  Lemma 3.1.

Under Assumption A, for any $m\geq 1$ the function $\psi_{m}(r;k)$ in (3.5)
possesses at least one zero point in
$\left(\frac{j_{m,s_{0}}}{k\sqrt{M_{1}}},\frac{j_{m,s_{0}+1}}{k\sqrt{M_{1}}}\right)$
for any $s_{0}\in\mathbb{N}$.

###### Proof.

The condition (A1) guarantees the non-degeneracy of $u$ in (3.1). Let

$\left\\{\begin{array}[]{lll}P_{m}(r;k)&=&M_{1}k^{2}-(m^{2}-\frac{1}{4})\frac{1}{r^{2}},\\\
Q_{m}(r;k)&=&\left(\frac{k^{2}\mathbf{n}^{2}}{\sigma}+\frac{\sigma^{\prime
2}}{4\sigma^{2}}-\frac{\sigma^{\prime\prime}}{2\sigma}\right)-\frac{\sigma^{\prime}}{2\sigma
r}+\left(m^{2}-\frac{1}{4}\right)\frac{1}{r^{2}},\end{array}\right.$ (3.6)

and it follows from (A2)-(A5) that

$P_{m}(r;k)\leq Q_{m}(r;k).$ (3.7)

Consider the following differential equations

$\displaystyle(\mathbf{P}1)~{}~{}\qquad\xi_{m}^{\prime\prime}+P_{m}(r;k)\xi_{m}=0,$
(3.8)
$\displaystyle(\mathbf{P}2)\qquad\psi_{m}^{\prime\prime}+Q_{m}(r;k)\psi_{m}=0.$
(3.9)

Then for each fixed $m\in\mathbb{N}$ , the solutions of the $(\mathbf{P}1)$
are denoted by $J_{m}(k\sqrt{M_{1}}r)$, whose roots are given by
$\\{\frac{j_{m,s_{0}}}{k\sqrt{M_{1}}}\\}_{s_{0}\in\mathbb{N}}$. It follows
from the Singular Strum theorem in [1] and (3.7) that the solution of
$(\mathbf{P}2)$ has at least one zero in
$\left(\frac{j_{m,s_{0}}}{k\sqrt{M_{1}}},\frac{j_{m,s_{0}+1}}{k\sqrt{M_{1}}}\right)$
for any $s_{0}\in\mathbb{N}$, denoted by $t_{m,s_{0}}(k)$; see the dashed line
in Fig 3. ∎

Figure 3. Schematic demonstration of the curve $t_{m,s_{0}}(k)$ and the point
$z_{m,s_{0}}$.

Since the solutions of $(\mathbf{P}1)$ and $(\mathbf{P}2)$ are continuous with
respect to the parameter $k$, there exists a number
$z_{m,s_{0}}\in\left(\frac{j_{m,s_{0}}}{\sqrt{M_{1}}},\frac{j_{m,s_{0}+1}}{\sqrt{M_{1}}}\right)$
such that

$\psi_{m}(1;z_{m,s_{0}})=0,$ (3.10)

which is the intersection of the dashed line $t_{m,s_{0}}(k)$ and $r=1$ in Fig
3.

Let $\beta_{m}=1$ in (3.3). Using the transmission condition, it holds that

$\alpha_{m}=\frac{J_{m}(k)}{\phi_{m}(1;k)}.$

Set $f_{m}(k)=\sigma(1)\frac{\partial\phi_{m}(1;k)}{\partial
r}J_{m}(k)-kJ^{\prime}_{m}(k)\phi_{m}(1;k)$. Using the recursive formula of
Bessel functions [17], we have

$f_{m}(k)=\sigma(1)\frac{\partial\phi_{m}(1;k)}{\partial
r}J_{m}(k)+(mJ_{m}(k)-kJ_{m-1}(k))\phi_{m}(1;k),\quad m\geq 1.$ (3.11)

Next, we find the roots of $f_{m}(k)$ on the interval
$\left(\frac{j_{m,s_{0}}}{\sqrt{M_{1}}},\frac{j_{m,s_{0}+2}}{\sqrt{M_{1}}}\right)$.

######  Lemma 3.2.

Under Assumption A, for any given $s_{0}\in\mathbb{N}$, there exists
$m_{0}(\mathbf{n},\sigma)\in\mathbb{N}$, depending on $\mathbf{n}$ and
$\sigma$, such that when $m>m_{0}(\mathbf{n},\sigma)$, the function $f_{m}(k)$
in (3.11) possesses at least one root in
$\left(\frac{j_{m,s_{0}}}{\sqrt{M_{1}}},\frac{j_{m,s_{0}+2}}{\sqrt{M_{1}}}\right)$.

###### Proof.

In the following, we let $a_{s}$ denote the $s$-th negative zero of the Airy
function [35]:

$a_{s}=-\left[\frac{3\pi}{8}(4s-1)\right]^{2/3}(1+\sigma_{s}),$

where $\sigma_{s}$ satisfies the following estimate:

$0\leq\sigma_{s}\leq 0.130\left[\frac{3\pi}{8}(4s-1.051)\right]^{-2}.$

By (1.2) in [35], it holds that

$m-\frac{a_{s}}{2^{1/3}}m^{1/3}<j_{m,s}<m-\frac{a_{s}}{2^{1/3}}m^{1/3}+\frac{3}{20}a_{s}^{2}\frac{2^{1/3}}{m^{1/3}},$
(3.12)

where we recall that $j_{m,s}$ denotes the $s$-th positive root of $J_{m}(t)$.
For each fixed $s_{0}\in\mathbb{N}$, when $m(s_{0})$ is large enough, we have

$\frac{j_{m,s_{0}}}{\sqrt{M_{1}}}<j_{m,1}^{\prime}.$

Consider the interval
$\left(\frac{j_{m,s_{0}}}{\sqrt{M_{1}}},\frac{j_{m,s_{0}+2}}{\sqrt{M_{1}}}\right)$,
there exist at least two consecutive zeros $z_{m,s_{0}}$ and $z_{m,s_{0}+1}$
of $\psi_{m}(1;k)$, as well as $\phi_{m}(1;k)$. This result together with the
monotonicity of $J_{m}$ in the interval $[0,j_{m,1}^{{}^{\prime}}]$ readily
yields that

$\begin{array}[]{ll}J_{m}(z_{m,s_{0}})J_{m}(z_{m,s_{0}+1})<J_{m}^{2}(j_{m,1}^{{}^{\prime}}).\end{array}$
(3.13)

By virtue of (3.13) and (3.11), it can be directly verified that

$\begin{split}&f_{m}(z_{m,s_{0}})f_{m}(z_{m,s_{0}+1})\\\
=&\sigma^{2}(1)J_{m}(z_{m,s_{0}})J_{m}(z_{m,s_{0}+1})\frac{\partial\phi_{m}(1;z_{m,s_{0}})}{\partial
r}\frac{\partial\phi_{m}(1;z_{m,s_{0}+1})}{\partial r}\\\
<&\sigma^{2}(1)J_{m}^{2}(j_{m,1}^{\prime})\frac{\partial\phi_{m}(1;z_{m,s_{0}})}{\partial
r}\frac{\partial\phi_{m}(1;z_{m,s_{0}+1})}{\partial r}.\end{split}$ (3.14)

Since $z_{m,s_{0}}$, $z_{m,s_{0}+1}$ are two consecutive roots,
$\frac{\partial\phi_{m}(1;z_{m,s_{0}})}{\partial r}$ and
$\frac{\partial\phi_{m}(1;z_{m,s_{0}+1})}{\partial r}$ have opposite signs.
Hence, we have

$f_{m}(z_{m,s_{0}})f_{m}(z_{m,s_{0}+1})<0.$ (3.15)

By applying Rolle’s theorem to (3.15), we immediately see that there exists at
least one zero point of $f_{m}(k)$ in $(z_{m,s_{0}},z_{m,s_{0}+1})$.

The proof is complete. ∎

Lemma 3.2 shows the existence of transmission eigenvalues to (3.1) associated
with spherically stratified media. It is noted that in [26], transmission
eigenvalues were calculated in a similar setup but assuming $\sigma\equiv 1$.
In what follows, for a fixed $s_{0}$, we let the transmission eigenvalue be
denoted by

$k_{l_{m}}:=k_{m,s_{0}}\in(z_{m,s_{0}},z_{m,s_{0}+1}),\quad
m=m_{0}+1,m_{0}+2,...,$ (3.16)

where $m_{0}=m_{0}(\mathbf{n},\sigma)$ is sufficiently large.

