# Weak coupling limit for the ground state energy
of the 2D Fermi polaron

David Mitrouskas

###### Abstract

We analyze the ground state energy for $N$ fermions in a two-dimensional box
interacting with an impurity particle via two-body point interactions. We
allow for mass ratios $M>1.225$ between the impurity mass and the mass of a
fermion and consider arbitrarily large box sizes while keeping the Fermi
energy fixed. Our main result shows that the ground state energy in the limit
of weak coupling is given by the polaron energy. The polaron energy is an
energy estimate based on trial states up to first order in particle-hole
expansion, which was proposed by Chevy in the physics literature. For the
proof we apply a Birman–Schwinger principle that was recently obtained by
Griesemer and Linden. One main new ingredient is a suitable localization of
the polaron energy.

##  1 Introduction and main result

It is a universal challenge in quamtum theory to understand the physics of few
particles immersed into a complex enviroment in terms of properties of quasi-
particles. A famous example of a quasi-particle is the Fröhlich polaron
developed in a series of influential works by Landau, Pekar and Fröhlich [16,
17, 27, 11]. They suggested to describe the motion of an electron through a
polarizable crystal in terms of a polaron, that is, a quasi-particle composed
of an electron dressed by a local deformation of the crystal. This picture
leads to a drastic simplification as the complex many-body problem is replaced
by a self-consistent non-linear one-body model, which is much more accessible
to computations. While the Fröhlich polaron is certainly the most prominent
example for a polaron model, the concept of quasi-particles and polarons has
turned out very useful far beyond its original application in the theory of
electrons moving through crystals. For instance the experimental realization
of impurities immersed into ultracold atomic gases during the last two decades
has triggered the invention and analysis of many new models such as the Fermi
polaron [5], the Bose polaron [15] and the angulon [31]. In the present work
we are interested in the two-dimensional Fermi polaron which is a popular
model in theoretical physics to describe strongly population imbalanced Fermi
gases at low temperature confined to the two-dimensional plane. In case of
extreme imbalance there is only a single particle interacting with a gas of
non-interacting fermions via a two-body short range interaction.

We consider $N$ identical fermions and an additional distinguished particle,
called impurity particle, in a two-dimensional box $\Omega=[-L/2,L/2]^{2}$
with periodic boundary conditions. The underlying Hilbert space is
$L^{2}(\Omega)\otimes\mathscr{H}_{N}$ where
$\mathscr{H}_{N}=\bigwedge^{N}L^{2}(\Omega)$ denotes the space of anti-
symmetric $N$-particle wave functions. For a short-range potential, the Pauli
principle suppresses the interaction among the fermions which is therefore
neglected. The Hamiltonian of the system is formally described by

$\displaystyle-\frac{1}{M}\Delta_{y}-\sum_{i=1}^{N}\Delta_{x_{i}}-g\sum_{i=1}^{N}\delta(x_{i}-y),$
(1.1)

where $y$ represents the coordinate of the impurity, $\Delta$ is the Laplace
operator and $M$ denotes the ratio between the mass of the impurity particle
and the mass of a fermion. The interaction is given by a Dirac-delta-potantial
$\delta(x)$ with coupling strength $g>0$. This model is known as the 2D Fermi
polaron and has been analyzed to a great extent in the physics literature, see
e.g. [5, 7, 28, 6, 29, 4, 25, 30, 26]. The Fermi polaron is of interest, among
other reasons, because of the occurrence of a pairing mechanism somewhat
analogous to the famous BCS–BEC crossover. In two space dimensions, one
expects a transition of the ground state as a function of the coupling
strength. While for weak coupling, the impurity particle is expected to be
surrounded by a cloud of particle-hole excitations, in the strong coupling
regime it is predicted that the impurity is closely bound by a single fermion
forming a molecular state.

Here we provide a rigorous analysis of the ground state energy in the limit of
weak coupling by which we confirm its asymptotic form conjectured in the
physics literature. From the mathematical point of view, our work is a
continuation of recent articles by Griesemer and Linden [19, 13, 14] in which
they provide a definition of the self-adjoint Hamiltonian $H$ associated with
the formal expression (1.1), derive a Birman-Schwinger type principle for this
Hamiltonian and prove stability of the Fermi polaron at zero density. The
Birman–Schwinger priciple characterizes the low energy spectrum by means of an
operator $\phi(\lambda)$ with spectral parameter $\lambda$. Compared to $H$
the operator $\phi(\lambda)$ is given more explicitly and thus provides a
suitable tool for the analysis of the low energies, in particular for upper
and lower bounds for the ground state energy $\inf\sigma(H)$. Two such upper
bounds, called polaron and molecule energy, respectively, were discussed in
[14]. Motivated by the derivation of these upper bounds, we shall provide a
matching lower bound for $\inf\sigma(H)$ in the limit of weak coupling.

###  1.1 The model

A possible approach to define the Fermi polaron is to start with a regularized
version of the point interaction and then remove the regularization in a
suitable sense [14]. Since this lays the foundation for our work, we provide a
short summary.

For reasons of convenience we describe the fermions in the formalism of second
quantization. This means that we think of $\mathscr{H}_{N}$ as the
$N$-particle sector of
$\mathcal{F}=\bigoplus_{n=0}^{\infty}\bigwedge^{n}L^{2}(\Omega)$, the
fermionic Fock space over $L^{2}(\Omega)$. We denote the vacuum state with
zero particles by $\left|0\right\rangle=(1,0,0,...)$ and define creation and
annihilation operators $a_{k}^{*},a_{k}:\mathcal{F}\to\mathcal{F}$ of plane
waves $\varphi_{k}(x)=L^{-1}e^{ikx}$, $k\in(2\pi/L)\mathbb{Z}^{2}$,

$\displaystyle(a_{k}\Psi)^{(n)}$
$\displaystyle=\sqrt{n+1}\int_{\Omega}\textnormal{d}x_{n+1}\overline{\varphi_{k}(x_{n+1})}\Psi^{(n+1)}(x_{1},...,x_{n+1}),$
(1.2) $\displaystyle(a_{k}^{*}\Psi)^{(n)}$
$\displaystyle=\frac{1}{\sqrt{n}}\sum_{j=1}^{n}(-1)^{j}\varphi_{k}(x_{j})\Psi^{(n-1)}(x_{1},...,x_{j-1},x_{j+1},...,x_{n})$
(1.3)

for $\Psi=(\Psi^{(n)})_{n\geq 0}\in\mathcal{F}$. The creation and annihilation
operators satisfy the usual canonical anti-commutation relations (CAR),

$\displaystyle a_{k}a^{*}_{l}+a_{l}^{*}a_{k}=\delta_{kl},\quad
a_{k}a_{l}+a_{l}a_{k}=0$ (1.4)

for all pairs $k,l\in(2\pi/L)\mathbb{Z}^{2}$.