######  Theorem 3.1.

Consider the same setup as Theorem 2.1 in $\mathbb{R}^{2}$ and assume that
Assumption A holds. Let $(u_{m},v_{m})$ be the pair of eigenfunctions
associated with $k_{l_{m}}$ in (3.16). We have

$\lim\limits_{m\rightarrow\infty}\frac{\|v_{m}\|_{L^{2}(\Omega_{\tau})}}{\|v_{m}\|_{L^{2}(\Omega)}}=0.$
(3.17)

###### Proof.

Let $\beta_{m}=1$ in (3.2). By direct calculations, we have

$\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}=\int_{\Omega_{\tau}}\left|J_{m}(k_{l_{m}}|x|)\right|^{2}\mathrm{d}x=2\pi\int_{0}^{\tau}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r,$

which in particular gives that

$\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}=2\pi\int_{0}^{1}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r\
\text{.}$

Combining

$\frac{j_{m,s_{0}}}{\sqrt{M_{1}}}\leq z_{m,s_{0}}\leq k_{l_{m}}\leq
z_{m,s_{0}+1}\leq\frac{j_{m,s_{0}+2}}{\sqrt{M_{1}}},\quad
m=m_{0}+1,m_{0}+2,...$

and the estimate (3.12), we know that for any $s_{0}\in\mathbb{N}$, there
exists $m_{1}(M_{1},\sigma)\in\mathbb{N}$ such that when
$m>m_{1}(M_{1},\sigma)$, the following estimate holds

$\frac{1}{\sqrt{M_{1}}}\leq\frac{k_{l_{m}}}{m}\leq\frac{1+\sqrt{M_{1}}}{2\sqrt{M_{1}}}.$

Similar to the arguments in the proof of Theorem 2.6 in [20], for any
$\tau\in(0,1)$, there exists $\delta(\tau,M_{1},\sigma)>0$ such that

$\begin{split}\frac{\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}}&\leq\displaystyle\frac{\int_{0}^{\tau}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{1}rJ^{2}_{m}(k_{l_{m}}r)\mathrm{d}r}\leq\displaystyle\frac{\tau^{2}J^{2}_{m}(k_{l_{m}}\tau)}{\displaystyle\frac{\frac{1}{2}J^{3}_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})+2k_{l_{m}}J^{\prime}_{m}(k_{l_{m}})}}\\\
&\leq\displaystyle
2\tau^{2}\left(\frac{J_{m}(k_{l_{m}}\tau)}{J_{m}(k_{l_{m}})}\right)^{2}\left(1+2k_{l_{m}}\frac{J^{\prime}_{m}(k_{l_{m}})}{J_{m}(k_{l_{m}})}\right)\\\
&\leq\displaystyle
36\left(\sqrt{M_{1}}-1\right)m^{4}\tau^{2}\left(\frac{J_{m}(k_{l_{m}}\tau)}{J_{m}(k_{l_{m}})}\right)^{2}\\\
&\leq\displaystyle
144\frac{\sqrt{M_{1}}}{(\sqrt{M_{1}}-1)^{2}}m^{4}\tau^{2}\Big{(}1-\delta(\tau,M_{1},\sigma)\Big{)}^{2m},\end{split}$

which readily implies (3.17).

The proof is complete. ∎

###  3.2. Three-dimensional result

The proof of Theorem 2.1 in three dimensions follows a similar argument to
that of the two-dimensional case in Theorem 3.1 . In what follows, we only
sketch the necessary modifications in what follows.

In three dimensions, we let
$x=(r\cos\theta_{1},r\sin\theta_{1}\cos\theta_{2},r\sin\theta_{1}\sin\theta_{2})\in\mathbb{R}^{3}$
denote the polar coordinate, where $r=|x|$, $\theta_{1}\in[0,\pi]$ and
$\theta_{2}\in[0,2\pi]$. Let $Y_{m}^{l}$ be the spherical harmonic function of
order $m$ and degree $l$, and $j_{m}(|x|):=\sqrt{\frac{\pi}{2r}}J_{m+1/2}(r)$
be the spherical Bessel function [17, 31]. The solutions to (3.1) in
$\mathbb{R}^{3}$ have the following Fourier expansions:

$\displaystyle u(x)$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}\alpha_{m}^{l}\phi_{m}(r;k)Y_{m}^{l}(\hat{x}),$
(3.18) $\displaystyle v(x)$
$\displaystyle=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}\beta_{m}^{l}j_{m}(kr)Y_{m}^{l}(\hat{x}),$

where $\alpha_{l}^{m},\beta_{l}^{m}\in\mathbb{C}$. Set

$u_{m}(x)=\alpha_{m}\phi_{m}(r;k)Y_{m}^{l}(\hat{x}),\quad
v_{m}(x)=\beta_{m}j_{m}(kr)Y_{m}^{l}(\hat{x}).$ (3.19)

Then we have

$\sigma\phi_{m}^{\prime\prime}+\left(\sigma^{\prime}+\frac{2\sigma}{r}\right)\phi_{m}^{\prime}+\left(k^{2}\mathbf{n}^{2}-\frac{m(m+1)}{r^{2}}\sigma\right)\phi_{m}=0.$
(3.20)

where the derivatives are associated to the variable $r$. Furthermore, we
assume that

$\psi_{m}(r;k)=r\sigma^{\frac{1}{2}}(r)\phi_{m}(r;k).$ (3.21)

Then

$\psi_{m}^{\prime\prime}+\left(\frac{k^{2}\mathbf{n}^{2}}{\sigma}-\frac{m(m+1)}{r^{2}}+\frac{\sigma^{\prime
2}}{4\sigma^{2}}-\frac{\sigma^{\prime}}{r\sigma}-\frac{\sigma^{\prime\prime}}{2\sigma}\right)\psi_{m}=0,$
(3.22)

which implies

$\psi_{m}^{\prime\prime}+\left\\{\left(\frac{k^{2}\mathbf{n}^{2}}{\sigma}+\frac{\sigma^{\prime
2}}{4\sigma^{2}}-\frac{\sigma^{\prime\prime}}{2\sigma}\right)-\frac{\sigma^{\prime}}{\sigma
r}-m\left(m+1\right)\frac{1}{r^{2}}\right\\}\psi_{m}=0.$ (3.23)

######  Lemma 3.3.

Under Assumption A, for any $m\geq 1$ the function $\psi_{m}(r;k)$ in (3.21)
possesses at least one zero point in
$\left(\frac{j_{m+\frac{1}{2},s_{0}}}{k\sqrt{M_{1}}},\frac{j_{m+\frac{1}{2},s_{0}+1}}{k\sqrt{M_{1}}}\right)$
for any $s_{0}\in\mathbb{N}$.

###### Proof.

The condition (A1) guarantees the non-degeneracy of $u$ in equation (3.1). Let

$\left\\{\begin{array}[]{lll}P_{m}(r;k)&=&M_{3}k^{2}-m(m+1)\frac{1}{r^{2}},\\\
Q_{m}(r;k)&=&\left(\frac{k^{2}\mathbf{n}^{2}}{\sigma}+\frac{\sigma^{\prime
2}}{4\sigma^{2}}-\frac{\sigma^{\prime\prime}}{2\sigma}\right)-\frac{\sigma^{\prime}}{\sigma
r}+m(m+1)\frac{1}{r^{2}},\end{array}\right.$ (3.24)

and it follows from (A2)-(A5) that

$P_{m}(r;k)\leq Q_{m}(r;k).$ (3.25)

Consider the following differential equations:

$\displaystyle(\mathbf{P}1)~{}~{}\qquad\xi_{m}^{\prime\prime}+P_{m}(r;k)\xi_{m}=0,$
(3.26)
$\displaystyle(\mathbf{P}2)\qquad\psi_{m}^{\prime\prime}+Q_{m}(r;k)\psi_{m}=0.$
(3.27)

Then for each fixed $m\in\mathbb{N}$ , the solutions of the $(\mathbf{P}1)$
are denoted by $\sqrt{r}J_{m+\frac{1}{2}}(k\sqrt{M_{1}}r)$, whose roots are
given by
$\\{\frac{j_{{m+\frac{1}{2}},s_{0}}}{k\sqrt{M_{1}}}\\}_{s_{0}\in\mathbb{N}}$.
It follows from the Singular Strum theorem in [1] and (3.7) that the solutions
of $(\mathbf{P}2)$ has at least one zero in
$\left(\frac{j_{{m+\frac{1}{2}},s_{0}}}{k\sqrt{M_{1}}},\frac{j_{{m+\frac{1}{2}},s_{0}+1}}{k\sqrt{M_{1}}}\right)$
for any $s_{0}\in\mathbb{N}$, denoted by $t_{{m+\frac{1}{2}},s_{0}}(k)$. ∎

Since the solutions of $(\mathbf{P}1)$ and $(\mathbf{P}2)$ are continuous with
respect to the parameter $k$, there exists a number
$z_{{m+\frac{1}{2}},s_{0}}\in\left(\frac{j_{{m+\frac{1}{2}},s_{0}}}{\sqrt{M_{1}}},\frac{j_{{m+\frac{1}{2}},s_{0}+1}}{\sqrt{M_{1}}}\right)$,
such that

$\psi_{m}(1;z_{{m+\frac{1}{2}},s_{0}})=0.$ (3.28)

Let $\beta_{m}=1$ in (3.19). For the transmission condition, it holds that

$\alpha_{m}=\frac{\sqrt{\frac{\pi}{2}}J_{m+\frac{1}{2}}(k)}{\phi_{m}(1;k)}.$
(3.29)

Set $f_{m}=\sigma(1)\frac{\partial\phi_{m}(1;k)}{\partial
r}j_{m}(k)-kj_{m}^{\prime}(k)\phi_{m}(1;k)$. Using the recursive formula of
the Bessel functions [17], we have

$f_{m}(k)=\sigma(1)\frac{\partial\phi_{m}}{\partial
r}(k,1)J_{m+\frac{1}{2}}(k)+((m+1)J_{{m+\frac{1}{2}}}(k)-kJ_{m-\frac{1}{2}}(k))\phi_{m}(k,1),\quad
m\geq 1.$ (3.30)

Next, we find the roots of $f_{m}(k)$ within the interval
$\left(\frac{j_{{m+\frac{1}{2}},s_{0}}}{\sqrt{M_{1}}},\frac{j_{{m+\frac{1}{2}},s_{0}+2}}{\sqrt{M_{1}}}\right)$.

######  Lemma 3.4.

Under Assumption A, for any given $s_{0}\in\mathbb{N}$, there exists
$m_{0}(\mathbf{n},\sigma)\in\mathbb{N}$, depending on $\mathbf{n}$ and
$\sigma$, such that when $m>m_{0}(\mathbf{n},\sigma)$, the function $f_{m}(k)$
in (3.30) possesses at least one root in
$\left(\frac{j_{{m+\frac{1}{2}},s_{0}}}{\sqrt{M_{1}}},\frac{j_{{m+\frac{1}{2}},s_{0}+2}}{\sqrt{M_{1}}}\right)$.

###### Proof.

For any given $s_{0}\in\mathbb{N}$, we have for $m(s_{0})\in\mathbb{N}$ large
enough that

$\frac{j_{{m+\frac{1}{2}},s_{0}}}{\sqrt{M_{1}}}<j_{{m+\frac{1}{2}},1}^{\prime}.$
(3.31)

Consider the interval
$\left(\frac{j_{{m+\frac{1}{2}},s_{0}}}{\sqrt{M_{1}}},\frac{j_{{m+\frac{1}{2}},s_{0}+2}}{\sqrt{M_{1}}}\right)$.
There exist at least two consecutive zeros $z_{{m+\frac{1}{2}},s_{0}}$ and
$z_{{m+\frac{1}{2}},s_{0}+1}$ of $\psi_{m}(1;k)$, as well as $\phi_{m}(1;k)$.
Using such a fact, together with the monotonicity of $J_{m+\frac{1}{2}}$
within the interval $[0,j_{{m+\frac{1}{2}},1}^{{}^{\prime}}]$, one can show
that

$\begin{array}[]{ll}J_{{m+\frac{1}{2}}}(z_{{m+\frac{1}{2}},s_{0}})J_{{m+\frac{1}{2}}}(z_{{m+\frac{1}{2}},s_{0}+1})<J_{m+\frac{1}{2}}^{2}(j_{{m+\frac{1}{2}},1}^{{}^{\prime}}).\end{array}$
(3.32)

By virtue of (3.32) and (3.30), one can show that

$\begin{split}&f_{m}(z_{{m+\frac{1}{2}},s_{0}})f_{m}(z_{{m+\frac{1}{2}},s_{0}+1})\\\
=&\sigma^{2}(1)J_{m+\frac{1}{2}}(z_{{m+\frac{1}{2}},s_{0}})J_{m+\frac{1}{2}}(z_{{m+\frac{1}{2}},s_{0}+1})\frac{\partial\phi_{m}(1;z_{{m+\frac{1}{2}},s_{0}})}{\partial
r}\frac{\partial\phi_{m}(1;z_{{m+\frac{1}{2}},s_{0}+1})}{\partial r}\\\
<&\sigma^{2}(1)J_{m+\frac{1}{2}}^{2}(j_{{m+\frac{1}{2}},1}^{\prime})\frac{\partial\phi_{m}(1;z_{{m+\frac{1}{2}},s_{0}})}{\partial
r}\frac{\partial\phi_{m}(1;z_{{m+\frac{1}{2}},s_{0}+1})}{\partial
r}.\end{split}$ (3.33)

Since $z_{{m+\frac{1}{2}},s_{0}}$, $z_{{m+\frac{1}{2}},s_{0}+1}$ are two
consecutive roots, it is clear that
$\frac{\partial\phi_{m}(1;z_{{m+\frac{1}{2}},s_{0}})}{\partial r}$ and
$\frac{\partial\phi_{m}(1;z_{{m+\frac{1}{2}},s_{0}+1})}{\partial r}$ have
opposite signs. Hence,

$f_{m}(z_{{m+\frac{1}{2}},s_{0}})f_{m}(z_{{m+\frac{1}{2}},s_{0}+1})<0.$ (3.34)

By applying Rolle’s theorem to (3.34), one sees that there exists at least one
root of $f_{m}(k)$ in
$(z_{{m+\frac{1}{2}},s_{0}},z_{{m+\frac{1}{2}},s_{0}+1})$.

The proof is complete. ∎

In the following, for a given $s_{0}\in\mathbb{N}$, we let
$k_{l_{m}}:=k_{m,s_{0}}\in(z_{{m+\frac{1}{2}},s_{0}},z_{{m+\frac{1}{2}},s_{0}+1}),$
$m=m_{0}+1,m_{0}+2,...,$ denote the transmission eigenvalues determined in
Lemma 3.4, where $m_{0}=m_{0}(\mathbf{n},\sigma)$ is chosen to be sufficiently
large.

######  Theorem 3.2.

Consider the same setup as Theorem 2.1 in $\mathbb{R}^{3}$ and assume that
$\mathbf{n}>1$ and $\tau\in(0,1)$ is fixed. Let $(w_{m},v_{m})$ be the pair of
eigenfunctions associated with $k_{l_{m}}$ defined above. Then it holds that

$\lim\limits_{m\rightarrow\infty}\frac{\|v_{m}\|_{L^{2}(\Omega_{\tau})}}{\|v_{m}\|_{L^{2}(\Omega)}}=0.$
(3.35)

###### Proof.

Let $\beta_{m}=1$ in (3.2). By direct calculations, one has that

$\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}=\int_{\Omega_{\tau}}\left|j_{m}(k_{l_{m}}|x|)\right|^{2}\mathrm{d}x=\pi^{2}\int_{0}^{\tau}rJ^{2}_{m+\frac{1}{2}}(k_{l_{m}}r),\mathrm{d}r.$

which in particular gives that

$\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}=\pi^{2}\int_{0}^{1}rJ^{2}_{m+\frac{1}{2}}(k_{l_{m}}r)\mathrm{d}r\
\text{.}$

Combining

$\frac{j_{{m+\frac{1}{2}},s_{0}}}{\sqrt{M_{1}}}\leq
z_{{m+\frac{1}{2}},s_{0}}\leq k_{l_{m}}\leq
z_{{m+\frac{1}{2}},s_{0}+1}\leq\frac{j_{{m+\frac{1}{2}},s_{0}+2}}{\sqrt{M_{1}}},\quad
m=m_{0}+1,m_{0}+2,...$

and the estimate (3.12), we know that for any $s_{0}\in\mathbb{N}$, there
exists $m_{3}(M_{1},\sigma)\in\mathbb{N}$ such that when
$m>m_{3}(M_{1},\sigma)$, the following estimate holds

$\frac{1}{\sqrt{M_{1}}}\leq\frac{k_{l_{m}}}{m}\leq\frac{1+\sqrt{M_{1}}}{2\sqrt{M_{1}}}.$

Similar to the arguments in the proof of Theorem 2.6 in [20], for any
$\tau\in(0,1)$, there exists $\delta(\tau,M_{1},\sigma)>0$ such that