For any number $E_{B}<0$ we introduce the inverse coupling constant

$\displaystyle g_{n}^{-1}=\sum_{k^{2}\leq
n}\frac{1}{(1+\frac{1}{M})k^{2}-E_{B}}$ (1.5)

and define the sequence of regularized Hamiltonians
$(H_{n})_{n\in\mathbb{N}}$, acting on $L^{2}(\Omega)\otimes\mathscr{H}_{N}$,
by

$\displaystyle
H_{n}=-\frac{1}{M}\Delta_{y}+\sum_{k}k^{2}a_{k}^{*}a_{k}-g_{n}\sum_{k^{2},l^{2}\leq
n}\,e^{i(k-l)y}\,a_{l}^{*}a_{k}.$ (1.6)

If not stated otherwise, sums run over the two-dimensional momentum lattice
$(2\pi/L)\mathbb{Z}^{2}$ with possible restrictions indicated, e.g., as
$k^{2}\leq n$.

The following statement proves the existence of the self-adjoint Hamiltonian
describing the 2D Fermi polaron.

######  Proposition 1.1.

(see [14, Theorem 6]) For given $L>0$, $N\geq 1$, $M>0$ and $E_{B}<0$ there
exists a self-adjoint Hamiltonian $H:D(H)\subseteq
L^{2}(\Omega)\otimes\mathscr{H}_{N}\to L^{2}(\Omega)\otimes\mathscr{H}_{N}$
such that $H_{n}\to H$ in strong resolvent sense as $n\to\infty$. $H$ is
bounded from below.

######  Remarks.

a

1.1. From Proposition 5.1 [14] we know that the spectrum $\sigma(H)$ is purely
discrete.

1.2. The choice of $g_{n}$ ensures the following renormalization condition
(note that $g_{n}$ has a logarithmic divergence as $n\to\infty$): For $N=1$
the Hamiltonian $H$ has exactly one negative eigenvalue which coincides with
$E_{B}<0$. Hence the number $E_{B}$ corresponds to the binding energy of the
$1+1$-particle model and can be used as a suitable coupling parameter of the
point interaction.

The goal of this work is to derive an asymptotic formula for the ground state
energy $\min\sigma(H)$ in the limit of weak coupling $E_{B}\nearrow 0$ or in
the limit of large density $NL^{-2}\to\infty$ (the two limits turn out to be
closely connected). Instead of working with the particle number $N$ as a free
parameter, it is more convenient to fix a chemical potential $\mu>0$ and then
choose the number of fermions by $N=N(\mu)$ with

$\displaystyle
N(\mu)=\left|\left\\{k\in\kappa\mathbb{Z}^{2}\,:\,k^{2}\leq\mu\right\\}\right|,\quad\kappa=\frac{2\pi}{L}.$
(1.7)

Since the number of fermions now coincides with the number of eigenvalues of
$-\Delta$ that are less or equal than $\mu$, counting multiplicities, the
parameter $\mu$ plays the role of the Fermi energy. We write the ground state
energy of $H$ as a function of $\mu$ and $E_{B}$ as
$E(\mu,E_{B})=\min\sigma(H)$ and introduce the energy of $N(\mu)$ non-
interacting fermions inside the box $\Omega$,111The energies $E(\mu,E_{B})$
and $E_{0}(\mu)$ depend of course also on $L$ but we omit this in our
notation.

$\displaystyle E_{0}(\mu)=\sum_{k^{2}\leq\mu}k^{2}.$ (1.8)

Our main result, Theorem 1.2, shows that the energy difference
$E(\mu,E_{B})-E_{0}(\mu)$ is given at leading order by the polaron energy
$e_{\rm P}(\mu,E_{B})$, that is

$\displaystyle\frac{E(\mu,E_{B})-E_{0}(\mu)}{e_{\rm
P}(\mu,E_{B})}=1+o(1)\quad\text{as}\quad\frac{\mu}{|E_{B}|}\to\infty.$ (1.9)

The polaron energy $e_{\rm P}(\mu,E_{B})<0$ is the lowest solution to the
polaron equation

$\displaystyle e_{\rm
P}(\mu,E_{B})=-\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\frac{1}{G(k,-k^{2}-e_{\rm
P}(\mu,E_{B}))},$ (1.10)

where $G(q,\tau)$ is defined for $\tau>-\mu$ and $q\in\kappa\mathbb{Z}^{2}$ by

$\displaystyle
G(q,\tau)=\frac{1}{L^{2}}\sum_{k}\left(\frac{1}{(1+\frac{1}{M})k^{2}-E_{B}}-\frac{\chi_{(\mu,\infty)}(k^{2})}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau}\right).$
(1.11)

Here $\chi_{(\mu,\infty)}(s)$ denotes the characteristic function
$\chi_{(\mu,\infty)}(s)=1$ for $s>\mu$ and $\chi_{(\mu,\infty)}(s)=0$
otherwise. That (1.10) admits a lowest negative solution was shown in [14,
Proposition 7.1]. Let us mention that our main result (1.9) holds in
particular in the thermodynamic limit, i.e. after taking the limit
$L\to\infty$.

The polaron equation (1.10) was proposed in [5] based on a formal variational
calculation with trial states $w_{\rm P}\in
L^{2}(\Omega)\otimes\mathscr{H}_{N(\mu)}$ of the form

$\displaystyle w_{\rm
P}=\alpha_{0}\varphi_{0}\otimes\left|\rm{FS}_{\mu}\right\rangle+\sum_{k^{2}\leq\mu}\sum_{l^{2}>\mu}\alpha_{k,l}\varphi_{k-l}\otimes
a^{*}_{l}a_{k}\left|\rm{FS}_{\mu}\right\rangle$ (1.12)

where $\alpha_{0},\alpha_{k,l}\in\mathbb{C}$, $\varphi_{k}(y)=L^{-1}e^{iky}$
and

$\displaystyle\left|\rm{FS}_{\mu}\right\rangle=\prod_{k^{2}\leq\mu}a_{k}^{*}\left|0\right\rangle$
(1.13)

denotes the ground state of the kinetic operator
$\sum_{k}k^{2}a_{k}^{*}a_{k}\restriction\mathscr{H}_{N(\mu)}$ (called the
Fermi sea). A rigorous proof of the upper bound $E(\mu,E_{B})\leq
E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$ was given in [14] utilizing a generalized
Birman–Schwinger principle for the Hamiltonian $H$ (see Section 2).