$\begin{split}\frac{\left\|v_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|v_{m}\right\|_{L^{2}(\Omega)}^{2}}&\leq\frac{\int_{0}^{\tau}rJ^{2}_{m+\frac{1}{2}}(k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{1}rJ^{2}_{m+\frac{1}{2}}(k_{l_{m}}r)\mathrm{d}r}\leq\displaystyle\frac{\tau^{2}J^{2}_{m+\frac{1}{2}}(k_{l_{m}}\tau)}{\frac{\frac{1}{2}J^{3}_{m+\frac{1}{2}}(k_{l_{m}})}{J_{m+\frac{1}{2}}(k_{l_{m}})+2k_{l_{m}}J^{\prime}_{m+\frac{1}{2}}(k_{l_{m}})}}\\\
&\leq
2\tau^{2}\left(\frac{J_{m+\frac{1}{2}}(k_{l_{m}}\tau)}{J_{m+\frac{1}{2}}(k_{l_{m}})}\right)^{2}\left(1+2k_{l_{m}}\frac{J^{\prime}_{m+\frac{1}{2}}(k_{l_{m}})}{J_{m+\frac{1}{2}}(k_{l_{m}})}\right)\\\
&\leq
36(\sqrt{M_{1}}-1)m^{4}\tau^{2}\left(\frac{J_{m+\frac{1}{2}}(k_{l_{m}}\tau)}{J_{m+\frac{1}{2}}(k_{l_{m}})}\right)^{2}\\\
&\leq
144\frac{\sqrt{M_{1}}}{(\sqrt{M_{1}}-1)^{2}}m^{4}\tau^{2}(1-\delta(\tau,M_{1},\sigma))^{2m},\end{split}$
(3.36)

which readily gives (3.35).

The proof is complete. ∎

##  4\. Proof of Theorem 2.2

This section is devoted to the proof of Theorem 2.2. In what follows, we
assume that $\sigma$ and $\mathbf{n}$ are both constant. We shall construct a
sequence of transmission eigenvalues $k_{l_{m}}$ and prove that the
corresponding eigenfunctions $(u_{m},v_{m})$ are both boundary-localized.

First, we know that the solutions to (3.1) have the following Fourier
representations:

$\displaystyle
u(x)=\sum_{m=0}^{\infty}\alpha_{m}J_{m}(\frac{k\mathbf{n}r}{\sqrt{\sigma}})e^{\mathrm{i}m\theta},\quad
v(x)=\sum_{m=0}^{\infty}\beta_{m}J_{m}(kr)e^{\mathrm{i}m\theta},\quad N=2,$
(4.1) $\displaystyle
u(x)=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}\alpha_{m}^{l}j_{m}(\frac{k\mathbf{n}r}{\sqrt{\sigma}})Y_{m}^{l}(\hat{x}),\quad
v(x)=\sum_{m=0}^{\infty}\sum_{l=-m}^{m}\beta_{m}^{l}j_{m}(kr)Y_{m}^{l}(\hat{x}),\quad
N=3.$

Set

$\displaystyle
u_{m}(x)=\alpha_{m}J_{m}(\frac{k\mathbf{n}r}{\sqrt{\sigma}})e^{\mathrm{i}m\theta},\quad
v_{m}(x)=\beta_{m}J_{m}(kr)e^{\mathrm{i}m\theta},\quad N=2,$ (4.2)
$\displaystyle
u_{m}(x)=\alpha_{m}j_{m}(\frac{k\mathbf{n}r}{\sqrt{\sigma}})Y_{m}^{l}(\hat{x}),\quad
v_{m}(x)=\beta_{m}j_{m}(kr)Y_{m}^{l}(\hat{x}),\quad N=3.$

Let $\beta_{m}=1$ in (4.2), for the transmission condition, then set

$\displaystyle
f^{(2)}_{m}(k)=J_{m}(\frac{k\mathbf{n}}{\sqrt{\sigma}})J^{\prime}_{m}(k)-\sqrt{\sigma}\mathbf{n}J^{\prime}_{m}(\frac{k\mathbf{n}}{\sqrt{\sigma}})J_{m}(k),\quad
N=2,$ (4.3) $\displaystyle
f^{(3)}_{m}(k)=J_{m+\frac{1}{2}}(\frac{k\mathbf{n}}{\sqrt{\sigma}})j^{\prime}_{m}(k)-\sqrt{\sigma}\mathbf{n}j^{\prime}_{m}(\frac{k\mathbf{n}}{\sqrt{\sigma}})J_{m+\frac{1}{2}}(k),\quad
N=3.$

######  Lemma 4.1.

Suppose that $\mathbf{n}>\sqrt{\sigma}$. For any given $s_{0}\in\mathbb{N}$,
there exists $m_{0}(\mathbf{n},\sigma)\in\mathbb{N}$, depending on
$\mathbf{n}$ and $\sigma$, such that when $m>m_{0}(\mathbf{n},\sigma)$,
$f^{(2)}_{m}(k)$ in (3.30) possesses at least one root in
$\left(\frac{\sqrt{\sigma}j_{{m},s_{0}}}{\mathbf{n}},\frac{\sqrt{\sigma}j_{{m},s_{0}+1}}{\mathbf{n}}\right)$
and $f^{(3)}_{m}(k)$ in (3.30) possesses at least one root in
$\left(\frac{\sqrt{\sigma}j_{{m+\frac{1}{2}},s_{0}}}{\mathbf{n}},\frac{\sqrt{\sigma}j_{{m+\frac{1}{2}},s_{0}+1}}{\mathbf{n}}\right)$.

###### Proof.

First, we have by direct calculations that

$\displaystyle
f^{(2)}_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}}}{\mathbf{n}}\right)f^{(2)}_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}+1}}{\mathbf{n}}\right)$
(4.4) $\displaystyle=$
$\displaystyle\sigma\mathbf{n}^{2}J_{m-1}\left(j_{m,s_{0}})J_{m-1}(j_{m,s_{0}+1}\right)\cdot
J_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}}}{\mathbf{n}}\right)J_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}+1}}{\mathbf{n}}\right),$
$\displaystyle
f^{(3)}_{m}\left(\frac{\sqrt{\sigma}j_{{m+\frac{1}{2}},s_{0}}}{\mathbf{n}}\right)f^{(3)}_{m}\left(\frac{\sqrt{\sigma}j_{{m+\frac{1}{2}},s_{0}+1}}{\mathbf{n}}\right)$
$\displaystyle=$
$\displaystyle\sigma\mathbf{n}^{2}J_{m-\frac{1}{2}}(j_{m+\frac{1}{2},s_{0}})J_{m-\frac{1}{2}}(j_{m+\frac{1}{2},s_{0}+1})\cdot
J_{m+\frac{1}{2}}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}}}{\mathbf{n}}\right)J_{m+\frac{1}{2}}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}+1}}{\mathbf{n}}\right).$

For a fixed $s_{0}\in\mathbb{N}$, we have for $m(s_{0})$ large enough that

$\displaystyle
J_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}}}{\mathbf{n}}\right)J_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}+1}}{\mathbf{n}}\right)<J^{2}_{m}(j^{\prime}_{m,1}),$
(4.5) $\displaystyle
J_{m+\frac{1}{2}}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}}}{\mathbf{n}}\right)J_{m+\frac{1}{2}}\left(\frac{\sqrt{\sigma}j_{{m+\frac{1}{2}},s_{0}+1}}{\mathbf{n}}\right)<J^{2}_{m+\frac{1}{2}}(j^{\prime}_{m+\frac{1}{2},1}).$

Note that the zeros of $J_{\nu}$ are interlaced with those of $J_{\nu+1}$ for
$\nu\in\mathbb{R}_{+}$ (cf. [31]). One clearly has

$f^{(2)}_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}}}{\mathbf{n}}\right)f^{(2)}_{m}\left(\frac{\sqrt{\sigma}j_{{m},s_{0}+1}}{\mathbf{n}}\right)<0,\
\
f^{(3)}_{m}\left(\frac{\sqrt{\sigma}j_{{m+\frac{1}{2}},s_{0}}}{\mathbf{n}}\right)f^{(3)}_{m}\left(\frac{\sqrt{\sigma}j_{{m+\frac{1}{2}},s_{0}+1}}{\mathbf{n}}\right)<0,$
(4.6)

which readily yields the desired results in the statement of the lemma by
Rolle’s theorem. ∎

######  Theorem 4.1.

Assume that $\mathbf{n}^{2}>\sigma$ and $\tau\in(0,1)$ is fixed. Let
$(u_{m},v_{m})$ be the pair of eigenfunctions associated with $k_{l_{m}}$ in
(4.2). We have

$\lim\limits_{m\rightarrow\infty}\frac{\|u_{m}\|_{L^{2}(\Omega_{\tau})}}{\|u_{m}\|_{L^{2}(\Omega)}}=0,\quad\lim\limits_{m\rightarrow\infty}\frac{\|v_{m}\|_{L^{2}(\Omega_{\tau})}}{\|v_{m}\|_{L^{2}(\Omega)}}=0.$
(4.7)

###### Proof.

For $v_{m}$, it is similar to the nonconstant case in the previous section.
Hence, we only prove that $u_{m}$ is boundary-localized in the two dimensions
and the three-dimensional case can be proved by following similar arguments.