In the physics literature the polaron energy is considered to be a good
approximation in the weak coupling limit $E_{B}\nearrow 0$ as well as in the
large density limit $\mu\to\infty$ [5, 7, 25]. In the regime of strong
coupling $E_{B}\to-\infty$, it is expected that the ground state undergoes a
transition to states in which the impurity is tightly bound by a single
fermion. This behavior is represented by the so-called molecule or dimer
ansatz [6, 29, 25]. In contrast to the latter, the polaron state (1.12) is
interpreted as an impurity that is surrounded by weak density fluctuations in
the Fermi sea. The two classes of trial states were investigated extensively
in the physics literature leading to indications for the anticipated
difference between the shape of the ground state in the weak and strong
coupling limits (see, e.g., the literature quoted in the previous section).
For this reason the Fermi polaron is also discussed in the context of the
BCS–BEC crossover. Most results in the physics literature, however, are based
on variational estimates using suitable classes of trial states. We remark
that this can only justify upper bounds for the ground state energy, whereas
here we provide a corresponding lower bound.

The Fermi polaron has been studied also in three dimensions. The problem of
defining a semi bounded self-adjoint Hamiltonian in this case was solved in
[22, 8, 23]. Contrary to the 2D model, it is known that the Hamiltonian is
semi-bounded in three dimensions only if $M\geq M_{*}$ for some critial mass
ratio $M_{*}>0$. Rigorous results concerning the ground state energy mostly
addressed the question of stability and the existence of a lower bound that is
uniform in the particle number $N$. In [23] it was shown that at zero density
there is such a uniform lower bound under the condition that $M>0.36$. In a
more recent work, Moser and Seiringer generalized their findings to the
positive density setup by proving that the energy shift caused by the impurity
particle depends only on the average density and the interaction strength but
not on the size of the system [24]. The question whether the polaron energy
describes the correct asymptotic form of the ground state energy similar to
(1.9) is still open for the three-dimensional model.

Quantum models with $N+1$ particles interacting via two-body point
interactions have been studied in the mathematical literature from various
points of views. Besides the works already quoted, we refer to [9, 10, 3, 22,
8, 21] and references therein.

###  1.2 Main result

We are now ready to state our main result which provides an asymptotic
estimate for the ground state energy $\min\sigma(H)$ of the 2D Fermi polaron.

######  Theorem 1.2.

Set $M>1.225$ and for $L>0$ and $\mu>0$, fix the number of particles $N(\mu)$
by (1.7). Moreover, let the Hamiltonian $H$ be the limit operator of
$(H_{n})_{n\in\mathbb{N}}$ as stated in Proposition 1.1. Then the ground state
energy $E(\mu,E_{B})=\min\sigma(H)$ and the lowest solution $e_{\rm
P}(\mu,E_{B})<0$ of the polaron equation (1.10) satisfy the following
property. There exist constants $c_{0},C>0$ (possibly depending on $M$) such
that

$\displaystyle\big{|}E(\mu,E_{B})-E_{0}(\mu)-e_{\rm P}(\mu,E_{B})\big{|}\,$
$\displaystyle\leq C\,\frac{|e_{\rm P}(\mu,E_{B})|}{\log(\mu/|E_{B}|)}$ (1.14)

for all $L>0$, $\mu>0$ and $E_{B}<0$ with $L^{2}|E_{B}|\geq 1$ and
$\mu/|E_{B}|\geq c_{0}$.

######  Remarks.

a

1.3. In Lemma 3.3 we show that $e_{\rm P}(\mu,E_{B})=O(\mu/\log(\mu/|E_{B}|))$
as $\mu/|E_{B}|\to\infty$.

1.4. The condition $L^{2}|E_{B}|\geq 1$ characterizes the range of parameters
in which the two-body binding energy $E_{B}$ is at least of the order of the
minimal kinetic excitation energy which equals $(2\pi/L)^{2}$. In this sense
our analysis is beyond the perturbative regime.

1.5. Since the constant on the right side of (1.14) does not depend on $L>0$,
we can directly infer a statement about the ground state energy in the
thermodynamic limit,

$\displaystyle\limsup_{L\to\infty}\bigg{|}\frac{E(\mu,E_{B})-E_{0}(\mu)}{e_{\rm
P}(\mu,E_{B})}-1\bigg{|}\,$ $\displaystyle\leq\frac{C}{\log(\mu/|E_{B}|)}$
(1.15)

for all $\mu/|E_{B}|\geq c_{0}$.

1.6. The condition $M>1.225$ is related to the problem of stability (of second
kind), that is, to find a uniform lower bound for the ground state energy in
the thermodynamic limit $L\to\infty$. While it is known that $H$ is bounded
from below for all $M>0$ [9, 14], it is unclear whether a uniform bound exists
when $M\leq 1.225$. This is an unsolved problem also in the case of zero
density, see [13].

1.7. The upper bound in (1.14) was proven in [14]. For the convenience of the
reader, we give a brief sketch of the argument in Section 2.2. The novel
contribution of the present work is the derivation of the lower bound.

1.8. A similar result was obtained in [20] for the case of an infinitely heavy
impurity, formally corresponding to $M=\infty$. In this case the $N$ fermions
interact with an external delta potential which simplifies the analysis
significantly.

The rest of the article is organized as follows. In the next section we
introduce the Birman–Schwinger operator $\phi(\lambda)$ associated to the
Hamiltonian $H$ and state the corresponding Birman–Schwinger principle. Upper
and lower bounds for the ground state energy follow from suitable bounds for
$\phi(\lambda)$. In Section 2.2 we recall how to obtain the upper bound in
(1.14). Sections 3–6 are about the matching lower bound. They account for the
main part of this work. In Section 3 we derive a localization of the polaron
energy inside a suitable subspace of the Hilbert space. In the two subsequent
sections we provide lower bounds for the Birman–Schwinger operator on the
localization subspace and its orthogonal complement. On the localization
subspace, we obtain a perturbed polaron equation whose solution we compare to
the polaron energy, see Section 4. In Section 5 we analyze the
Birman–Schwinger operator on the orthogonal complement of the localization
subspace. The lower bound on this subspace can be understood as a proof of
stability of the Fermi polaron at positive density which generalizes analogous
findings for the zero density model [14]. In Section 5.1 we combine the
obtained results to conclude the proof of Theorem 1.2. The last section
contains the proof of a technical lemma that is used several times throughout
the article.

##  2 Preliminaries and upper bound

In this section we discuss the Birman–Schwinger principle for the Hamiltonian
$H$ which provides a suitable tool for the analysis of upper and lower bounds
for $E(\mu,E_{B})=\min\sigma(H)$.

###  2.1 The Birman–Schwinger operator $\phi(\lambda)$

Our starting point for the proof of Theorem 1.2 is a Birman–Schwinger type
principle for the operator $H$. This is the second result from [14] which is
important for our analysis. For the precise statement, let us introduce the
resolvent set $\rho(H_{0})\subset\mathbb{C}$ of the non-interacting
Hamiltonian

$\displaystyle H_{0}=\Big{(}-\frac{1}{M}\Delta_{y}+T\Big{)}\restriction
L^{2}(\Omega)\otimes\mathscr{H}_{N(\mu)},$ (2.1)

with $T=\sum_{k}k^{2}a_{k}^{*}a_{k}$ the kinetic energy operator on the
fermionic Fock space.

######  Proposition 2.1.

(See [14, Sections 5 and 6]) There exists a family of operators
$\phi(\lambda)$, $\lambda\in\rho(H_{0})$, acting on
$L^{2}(\Omega)\otimes\mathscr{H}_{N(\mu)-1}$ with $\lambda$-independent domain
$\mathscr{D}$, such that for all real-valued $\lambda$, $\phi(\lambda)$ is
essentially self-adjoint and its closure (denoted again by $\phi(\lambda)$)
satisfies

$\displaystyle\inf\sigma(\phi(\lambda))\leq 0\qquad\Leftrightarrow\qquad
E(\mu,E_{B})\leq\lambda,$ (2.2)

with equality on one side implying equality on both sides. Moreover the
$\phi(\lambda)$ form an analytic family of type (A) and for
$\lambda\in\mathbb{R}_{-}\cup(\mathbb{C}\setminus\mathbb{R})\subset\rho(H_{0})$
they are given explicitly by

$\displaystyle\phi(\lambda)$
$\displaystyle=F(i\nabla_{y},T-\lambda)+\frac{1}{L^{2}}\sum_{k,l}a_{l}^{*}e^{iky}\frac{1}{-\frac{1}{M}\Delta_{y}+T+k^{2}+l^{2}-\lambda}e^{-ily}a_{k}$
(2.3)

where

$\displaystyle
F(q,\tau)=\frac{1}{L^{2}}\sum_{k}\bigg{(}\frac{1}{mk^{2}-E_{B}}-\frac{1}{\frac{1}{M}q^{2}+k^{2}+\tau}\bigg{)},\quad
m=\frac{M+1}{M}.$ (2.4)

Remarks.

2.1 Note that while $H$ is defined on the Hilbert space
$L^{2}(\Omega)\otimes\mathscr{H}_{N(\mu)}$, the Birman–Schwinger operator
$\phi(\lambda)$ acts on $L^{2}(\Omega)\otimes\mathscr{H}_{N(\mu)-1}$. We also
remark that the domain $\mathscr{D}$ is given by the set of all finite linear
combinations of states of the form
$\varphi_{q}\otimes\varphi_{k_{1}}\wedge\ldots\wedge\varphi_{k_{N(\mu)-1}}$
with $q,k_{1},...,k_{N(\mu)-1}\in\kappa\mathbb{Z}^{2}$ and $\varphi_{k}$ the
normalized plane waves in $L^{2}(\Omega)$.

2.2. The operator defined in (2.3) coincides with the Birman–Schwinger
operator $\phi(z)$ from [14, Lemma 6.3] up to a multiplicative factor
$L^{-2}$. Apart from renaming $z$ into $\lambda$, we write the impurity degree
of freedom in first quantization whereas in [14], all degrees of freedom are
expressed in second quantization. Proposition 2.1 is a direct consequence of
the statements from [14, Section 5 and Lemma 6.3].

For explicit computations it is useful to invert the normal order of creation
and annihilation operators in (2.3) when $k^{2},l^{2}\leq\mu$. With
$G(q,\tau)$ defined in (1.11), this leads for
$\lambda\in\mathbb{R}_{-}\cup(\mathbb{C}\setminus\mathbb{R})$ to

$\displaystyle\phi(\lambda)$
$\displaystyle=G(i\nabla_{y},T-\lambda)-\frac{1}{L^{2}}\sum_{k^{2},l^{2}\leq\mu}a_{k}e^{iky}\frac{1}{-\frac{1}{M}\Delta_{y}+T-\lambda}e^{-ily}a_{l}^{*}$
$\displaystyle\quad\quad\quad-\bigg{(}\frac{1}{L^{2}}\sum_{\begin{subarray}{c}k^{2}\leq\mu\\\
l^{2}>\mu\end{subarray}}e^{iky}a_{k}a_{l}^{*}\frac{1}{-\frac{1}{M}\Delta_{y}+T+l^{2}-\lambda}e^{-ily}+\text{h.c.}\bigg{)}$
$\displaystyle\quad\quad\quad+\frac{1}{L^{2}}\sum_{k^{2},l^{2}>\mu}a_{l}^{*}e^{iky}\frac{1}{-\frac{1}{M}\Delta_{y}+T+k^{2}+l^{2}-\lambda}e^{-ily}a_{k}$
(2.5)

understood as an operator on $L^{2}(\Omega)\otimes\mathscr{H}_{N(\mu)-1}$.
Through analytic continuation the above identity extends to
$\lambda<E_{0}(\mu)$. This explicit expression of $\phi(\lambda)$ will be the
main object to be analyzed.

To arrive at (2.1) we made use of the CAR and the pull-through formula, which
for suitable functions $f:\kappa\mathbb{Z}^{2}\times\mathbb{R}\to\mathbb{C}$
reads

$\displaystyle a_{k}f(P_{\rm f},T)$ $\displaystyle=f(P_{\rm
f}+k,T+k^{2})a_{k},\quad a_{k}^{*}f(P_{\rm f},T)=f(P_{\rm
f}-k,T-k^{2})a_{k}^{*}.$ (2.6)

Here $P_{\rm f}=\sum_{k}ka_{k}^{*}a_{k}$ denotes the momentum operator of the
fermions.