Since
$u_{m}(x)=J_{m}(\frac{k\mathbf{n}r}{\sqrt{\sigma}})e^{\mathrm{i}m\theta}$, one
has

$\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}>\frac{\mathbf{n}}{\sqrt{\sigma}}\frac{\sqrt{\sigma}j_{m,s_{0}}}{\mathbf{n}}>m(1+2(\frac{s_{0}+1}{m})^{\frac{2}{3}}).$
(4.8)

Next, we show that for $m$ sufficiently large, it holds that
$\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}\tau<j^{\prime}_{m,1}$. In fact, one
can first deduce that

$\frac{m\sqrt{\sigma}}{\mathbf{n}k_{l_{m}}}<\frac{j^{\prime}_{m,1}\sqrt{\sigma}}{\mathbf{n}k_{l_{m}}}<\frac{m(1+3(\frac{1}{m})^{\frac{2}{3}}+2(\frac{1}{m})^{\frac{4}{3}})}{m(1+2(\frac{s_{0}+1}{m})^{\frac{2}{3}})},$
(4.9)

which gives that

$\lim\limits_{m\rightarrow\infty}\frac{\sqrt{\sigma}j^{\prime}_{m,1}}{\mathbf{n}k_{l_{m}}}=1.$
(4.10)

Hence, for $\varepsilon=\frac{1}{2}(1-\tau)$, there exists a
$m_{0}\in\mathbb{N}$ such that when $m>m_{0}$, we have

$\frac{\sqrt{\sigma}j^{\prime}_{m,1}}{\mathbf{n}k_{l_{m}}}>1-\varepsilon>\frac{1}{2}(1+\tau)>\tau.$
(4.11)

That is, $\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}\tau<j^{\prime}_{m,1}$.
Next similar to the arguments in the proof of Theorem 2.6 in [20], we have

$\begin{array}[]{ll}\frac{\left\|u_{m}\right\|_{L^{2}(\Omega_{\tau})}^{2}}{\left\|u_{m}\right\|_{L^{2}(\Omega)}^{2}}&\leq\frac{\int_{0}^{\tau}rJ^{2}_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}r)\mathrm{d}r}{\int_{0}^{1}rJ^{2}_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}r)\mathrm{d}r}\\\
&\leq\displaystyle\frac{\tau^{2}J^{2}_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}\tau)}{\frac{\frac{1}{2}J^{3}_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}})}{J_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}})+2k_{l_{m}}J^{\prime}_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}})}}\\\
&\leq
2\tau^{2}\left(\frac{J_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}\tau)}{J_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}})}\right)^{2}\left(1+2k_{l_{m}}\frac{J^{\prime}_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}})}{J_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}})}\right)\\\
&\leq
36(\sqrt{M_{1}}-1)m^{4}\tau^{2}\left(\frac{J_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}}\tau)}{J_{m}(\frac{\mathbf{n}}{\sqrt{\sigma}}k_{l_{m}})}\right)^{2}\\\
&\leq
144\frac{\sqrt{M_{1}}}{(\sqrt{M_{1}}-1)^{2}}m^{4}\tau^{2}(1-\delta(\tau,M_{1},\sigma))^{2m}.\end{array}$
(4.12)

which readily proves the first limit in (4.7).

The proof is complete.

∎

## Acknowledgments

The research of H. Liu was supported by the Hong Kong RGC General Research
Funds (projects 11300821, 12301420 and 12302919) and NSFC/RGC Joint Research
Fund (project N_CityU101/21). The research of J. Zhang was supported by the
Natural Science Foundation of Jiangsu Province (grant no. BK20210540), the
Natural Science Foundation of the Jiangsu Higher Education Institutions of
China (grant no. 21KJB110015). The research of Y. Jiang and K. Zhang was
supported in part by China Natural National Science Foundation (grant no.
11871245 and 11971198), the National Key R&D Program of China (grant no.
2020YFA0713601), and by the Key Laboratory of Symbolic Computation and
Knowledge Engineering of Ministry of Education, Jilin University, China.

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# Generic free fermions with nearest neighbour interactions

Jonathon Riddell  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK  School of Physics and Astronomy,
University of Nottingham, Nottingham, NG7 2RD, UK  Centre for the Mathematics
and Theoretical Physics of Quantum Non-Equilibrium Systems, University of
Nottingham, Nottingham, NG7 2RD, UK  Bruno Bertini  School of Physics and
Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7
2RD, UK  Centre for the Mathematics and Theoretical Physics of Quantum Non-
Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK

###### Abstract

Generic free fermions are free fermions with a single particle spectrum that
satisfies the $q$ no resonance condition, i.e., where equal sums of single-
particle energies are unique. This property guaranties that they have no
degeneracies and gives them relaxation properties more similar to those of
generic systems. In this article we provide a minimal example of a generic
free fermionic model with nearest neighbour interactions — a tight-binding
model with complex hopping. Using some standard results from number theory we
prove that this model fulfils the $q$ no resonance condition when the number
of lattice sites is prime. Whenever this is not the case one can recover the
$q$ no resonance condition by adding hopping terms between sites corresponding
to the divisors of the number of sites. We further discuss its many-body
spectral statistics and show that local probes — like the ratio of consecutive
level spacings — look very similar to what is expected for the Poisson
statistics. We however demonstrate that free fermion models can never have
Poisson statistics with an analysis of the moments of the spectral form
factor.

##  I Introduction

In the last two decades there has been a large amount of progress classifying
and understanding the dynamics generated by isolated quantum many body
Hamiltonians. A rich literature is forming discussing properties of quantum
systems in the context of equilibration, thermalization, and information
spreading Polkovnikov  _et al._ (2011); Gogolin and Eisert (2016); Calabrese
_et al._ (2016); Vidmar and Rigol (2016); Essler and Fagotti (2016); Doyon
(2020); Bastianello  _et al._ (2022); Alba  _et al._ (2021); Ueda (2020);
Fisher  _et al._ (2023). This also lead us to revisit concepts like quantum
chaos, integrability, and localization in the quantum many-body setting Hayden
and Preskill (2007); Sekino and Susskind (2008); Kos  _et al._ (2018); Chan
_et al._ (2018a); Luca D’Alessio and Rigol (2016); Abanin  _et al._ (2019);
Sierant  _et al._ (2024).

The field has made several advances in terms of understanding late time
dynamics and the corresponding equilibrium values of dynamical quantities such
as expectation values, correlation functions, entanglement measures, out of
time ordered correlators and more. Traditionally there have been two main
approaches for characterising these quantities. The first consists in studying
the problem directly in the thermodynamic limit and then explicitly take the
limit of large times. This approach is particularly convenient for analytic
studies and relies on some special simplifications occurring in the
thermodynamic limit Essler and Fagotti (2016); Fisher  _et al._ (2023). The
second approach instead consists of studying infinite time averages in finite
systems. This second approach is more suited to numerical investigations and
does not require any special simplification. Instead, late time properties are
studied through dephasing arguments based on some typicality assumptions on
the spectrum of the system and, sometimes, the functional form of the
observables in the energy eigenbasis. Generally one assumes the $q$ no-
resonance condition Mark  _et al._ (2023); Riddell  _et al._ (2023); Kaneko
_et al._ (2020); Riddell and Pagliaroli (2023); Srednicki (1999)

$\sum_{j\in M}E_{j}=\sum_{k\in N}E_{k}\,\,\text{ s.t }\,\,|M|=|N|\implies
M=N,$ (1)

where $M,N$ are sets of indices with $q$ entries, identifying the energy
values of the model. This allows one to make progress when taking the infinite
time average. Such a simple assumption on the spectrum allows one to make
strong statements on equilibration and quantum recurrence times Riddell  _et
al._ (2023) with fixed $q$ assumptions being the most common in the literature
Huang and Harrow (2020); Riddell and Sørensen (2020); Goldstein  _et al._
(2013); García-Pintos  _et al._ (2017); Wilming  _et al._ (2018); Torres-
Herrera and Santos (2014); Chen  _et al._ (2023); Lezama  _et al._ (2023);
Reimann and Kastner (2012); Short (2011); Alhambra  _et al._ (2020). Late time
properties can typically be deduced with few additional assumptions like
eigenstate typicality/thermalization, chaos, large inverse participation
ratios etc Short (2011); Alhambra  _et al._ (2020); Huang  _et al._ (2017);
Yoshida and Yao (2019); Srednicki (1999). Eq. (1) is a strong assumption on
the spectrum and is generally assumed to hold for quantum chaotic models.
Recently the spectral statistics of (1) was studied, and indeed the local
spectral statistics for many body chaotic (or random) Hamiltonians for the
condition when $q>1$ are Poissonian, indicating that violations of the
conditions should be rare or non-existent Riddell and Pagliaroli (2023).
Probing whether or not a given model violates the condition in Eq. (1) is
however generically unfeasible, both analytically and numerically.