###  2.2 Upper bound

We show how to use Proposition 2.1 to obtain an upper bound for
$E(\mu,E_{B})$. This resembles the analysis performed in the first part of
Section 7 [14]. For an upper bound, it is sufficient to find a trial state $w$
and a suitable $\lambda$ that satisfy $\langle w,\phi(\lambda)w\rangle\leq 0$.
As such we choose $\lambda=E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$ and the wave
function

$\displaystyle w=\sum_{k^{2}\leq\mu}\frac{1}{G(k,-k^{2}-e_{\rm
P}(\mu,E_{B}))}\varphi_{k}\otimes a_{k}\left|{\rm FS}_{\mu}\right\rangle.$
(2.7)

With the aid of (2.1), a straightforward computation leads to

$\displaystyle\big{\langle}w,\phi(E_{0}(\mu)+e_{\rm
P}(\mu,E_{B}))w\big{\rangle}$
$\displaystyle=\sum_{k^{2}\leq\mu}\frac{1}{G(k,-k^{2}-e_{\rm
P}(\mu,E_{B}))}\bigg{[}1+\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\frac{1}{G(k,-k^{2}-e_{\rm
P}(\mu,E_{B}))}\cdot\frac{1}{e_{\rm P}(\mu,E_{B})}\bigg{]},$ (2.8)

which is identically zero because of (1.10). By Proposition 2.1 this implies
the upper bound

$\displaystyle E(\mu,E_{B})\leq E_{0}(\mu)+e_{\rm P}(\mu,E_{B}).$ (2.9)

###  2.3 Momentum decomposition of $\phi(\lambda)$

For the analysis of the lower bound it is convenient to make use of the
translational invariance of the model, in particular, that $\phi(\lambda)$
commutes with the total momentum operator $P_{\rm tot}=-i\nabla_{y}+P_{\rm f}$
with $P_{\rm f}=\sum_{k}ka_{k}^{*}a_{k}\restriction\mathscr{H}_{N(\mu)-1}$.
This guarantees a total momentum decomposition of $\phi(\lambda)$, meaning
that there is a unitary map

$\displaystyle
V:L^{2}(\Omega)\otimes\mathscr{H}_{N(\mu)-1}\to\bigoplus_{p\in\kappa\mathbb{Z}^{2}}\mathscr{H}_{N(\mu)-1}$
(2.10)

that diagonalizes $P_{\rm tot}$ by eliminating the $y$ coordinate in favor of
the total momentum $p\in\kappa\mathbb{Z}^{2}$. This unitary is given by
$(Vw)_{p}=(\langle\varphi_{p}|\otimes 1_{\mathscr{H}_{N(\mu)-1}})\,e^{iP_{\rm
f}y}w$ where $\langle\varphi_{p}|\otimes 1_{\mathscr{H}_{N(\mu)-1}}$ shall
indicate to take the scalar product in the coordinate $y$ with the plane wave
$\varphi_{p}\in L^{2}(\Omega)$. To see that the parameter $p$ describes the
total momentum, use $(VP_{\rm tot}w)_{p}=p(Vw)_{p}$ to verify

$\displaystyle\big{\langle}w,P_{\rm
tot}w\big{\rangle}=\sum_{p\in\kappa\mathbb{Z}^{2}}p\,\big{\langle}(Vw)_{p},(Vw)_{p}\big{\rangle}.$
(2.11)

The map $V$ is called Lee–Low–Pines transformation [18] and its inverse is
given by $V^{*}(w_{p})=e^{-iP_{\rm f}y}(\varphi_{p}\otimes w_{p})$.

From this definition it is not difficult to check that $\phi(\lambda)$ in
(2.1) transforms into
$V\phi(\lambda)V^{*}=\sum_{p\in\kappa\mathbb{Z}^{2}}\phi_{p}(\lambda)$ where

$\displaystyle\phi_{p}(\lambda)=G(p-P_{\rm
f},T-\lambda)-H_{p}(\lambda)-X_{p}(\lambda)+P_{p}(\lambda)$ (2.12)

is defined as an operator on $\mathscr{H}_{N(\mu)-1}$ with

$\displaystyle H_{p}(\lambda)$
$\displaystyle=a(\eta)\frac{1}{\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda}a^{*}(\eta),$ (2.13) $\displaystyle X_{p}(\lambda)$
$\displaystyle=a(\eta)A^{*}_{p}(\lambda)+A_{p}(\lambda)a^{*}(\eta),$ (2.14)
$\displaystyle P_{p}(\lambda)$
$\displaystyle=\frac{1}{L^{2}}\sum_{k^{2},l^{2}>\mu}a_{l}^{*}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-k-l)^{2}+T+k^{2}+l^{2}-\lambda}a_{k},$ (2.15)

and

$\displaystyle\quad a(\eta)=\frac{1}{L}\sum_{k^{2}\leq\mu}a_{k},\quad
A_{p}(\lambda)$
$\displaystyle=\frac{1}{L}\sum_{k^{2}>\mu}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-k)^{2}+T+k^{2}-\lambda}a_{k}.$ (2.16)

Note that the first summand in $\phi_{p}(\lambda)$ defines an unbounded
operator whereas the three other terms can be shown to be bounded operators.
The domain of essential self-adjointness of $\phi_{p}(\lambda)$ is the dense
subspace consisting of all finite linear combinations of states of the form
$\varphi_{k_{1}}\wedge\ldots\wedge\varphi_{k_{N(\mu)-1}}$ with
$k_{1},...,k_{N(\mu)-1}\in\kappa\mathbb{Z}^{2}$.

##  3 Localization of the polaron energy

By proposition 2.1 the lower bound $E(\mu,E_{B})\geq\lambda$ is equivalent to
$\phi(\lambda)\geq 0$. The next four sections are therefore devoted to the
analysis of the condition $\phi_{p}(\lambda)\geq 0$ for the operator (2.12)
with $p\in\kappa\mathbb{Z}^{2}$. In view of the upper bound (2.2) it is
sufficient to consider $\lambda\leq E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$ from now
on.

To prepare our first main statement we need to introduce a suitable orthogonal
projector in the Hilbert space $\mathscr{H}_{N(\mu)-1}$. For its definition
let us give names to the subsets of the momentum lattice
$\kappa\mathbb{Z}^{2}$ that correspond to hole and particle momenta w.r.t. the
Fermi sea,

$\displaystyle\Lambda_{\rm
h}=\big{\\{}k\in\kappa\mathbb{Z}^{2}:\,k^{2}\leq\mu\big{\\}},\quad\quad\Lambda_{\rm
p}=\big{\\{}k\in\kappa\mathbb{Z}^{2}\,:\,k^{2}>\mu\big{\\}}.$ (3.1)

Moreover for $\varepsilon>0$ we set

$\displaystyle\Lambda_{{\rm p},\varepsilon}^{\leq}=\left\\{k\in\Lambda_{\rm
p}\,:\,\mu<k^{2}\leq\left(1+\frac{1}{\varepsilon\log\widetilde{\mu}}\right)\mu\right\\}$
(3.2)

and define the orthogonal projectors $\Pi_{\varepsilon}$ and
$\Pi^{\perp}_{\varepsilon}=I-\Pi_{\varepsilon}$ through

$\displaystyle\text{Ran}(\Pi_{\varepsilon})=\overline{\text{lin}}\big{\\{}a^{*}_{l_{1}}...a_{l_{m-1}}^{*}a_{k_{1}}...a_{k_{m}}\left|\rm{FS}_{\mu}\right\rangle\,:\,m\geq
1,\ k_{1},...,k_{m}\in\Lambda_{\rm h},\ l_{1},....,l_{m-1}\in\Lambda_{{\rm
p},\varepsilon}^{\leq}\big{\\}}.$ (3.3)

Here $\overline{\text{lin}}V$ stands for the closure in
$\mathscr{H}_{N(\mu)-1}$ of the linear hull of the subset
$V\subseteq\mathscr{H}_{N(\mu)-1}$. For a better understanding of
$\text{Ran}(\Pi_{\varepsilon})$ and its orthogonal complement, let us recall
that the set of all anti-symmetric products of $N(\mu)-1$ plane waves,

$\displaystyle
D=\big{\\{}a^{*}_{l_{1}}...a_{l_{m-1}}^{*}a_{k_{1}}...a_{k_{m}}\left|\rm{FS}_{\mu}\right\rangle\,:\,m\geq
1,\ k_{1},...,k_{m}\in\Lambda_{\rm h},\ l_{1},...,l_{m-1}\in\Lambda_{\rm
p}\big{\\}},$ (3.4)

is a total set of the Hilbert space $\mathscr{H}_{N(\mu)-1}$, i.e.
$\overline{\text{lin}}D=\mathscr{H}_{N(\mu)-1}$. A comparison with (3.3) shows
that $\Pi_{\varepsilon}$ projects on all states in $\mathscr{H}_{N(\mu)-1}$
that have particle modes occupied solely in the momentum lattice region
$\Lambda_{{\rm p},\varepsilon}^{\leq}$ (this includes all states with zero
particle modes occupied), whereas the range of $\Pi^{\perp}_{\varepsilon}$
consists of states that have at least one mode occupied in $\Lambda_{{\rm
p},\varepsilon}^{>}=\Lambda_{\rm p}\setminus\Lambda_{{\rm
p},\varepsilon}^{\leq}$.

In the next proposition we provide a lower bound for $\phi_{p}(\lambda)$ in
terms of two operators that act only on $\text{Ran}(\Pi_{\varepsilon})$ and
$\text{Ran}(\Pi^{\perp}_{\varepsilon})$, respectively. The physical meaning of
the two subspaces is the following: On $\text{Ran}(\Pi^{\perp}_{\varepsilon})$
it is not clear how to obtain a suitable $L$-independent bound for the
operator $P_{p}(\lambda)$ which is one of the main obstacles in the analysis.
On this subspace we estimate the negative part of $P_{p}(\lambda)$ in terms of
$G(p-P_{\rm f},T-\lambda)$. This is closely connected to the problem of
obtaining a lower bound of $H$ uniformly in the system size $L\to\infty$. Such
a bound, though necessary for the proof of Theorem 1.2, is however not much
related to the asymptotic form of $E(\mu,E_{B})-E_{0}(\mu)$. The latter will
be determined on $\text{Ran}(\Pi_{\varepsilon})$ on which the operator
$P_{p}(\lambda)$ is easily estimated with a suitable uniform bound. Hence on
this subspace all of $G(p-P_{\rm f},T-\lambda)$ is available (and needed) for
the analysis of the correct energy asymptotics. This explains the motivation
behind the following decomposition of $\phi_{p}(\lambda)$. Since the operator
$G(p-P_{\rm f},T-\lambda)$ is needed on both subspaces separately, it is an
important step in our argument.

######  Proposition 3.1.

There are constants $c_{0},\varepsilon_{0}>0$ such that for all
$p\in\kappa\mathbb{Z}^{2}$, $L^{2}|E_{B}|\geq 1$, $\mu/|E_{B}|\geq c_{0}$ and
$\varepsilon\in(0,\varepsilon_{0})$, it holds that
$\phi_{p}(\lambda)\geq\Phi_{p}(\lambda,\varepsilon)+\Psi_{p}(\lambda,\varepsilon)$,
with

$\displaystyle\Phi_{p}(\lambda,\varepsilon)\,$
$\displaystyle=\,\Pi_{\varepsilon}\big{(}G(p-P_{\rm
f},T-\lambda)\,-\,H_{p}(\lambda)-X_{p}(\lambda)-\varepsilon^{-1}\big{)}\Pi_{\varepsilon},$
(3.5) $\displaystyle\Psi_{p}(\lambda,\varepsilon)\,$
$\displaystyle=\,\Pi^{\perp}_{\varepsilon}\big{(}G(p-P_{\rm
f},T-\lambda)\,+\,P_{p}(\lambda)\,-K(\varepsilon,\widetilde{\mu})\big{)}\Pi^{\perp}_{\varepsilon},$
(3.6)

and
$K(\varepsilon,\widetilde{\mu})=\varepsilon^{-1/2}(\varepsilon^{-1/2}+\sqrt{\log\widetilde{\mu}}+\varepsilon\log\widetilde{\mu})$.
We use the notation $\widetilde{\mu}=\mu/|E_{B}|$.

Remark 3.1. From the discussion above it is clear that the $\varepsilon$\- and
$\widetilde{\mu}$-dependent errors in (3.5) and (3.6) have physically
different meanings. Eventually only the error in (3.5) enters the constant on
the right side of (1.14). For that reason, we do not optimize the error terms
as $\widetilde{\mu}\to\infty$, and always consider $\varepsilon$ sufficiently
small but fixed w.r.t. $\widetilde{\mu}$ and $L>0$.

Before we come to the proof of the proposition, we state two helpful results
about the asymptotics of $G(q,\tau)$ and $e_{\rm P}(\mu,E_{B})$.

###  3.1 Asymptotics of $G(q,\tau)$ and $e_{\rm P}(\mu,E_{B})$

As the following bound will be used several times, we note that it follows
easily with the aid of Lemma A.1,

$\displaystyle\bigg{|}\frac{1}{L^{2}}\sum_{a\mu\leq
k^{2}<b\mu}1-\frac{(b-a)\mu}{4\pi}\bigg{|}\leq\frac{2}{\pi
L}\big{(}\sqrt{a\mu}+\sqrt{b\mu}\big{)}+\frac{6}{L^{2}}$ (3.7)

for any $b>a\geq 0$.

The first lemma of this section tells us the error for replacing the sum in
$G(q,\tau)$ by the corresponding integral which can be evaluated explicitly.