In this document we focus our attention on a more analytically tractable set
of models: free fermionic systems, whose spectral properties have lately been
at a centre of a resurgence of interest Winer  _et al._ (2020); Liao  _et al._
(2020); Łydżba  _et al._ (2021a, b); Šuntajs  _et al._ (2023). Specifically,
we consider free fermion models defined by

$\hat{H}=\sum_{n,m=1}^{L}M_{n,m}\hat{f}_{n}^{\dagger}\hat{f}^{\phantom{{\dagger}}}_{m},$
(2)

with $M$ Hermitian. One can always solve this model by diagonalising the
matrix $M=O\epsilon O^{\dagger}$, where $O$ is unitary and $\epsilon$ is a
diagonal matrix with real entries. Therefore we can write

$\hat{H}=\sum_{k=1}^{L}\epsilon_{k}d_{k}^{\dagger}d^{\phantom{{\dagger}}}_{k},$
(3)

where $d_{k}^{\dagger}=\sum_{n=1}^{L}O_{n,k}\hat{f}_{n}^{\dagger}$. We make
two demands of this model. First we ask it to have no non-trivial resonances
in the eigenmode spectrum. This property is referred to as the single particle
$q$ no resonance condition. More precisely, letting $R_{q}$ and
$R_{q}^{\prime}$ be two sets of exactly $q$ eigenmodes $\epsilon_{j}$, we say
the model satisfies the single particle $q$ no resonance condition if

$\sum_{j\in R_{q}}\epsilon_{j}=\sum_{j\in
R_{q}^{\prime}}\epsilon_{j}\quad\implies\quad R_{q}=R_{q}^{\prime}\,.$ (4)

The second property we require from our free model is extensivity, i.e.,

$O_{n,k}=\frac{c_{n,k}}{\sqrt{L}},$ (5)

with $c_{n,k}=\mathcal{O}(1)$. throughout this text we refer to models that
satisfy these conditions as generic free fermion models.

These two properties imply the free model has exponentially long quantum
recurrence times, and the concentration of equilibration is exponentially
tight in system size Riddell  _et al._ (2023). This is to be compared to the
typical tight binding model which has a quantum recurrence time that is linear
in size system and a concentration around equilibrium that is a power law
Lezama  _et al._ (2023); Venuti and Zanardi (2013); Solano-Carrillo (2015);
Riddell and Sørensen (2020); Kaminishi  _et al._ (2015); Kaminishi and Mori
(2019); Gluza2016; Malabarba2015. Therefore, a model that satisfies Eqs. (4)
and (5) has late time properties more similar to those of a generic system
than typically studied free ones. Although generic free models of this kind
can be constructed relatively easily if one allows for long range couplings,
the open question is whether such a model can be found if we restrict to local
interactions. In this article we resolve this question by providing the
example of a “minimal” generic free fermion model with nearest neighbour
interactions.

The model we construct has a non-degenerate many-body spectrum, a property
that the tight binding model lacks, and a relevant question concerns the
nature of its spectral correlations. First we show that under the ratio test
its spectrum is compatible with an underlying Poissonian statistics of the
energy levels. Therefore the system seems to follow the Berry Tabor conjecture
Berry and Tabor (1977) like interacting integrable systems. However, analysing
its spectral form factor, and more precisely its higher moments, we show that
this is not the case. The spectral statistics is Poissonian only at the level
of two point correlations: higher point correlations can never be Poissonian
in free systems.

The rest of this manuscript is laid out as follows. In Sec. II we construct
the minimal generic free fermion model and show that it satisfies (4) and (5).
In Sec. III we show that, contrary to the tight binding model, its level
spacing distribution appears Poissonian. In Sec. IV we characterise its
spectral statistics using the spectral form factor and its higher moments.
Finally, Sec. V contains our conclusions.

##  II Minimal model

In this section we show that to achieve a generic free fermion model it is
sufficient to add a parity-breaking term to the tight binding model. Namely we
consider

$\hat{H}=\frac{\alpha+i\beta}{2}\sum_{n=1}^{L}\hat{f}_{n}^{\dagger}\hat{f}^{\phantom{{\dagger}}}_{n+1}+\frac{\gamma}{2}\sum_{n=1}^{L}\hat{f}_{n}^{\dagger}\hat{f}^{\phantom{{\dagger}}}_{n}+{\rm
h.c.}.$ (6)

In the spin language (i.e. via a standard Jordan Wigner transformation) this
system is mapped into an XX chain with a magnetic field and a
Dzyaloshinskii–Moriya interaction of the form

$\propto\sum_{j}\sigma^{x}_{j}\sigma^{y}_{j+1}-\sigma^{x}_{j}\sigma^{y}_{j+1}.$
(7)

Because of the translational invariance of (6), the diagonal form (3) is
simply attained by Fourier transform, i.e. by choosing

$O_{n,k}=\frac{e^{i\frac{2\pi}{L}kj}}{\sqrt{L}},$ (8)

and gives a dispersion relation of the form

$\epsilon_{k}=\alpha\cos\left({\frac{2\pi}{L}k}\right)+\beta\sin\left({\frac{2\pi}{L}k}\right)+\gamma.$
(9)

Note that the $O_{n,k}$ in Eq. (8) immediately satisfies our second condition
(5). Instead, the validity of (4) is assessed by the following

######  Property 1.

When the couplings $\alpha,\beta$ and $\gamma$ are _incommensurate_ the
dispersion relation (9) fulfils (4) _iff $L$ is prime_.

We begin by giving a more precise definition of incommensurate. Let

$\displaystyle\mathcal{C}_{L}=\left\\{\cos\left(\frac{2\pi}{L}k_{1}\right)+\ldots+\cos\left(\frac{2\pi}{L}k_{n}\right),\right.$
(10)
$\displaystyle\left.\phantom{\left(\frac{2\pi}{L}\right)}n\in\mathbb{Z}_{L},\quad
1\leq k_{1}<\cdots<k_{n}\leq L\right\\},$

be the finite set of points achieved by summing any group of cosine terms
generated by our finite quantised wave-numbers. Likewise we denote by
$\mathcal{S}_{L}$ set of sums of sines. We demand that $\alpha,\beta,\gamma$
be chosen such that for any choice of
$x\in\mathcal{C}_{L},y\in\mathcal{S}_{L}$ we have that ${x\alpha}/{y\beta}$
and $({x\alpha}+{y\beta})/\gamma$ are irrational.

This gives us two important properties. First, sums of cosine and sine terms
will not cancel each other. Second, sums of different numbers of eigenmodes
also cannot be equal. As we now show these properties are enough to ensure
property (4). In fact they lead to an even stronger feature of the single-
particle spectrum called _rational independence_. Namely they imply that

$\sum_{k=1}^{L}a_{k}\epsilon_{k}=0,\qquad a_{k}\in\mathbb{Q},$ (11)

has no solutions except for the trivial case when $a_{k}=0$ for all $k$.

To see this we note that, due to the incommensurability of $\alpha,\beta$ and
$\gamma$, we can break Eq. (11) into the following three equations

$\displaystyle\displaystyle\sum_{k=1}^{L}a_{k}\cos\left(\frac{2\pi}{L}k\right)=0,$
(12)
$\displaystyle\displaystyle\sum_{k=1}^{L}a_{k}\sin\left(\frac{2\pi}{L}k\right)=0,$
(13) $\displaystyle\displaystyle\sum_{k=1}^{L}a_{k}=0.$ (14)

Focussing on the first two equations and combining them into one we arrive at

$P(\xi)\equiv\sum_{k=0}^{L-1}a_{k}\xi^{k}=0,$ (15)

where we set $\xi\equiv\exp\left(i{2\pi}/{L}\right)$. The third equation is
then rewritten as

$P(1)=0.$ (16)

The key observation now is that Eq. (15) defines a polynomial with integer
coefficients that has a zero at the first $L$-th root of unity. This family of
polynomials is well studied in mathematics, with a particularly important role
played by the so called $L$-th _cyclotomic polynomial_ Weisstein (2002), which
we denote by $\Phi_{L}(z)$. The latter is the unique polynomial with roots at
_primitive_ $L$-th roots of unity 111Namely, they are not $N$-roots of unity
for some $N<L$. that is irreducible in $\mathbb{Q}$, i.e. it cannot be
factored as the product of two non-constant polynomials with rational
coefficients.