######  Lemma 3.2.

There are constant $c_{0},C>0$ such that

$\displaystyle\left|G(q,\tau)-\frac{1}{4\pi
m}\,\log\Bigg{(}\frac{\frac{q^{2}}{M+1}+m\mu+\tau}{|E_{B}|}\Bigg{)}\right|\leq
C\bigg{(}1+\frac{\mu}{(\mu+\tau)\log(\mu/|E_{B}|)}\bigg{)}^{3}$ (3.8)

for all $q\in\mathbb{R}^{2}$, $\tau>-\mu$, $L^{2}|E_{B}|\geq 1$ and
$\mu/|E_{B}|\geq c_{0}$. Recall $m=1+1/M$.

The proof of the lemma is postponed to Section 6. Utilizing this lemma, we can
derive an asymptotic formula for $e_{\rm P}(\mu,E_{B})$ as
$\mu/|E_{B}|\to\infty$. The precise statement is

######  Lemma 3.3.

There are constants $c_{0},C>0$ such that the polaron energy satisfies

$\displaystyle\left|e_{\rm
P}(\mu,E_{B})+\big{(}1+\frac{1}{M}\big{)}\frac{\mu}{\log(\mu/|E_{B}|)}\right|\,\leq\,C\,\frac{\mu}{(\log(\mu/|E_{B}|))^{2}}$
(3.9)

for all $L^{2}|E_{B}|\geq 1$ and $\mu/|E_{B}|\geq c_{0}$.

###### Proof.

Let us set $z_{\rm P}=|e_{\rm P}(\mu,E_{B})|$ and
$\widetilde{\mu}=\mu/|E_{B}|$. To prove suitable upper and lower bounds for
$z_{\rm P}$ we first show $z_{\rm P}\leq\mu$ for all $\mu/|E_{B}|\geq c_{0}$
given that the constant $c_{0}$ is chosen large enough.

Consider the set of parameters for which $z_{\rm P}$ exceeds the value $\mu$,

$\displaystyle\mathcal{M}_{c_{0}}=\big{\\{}(L,\mu,E_{B})\,:z_{\rm P}>\mu\geq
c_{0}|E_{B}|\,\,\text{and}\,\,L^{2}|E_{B}|\geq
1\big{\\}}\subseteq\mathbb{R}_{+}\times\mathbb{R}_{+}\times\mathbb{R}_{-}.$
(3.10)

By monotonicity of $G(q,\tau)$ in the $\tau$ variable, we have
$G(k,-k^{2}+z_{\rm p})\geq G(k,0)$ for all $k^{2}\leq\mu$ and
$(L,\mu,E_{B})\in\mathcal{M}_{c_{0}}$. By Lemma 3.2 this implies

$\displaystyle G(k,0)\geq\frac{1}{4\pi m}\log(mc_{0})-C\bigg{(}1+\frac{1}{\log
c_{0}}\bigg{)}^{3}\geq\frac{1}{8\pi m}\log(mc_{0}).$ (3.11)

Inserting this into the polaron equation (1.10) and employing (3.7) leads to

$\displaystyle z_{\rm
P}\leq\frac{2m\mu}{\log(mc_{0})}\bigg{(}1+\frac{8}{\sqrt{c_{0}}}+\frac{24\pi}{c_{0}}\bigg{)},$
(3.12)

which implies $z_{\rm P}\leq\mu$ for $c_{0}$ large enough. Hence
$\mathcal{M}_{c_{0}}$ is empty and we can assume $z_{\rm P}\leq\mu$.

Utilizing again monotonicity of $G(q,\tau)$, we get $G(k,-k^{2}+z_{\rm P})\leq
G(k,\mu)$. By (3.8) we have for all $k^{2}\leq\mu$,

$\displaystyle G(k,\mu)\leq\frac{1}{4\pi
m}\,\log\Bigg{(}\frac{\frac{\mu}{M+1}+m\mu+\mu}{|E_{B}|}\Bigg{)}+C_{1}\leq\frac{1}{4\pi
m}\,\log\widetilde{\mu}+C_{2}$ (3.13)

for two constants $C_{1},C_{2}>0$. Using (1.10) together with (3.7), we obtain
the lower bound

$\displaystyle z_{\rm
P}\geq\frac{\mu}{m^{-1}\log\widetilde{\mu}+C_{2}}\bigg{(}1-\frac{C_{3}}{\sqrt{\widetilde{\mu}}}\bigg{)}\geq\frac{m\mu}{\log\widetilde{\mu}}-C/(\log\widetilde{\mu})^{2}.$
(3.14)

With the lower bound (3.14) we can estimate for $0\leq k^{2}\leq\mu$,

$\displaystyle G(k,-k^{2}+z_{\rm P})$ $\displaystyle\geq\frac{1}{4\pi
m}\log(m\widetilde{\mu}-\widetilde{\mu})-C_{1}\bigg{(}1+\frac{\mu}{z_{\rm
P}\log\widetilde{\mu}}\bigg{)}^{3}\geq\frac{1}{4\pi
m}\log\widetilde{\mu}-C_{2}.$ (3.15)

Similarly as above, using the polaron equation and (3.7), one finds

$\displaystyle z_{\rm
P}\leq\frac{m\mu}{\log\widetilde{\mu}}\bigg{(}\frac{1}{1-C_{2}/\log\widetilde{\mu}}\bigg{)}\bigg{(}1+\frac{C_{3}}{\sqrt{\widetilde{\mu}}}\bigg{)},$
(3.16)

which gives the desired upper bound. ∎

Remark 3.2. For $\lambda\leq E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$ it follows from
$T\restriction\mathscr{H}_{N(\mu)-1}\geq E_{0}(\mu)-\mu$ that there are
constants $c_{0},C>0$ such that

$\displaystyle\pm\Bigg{(}G(p-P_{\rm f},T-\lambda)-\frac{1}{4\pi
m}\,\log\Bigg{(}\frac{\frac{(p-P_{\rm
f})^{2}}{M+1}+m\mu+T-\lambda}{|E_{B}|}\Bigg{)}\Bigg{)}\leq C$ (3.17)

as operator inequalities on $\mathscr{H}_{N(\mu)-1}$ for all $L^{2}|E_{B}|\geq
1$ and $\mu/|E_{B}|\geq c_{0}$. A useful implication of this bound is

$\displaystyle G(p-P_{\rm
f},T-\lambda)\restriction\mathscr{H}_{N(\mu)-1}\geq\frac{1}{4\pi
m}\log(\mu/|E_{B}|)-C.$ (3.18)

###  3.2 Proof of Proposition 3.1

Since $G(p-P_{\rm f},T-\lambda)$ and $H_{p}(\lambda)$ both commute with the
projector $\Pi_{\varepsilon}$, we have

$\displaystyle\phi_{p}(\lambda)\,$
$\displaystyle=\,\Pi_{\varepsilon}\,\phi_{p}(\lambda)\,\Pi_{\varepsilon}\,+\,\Pi^{\perp}_{\varepsilon}\,\phi_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}+\Big{(}\Pi_{\varepsilon}\big{(}-\,X_{p}(\lambda)\,+\,P_{p}(\lambda)\big{)}\Pi^{\perp}_{\varepsilon}\,+\text{h.c.}\Big{)}.$
(3.19)

The statement of Proposition 3.1 is a consequence of the following estimates.
Note that for notational convenience we estimate the constant $C$ from above
by $\varepsilon^{-1/2}$.