Recalling some basic facts about the cyclotomic polynomial we have that it has
degree $\phi(L)\leq L-1$, where $\phi(L)$ is Euler’s totient function
Weisstein (2002). In particular, this means that its degree is strictly
smaller than $L-1$ whenever $L$ is not prime. Instead, for prime $L$ one has
Weisstein (2002)

$\Phi_{L}(z)=1+z+\cdots+z^{L-1}\,.$ (17)

Since by definition $\Phi_{L}(z)$ must divide $P(z)$ and the degree of $P(z)$
is at most $L-1$, for $L$ prime we have either $P(z)=0$ or $P(z)=\Phi_{L}(z)$.
Namely we either have $a_{k}=0$ for all $k$ or $a_{k}=1$ for all $k$.
Considering now (16) we see that it is only fulfilled by the former choice.
This proves that, for $L$ prime we have rational independence.

Next, let us show that Eq. (4) does not hold for non-prime $L$ by providing an
explicit counter example. Consider a system size $L=np$, with $p$ prime and
$n>1$, and let $w=\xi^{n}$. Then $w$ is a $p$-th primitive root of unity and,
therefore, it is a zero of the cyclotomic polynomial $\Phi_{p}(z)$. Namely

$\Phi_{p}(w)=w^{p-1}+w^{p-2}+\ldots 1=0.$ (18)

Expressing this equation in terms of $\xi$, we can write it as $Q(\xi)=0$,
where we introduced the polynomial

$Q(z)\equiv z^{n(p-1)}+z^{n(p-2)}+\ldots+1.$ (19)

Since $Q(\xi)=0$ we can multiply any $R(\xi)$ into $Q(\xi)$ giving
$R(\xi)Q(\xi)=0$. We can use this to construct suitable $P(z)$ fulfilling (15)
and (16).

Take for example

$R(z)=z-1,$ (20)

and set $P(z)=R(z)Q(z)$. This is a legitimate choice because $R(z)Q(z)$ has
degree $L-n+1\leq L-1$. Moreover, since $R(1)=0$ we have that this polynomial
also fulfils Eq. (16). ∎

To recover rational independence for non-prime $L$ one should add higher
harmonics to the dispersion relation. In particular, for a generic $L$ a
rational independent dispersion relation takes the form

$\epsilon_{k}=\sum_{d\mid L}\left(\alpha_{d}\cos\left({\frac{2\pi
d}{L}k}\right)+\beta_{d}\sin\left({\frac{2\pi d}{L}k}\right)\right)+\gamma$
(21)

where the sum is over all the divisors of $L$ strictly smaller than $L$ and
$\\{\alpha_{d},\beta_{d}\\}$, and $\gamma$ are again incommensurable. The
corresponding real space Hamiltonian is recovered by again taking the Fourier
transform in Eq. (8). The resulting model can be written similarly to Eq. (6)
with the addition of terms of the form

$\hat{H}_{d}=\frac{\alpha_{d}+i\beta_{d}}{2}\sum_{n=1}^{L}\hat{f}_{n}^{\dagger}\hat{f}^{\phantom{{\dagger}}}_{n+d}+{\rm
h.c.}.$ (22)

This point is discussed more extensively in Appendix A where we provide the
explicit example of $L=pq$ with $p$ and $q$ primes.

##  III Level spacing statistics of the many body spectrum

In this section we investigate the level spacing statistics of the many body
spectrum $E_{n}=\vec{n}\cdot\mathbf{\epsilon}$ where we take $\epsilon$ from
our minimal model in Eq. (9). In particular we perform the ratio test
introduced in Ref. Oganesyan and Huse (2007). This method avoids unfolding the
spectrum while still probing the spectrum’s similarity to random matrix
theory. Suppose we take $E_{n}$ and re-order it such that $E_{k}<E_{k+1}$ for
all $k$. Then we define gaps in the spectrum as $s_{k}=E_{k}-E_{k-1}$. The
ratio test is then concerned with the quantity

$r_{k}=\frac{\min\\{s_{k},s_{k+1}\\}}{\max\\{s_{k},s_{k+1}\\}}.$ (23)

A spectrum obeying Poisson statistics will follow the distribution Atas  _et
al._ (2013)

$p(r)=\frac{2}{(1+r)^{2}}.$ (24)

and have the mean $\langle r\rangle=2\ln 2-1\approx 0.38629436112$.

Figure 1: In the top figure we have a histogram of the ratio test with our
generic single particle spectrum $\epsilon_{k}$ for $L=23$. In purple we plot
the Poisson distribution. On the bottom figure we print the average of the
ratios for various prime system sizes.

In Fig. 1 we see the surprising result that this model looks approximately
Poisson both when calculating $\langle r\rangle$ and when overlaying its
distribution $p(r)$. The model clearly experiences some form of level
attraction but the gaps do not cluster around zero as one expects for a
typical tight binding model. At $L=23$ we observe $\langle r\rangle=0.36936$,
a value somewhat similar the expected Poisson result. One may be tempted to
conclude that the model has Poisson spectral statistics and is just
experiencing relatively slow convergence to that behaviour. We demonstrate in
the next section that this cannot be true by presenting a more extensive probe
of the spectral statistics. For comparison we plot the ratio test for $L=20$,
a non-prime system size, in Fig. 2. The ratios cluster around zero with a
large amount of exact degeneracies.

Figure 2: In this figure we have a histogram of the ratio test with our
generic single particle spectrum $\epsilon_{k}$ for $L=20$. In purple we plot
the Poisson distribution.

Note, in the case of the tight binding model, the vast majority of gaps are
zero and we would have an approximate delta function at $0$.

##  IV Moments of the spectral form factor

A more robust probe of the spectral statistics is given by the so called
spectral form factor (SFF) Kos  _et al._ (2018); Chan  _et al._ (2018b);
Bertini  _et al._ (2018); Friedman  _et al._ (2019); Šuntajs  _et al._ (2020);
Bertini  _et al._ (2021); Moudgalya  _et al._ (2021); Šuntajs  _et al._
(2021); Garratt and Chalker (2021a, b); Bertini  _et al._ (2022); Shivam  _et
al._ (2023); Chan  _et al._ (2022); Šuntajs  _et al._ (2023), which is the
Fourier transform of the spectral two-point function. The SFF is expressed as

$K(t,L)=\mathbb{E}[\left|\operatorname{\textnormal{Tr}}\left[{\left(U_{t}\right)}\right]\right|^{2}],$
(25)

where $U_{t}=e^{i\hat{H}t}$ and $\mathbb{E}[\cdot]$ is an average (either over
an ensemble of similar systems or in time) that removes non-universal features
Prange (1997). Contrary to the level spacing statistics this quantity tests
spectral correlations over all energy scales, not only over short ones.

To probe also higher point correlations among energy levels, here we consider
arbitrary higher moments of the SFF. Namely

$K_{q}(t,L)=\mathbb{E}[\left|\operatorname{\textnormal{Tr}}\left[{\left(U_{t}\right)}\right]\right|^{2q}],$
(26)

and take $\mathbb{E}[\cdot]$ to be the moving time-average over a time window
of size $\tau$

$\mathbb{E}[f(t)]=\frac{1}{\tau}\int_{t}^{t+\tau}f(s){\rm d}s.$ (27)

In fact, to make the expressions more readable we will always consider the
limit of large $\tau$ 222More precisely we take $t\gg\tau\gg 1$..

Let us now compute $K_{q}(t,L)$ in our minimal generic free fermionic model
and compare our results with the predictions of the Poissonian statistics,
which gives the following SFF higher moments for large $L$ Haake (2001)

${K_{q}(t,L)}=2^{qL}q!.$ (28)

We begin by considering $q=1$ and noting that the trace can be expressed as

$\operatorname{\textnormal{Tr}}\left[{\left(U_{t}\right)}\right]=\sum_{m=1}^{2^{L}}e^{it\vec{m}\cdot\vec{\epsilon}},$
(29)

where $\vec{m}$ is the $L$-bit binary representation of $m$ and
${\vec{\epsilon}}=(\epsilon_{1},\ldots,\epsilon_{L})$. This means that we can
write

$\left|\operatorname{\textnormal{Tr}}\left[{\left(U_{t}\right)}\right]\right|^{2}=\sum_{m,n}e^{it(\vec{m}-\vec{n})\cdot\vec{\epsilon}}.$
(30)

A term survives the time average and the $\tau\to\infty$ limit iff

$(\vec{m}-\vec{n})\cdot\vec{\epsilon}=0.$ (31)

This is clearly fulfilled when $\vec{m}=\vec{n}$. To see that this is the only
case we expand it as

$(\vec{m}-\vec{n})\cdot\vec{\epsilon}=\sum_{k}(m_{k}-n_{k})\epsilon_{k}=0.$
(32)

$m_{k}-n_{k}\in\mathbb{Z}$. It then follows that our only choice is for
$m_{k}-n_{k}=0$ because of the rational independence of $\epsilon_{k}$.
Putting all together we then have

${K_{1}(t,L)}=\prod_{k=1}^{L}\sum_{m_{k}}1=2^{L}.$ (33)

Therefore our spectrum gives the first moment expected for the Poisson
ensemble.

Considering now arbitrary higher moments we have

$\left|\operatorname{\textnormal{Tr}}\left[{\left(U_{t}\right)}\right]\right|^{2q}=\sum_{m_{i},n_{i}=1}^{2^{L}}e^{it\mathbf{\epsilon}\cdot\sum_{i=1}^{q}\left(\vec{m}_{i}-\vec{n}_{i}\right)}.$
(34)

We now wish to understand which terms survive time average and infinite $\tau$
limit. This is again when the argument of the exponential is zero

$\mathbf{\epsilon}\cdot\sum_{i=1}^{q}\left(\vec{m}_{i}-\vec{n}_{i}\right)=0,$
(35)

writing this component-wise gives

$\sum_{k=1}^{L}\epsilon_{k}\sum_{i=1}^{q}\left(m_{ik}-n_{ik}\right)=0.$ (36)

The number $\sum_{i=1}^{q}\left(m_{ik}-n_{ik}\right)$ takes integer values in
$[-q,q]$. Therefore, the rational independence of $\\{\epsilon_{k}\\}$ tells
us that the only solution to this equation is

$\sum_{i=1}^{q}\left(m_{ik}-n_{ik}\right)=0,\qquad\forall k\,.$ (37)

This means that there are precisely $q!$ ways for this term to survive the
average and we obtain

${K_{q}(L)}=\prod_{k=1}^{L}\sum_{m_{k}^{(i)}}q!=(q!2^{q})^{L}.$ (38)

This result agrees with Eq. (28) for $q=1$ but disagrees with it for $q>1$.
Note in particular that the disagreement comes because Eq. (38) is larger than
Eq. (28). This happens because, even though the Hamiltonian in Eq. (6) has a
non-degenerate spectrum for $\alpha,\beta,\gamma$ incommensurate and $L$
prime, the spectrum of

$\hat{H}_{q}=\sum_{j=0}^{q-1}\overbrace{I\otimes\cdots\otimes
I}^{j}\otimes\hat{H}\otimes I\otimes\cdots\otimes I,$ (39)

has far more degeneracies than

$\hat{H}_{{\rm int},q}=\sum_{j=0}^{q-1}\overbrace{I\otimes\cdots\otimes
I}^{j}\otimes\hat{H}_{\rm int}\otimes I\otimes\cdots\otimes I\,,$ (40)

where $\hat{H}_{\rm int}$ is an Hamiltonian with a generic Poisson distributed
spectrum, e.g. that of a Bethe Ansatz integrable model. Indeed, whereas
$\hat{H}_{{\rm int},q}$ has an obvious $q!$ degeneracy due to permutation
symmetry, $\hat{H}_{q}$ has a degeneracy of $(q!)^{L}$ because it is invariant
under permutations of each separate mode. Therefore, free fermionic models
cannot display Poisson statistics in the many body spectrum. However, thanks
to the rational independence of $\\{\epsilon_{k}\\}$, one can claim that our
model produces the smallest possible SFF moments for a free fermionic system.

##  V Conclusion

In this article we have identified a locally interacting free fermionic system
— a tight binding model with a parity breaking term — for which we could prove
rational independence of the single-particle spectrum, i.e., that for fixed
system size its energy eigenvalues are linearly independent over rational
numbers. Free fermionic systems with this property are known to have special
late time behaviour including “interacting-like” concentration of
equilibration and quantum recurrence times and are sometimes referred to as
generic free fermions: Here we presented the first example of such systems
that also has local interactions.

We have also shown that our local generic free fermion model has a non-
degenerate many body spectrum that under local probes of the spectrum (the
ratio test Oganesyan and Huse (2007)) looks strikingly Poissonian — this is
not the case when rational independence is lifted as seen in Fig. 1. Studying
the higher moments of the spectral form factor, however, we have shown that
the spectral statistics of our system agrees with the Poissonian prediction
only at the level of two-point spectral correlations. In fact, this is the
closest any free system can ever be to have Poissonian statistics. This shows
that the predictions of local probes should be taken with care.

An immediate direction for future research is to identify other dynamical
signatures of rational independent single-particle spectrum in the context of
local interactions. In particular, a fascinating question concerns the
interplay between rational independence and interacting perturbations. Since
rational independence reduces the number of thermalising channels — for a
fixed size there are no non-trivial energy conserving scattering processes —
systems with rational independence should anomalously long prethermalization
plateaus in finite volume.

###### Acknowledgements.

We thank Maurizio Fagotti for useful discussions. We acknowledge financial
support from the Royal Society through the University Research Fellowship No.
201101 and warmly acknowledge the hospitality of the Simons Center for
Geometry and Physics during the program “Fluctuations, Entanglements, and
Chaos: Exact Results” where part of the project has been carried out.

##  Appendix A Higher order harmonics

In this appendix we discuss how the inclusion of higher harmonics helps
achieving rational independence also for non-prime sizes.

First we note that our counter example from Sec. II does not apply if one adds
higher harmonics. To see this consider again $L=np$ with $p$ prime and add the
term

$\delta\cos\left(\frac{2\pi p}{L}k\right)+\eta\sin\left(\frac{2\pi
p}{L}k\right)$ (41)

to the dispersion relation (9). Here $\delta$ and $\eta$ are chosen to retain
incommensurability of the individual terms in the dispersion. This means that
we now have the following three equations

$\displaystyle P(1)=0,$ $\displaystyle P(\xi)=0,$ $\displaystyle
P(\xi^{p})=0.$ (42)

where we again set $\xi=\exp(i2\pi/L)$. Our choice $P(z)=(z-1)Q(z)$ in the
main text, with $Q(z)$ given in Eq. (19), does not satisfy the third equation.
Indeed, we have $w^{p}=\xi^{np}=1$ and $Q(\xi^{p})=p\neq 0$.

Second, we show that when $n$ is also prime we have rational independence if
we add harmonics of order $p$ and $n$. This can be seen as follows. Assuming
again incommensurability, we obtain that (11) can be decomposed into the
following four equations

$\displaystyle P(1)=0,$ $\displaystyle P(\xi)=0,$ $\displaystyle
P(\xi^{p})=0,$ $\displaystyle P(\xi^{n})=0.$ (43)

Considering the last three equations we have that $P(z)$ has zeros at the
primitive roots of unity $\xi$, $\xi^{p}$, and $\xi^{n}$. This means that it
must be proportional to the cyclotomic polynomials $\Phi_{np}(z)$,
$\Phi_{n}(z)$, and $\Phi_{p}(z)$. Namely we have

$P(z)=R_{1}(z)\Phi_{np}(z)=R_{2}(z)\Phi_{n}(z)=R_{3}(z)\Phi_{p}(z).$ (44)

for some $R_{j}(z)$, $j=1,2,3$. Since by definition each cyclotomic polynomial
is irreducible the above equation implies

$P(z)=R_{4}(z)\Phi_{n}(z)\Phi_{p}(z)\Phi_{np}(z).$ (45)

for some $R_{4}(z)$.

We now recall another important property of products of cyclotomic polynomials
Weisstein (2002)

$\prod_{d|n}\Phi_{d}(z)=z^{n}-1.$ (46)

This tells us that $\Phi_{n}(z)\Phi_{p}(z)\Phi_{np}(z)$ is by definition

$\Phi_{n}(z)\Phi_{p}(z)\Phi_{np}(z)=1+z+\dots+z^{pn-1},$ (47)

Since $pn-1$ is the maximal degree that $P(z)$ can have (cf. Eq. (15)) we then
conclude $R_{4}(z)=0$ or $R_{4}(z)=1$. The first choice gives rational
independence and the second is incompatible with the first of (43). This
concludes the proof.

Upon adding all $d$ order harmonics such that the step in (46) is reproduced,
this argument can be directly extended to arbitrary products of powers of
primes. Namely, to arbitrary $L\in\mathbb{N}$.

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