######  Lemma 3.4.

There are constants $c_{0},\varepsilon_{0},C>0$ such that for all
$p\in\kappa\mathbb{Z}^{2}$, $L^{2}|E_{B}|\geq 1$,
$\widetilde{\mu}=\mu/|E_{B}|\geq c_{0}$ and
$\varepsilon\in(0,\varepsilon_{0})$,

$\displaystyle\Pi_{\varepsilon}\,P_{p}(\lambda)\Pi_{\varepsilon}\,$
$\displaystyle\geq\,-\frac{C}{\sqrt{\varepsilon}}\,\Pi_{\varepsilon},$ (3.20)
$\displaystyle\Pi^{\perp}_{\varepsilon}\,H_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}\,$
$\displaystyle\leq\,C\varepsilon\log\widetilde{\mu}\
\Pi^{\perp}_{\varepsilon},$ (3.21)
$\displaystyle\Pi^{\perp}_{\varepsilon}\,X_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}\,$
$\displaystyle\leq\,C\sqrt{\log\widetilde{\mu}}\ \Pi^{\perp}_{\varepsilon},$
(3.22)

and

$\displaystyle\Pi_{\varepsilon}\,X_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}\,+\,\textnormal{h.c.}\,$
$\displaystyle\leq\,C\varepsilon^{1/2}\log\widetilde{\mu}\
\Pi^{\perp}_{\varepsilon}\,+\,C\varepsilon^{-1/2}\ \Pi_{\varepsilon},$ (3.23)
$\displaystyle\Pi_{\varepsilon}\,P_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}\,+\,\textnormal{h.c.}\,$
$\displaystyle\geq\,-\frac{C}{\sqrt{\varepsilon}}.$ (3.24)

###### Proof.

Line (3.20). Using $a_{k}\Pi_{\varepsilon}=0$ for all $k\in\Lambda_{\rm
p}\setminus\Lambda_{{\rm p},\varepsilon}^{\leq}$ together with the pull-
through formula (2.6), we obtain

$\displaystyle\Pi_{\varepsilon}\,P_{p}(\lambda)\,\Pi_{\varepsilon}$
$\displaystyle=\Pi_{\varepsilon}\Bigg{(}\frac{1}{L^{2}}\sum_{k,l\in\Lambda_{{\rm
p},\varepsilon}^{\leq}}a_{l}^{*}a_{k}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-l)^{2}+T+l^{2}-\lambda}\Bigg{)}\Pi_{\varepsilon}.$ (3.25)

By means of the commutation relations
$a_{k}a_{l}^{*}+a_{l}^{*}a_{k}=\delta_{kl}$ and

$\displaystyle\frac{1}{L^{2}}\sum_{l\in\Lambda_{{\rm
p},\varepsilon}^{\leq}}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-l)^{2}+T+l^{2}-\lambda}\,\geq\,0$ (3.26)

on $\mathscr{H}_{N(\mu)-1}$, we further get

$\displaystyle\eqref{eq: bound for Pi P Pi line
1}\geq\Pi_{\varepsilon}\Bigg{(}-\frac{1}{L^{2}}\sum_{k,l\in\Lambda_{{\rm
p},\varepsilon}^{\leq}}a_{k}\frac{1}{\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda}a_{l}^{*}\Bigg{)}\Pi_{\varepsilon}.$ (3.27)

For $f(\mu,E_{B})>0$ we proceed by estimating the right side from below in
terms of the operator

$\displaystyle-\frac{f(\mu,E_{B})}{2}\Bigg{(}\frac{1}{L^{2}}\sum_{k,l\in\Lambda_{{\rm
p},\varepsilon}^{\leq}}a_{k}a_{l}^{*}\Bigg{)}-\frac{1}{2f(\mu,E_{B})}\Bigg{(}\frac{1}{L^{2}}\sum_{k,l\in\Lambda_{{\rm
p},\varepsilon}^{\leq}}a_{k}\frac{1}{(\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda)^{2}}a_{l}^{*}\Bigg{)}$ (3.28)

acting on $\Pi_{\varepsilon}\mathscr{H}_{N(\mu)-1}$. In the first summand, we
use the CAR together with (3.7) for $a=\mu$,
$b=\mu(1+\frac{1}{\varepsilon\log\widetilde{\mu}})$, and further employ
$L^{2}|E_{B}|\geq 1$ and $\widetilde{\mu}\geq c_{0}$. This gives

$\displaystyle\frac{1}{L^{2}}\sum_{k,l\in\Lambda_{p,\varepsilon}^{\leq}}a_{k}a_{l}^{*}\leq\frac{1}{L^{2}}\sum_{k\in\Lambda_{p,\varepsilon}^{\leq}}1\leq\frac{C\mu}{\varepsilon\log\widetilde{\mu}}.$
(3.29)

In the second summand in (3.28), we use $T-\lambda>0$ on
$\mathscr{H}_{N(\mu)}$ in order to neglect the positive operator
$\frac{1}{M}(p-P_{\rm f})^{2}$ in the denominator. Then we use again the pull-
through formula and the CAR to obtain

$\displaystyle\frac{1}{L^{2}}\sum_{k,l\in\Lambda^{\leq}_{{\rm
p},\varepsilon}}a_{k}\frac{1}{(\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda)^{2}}a_{l}^{*}$
$\displaystyle\leq\frac{1}{L^{2}}\sum_{k,l\in\Lambda^{\leq}_{{\rm
p},\varepsilon}}a_{k}\frac{1}{(T-\lambda)^{2}}a_{l}^{*}$
$\displaystyle\leq\frac{1}{L^{2}}\sum_{k\in\Lambda^{\leq}_{{\rm
p},\varepsilon}}\frac{1}{(T+k^{2}-\lambda)^{2}}.$ (3.30)

Note that in the last step, we applied the inequality

$\displaystyle\frac{1}{L^{2}}\sum_{k,l\in\Lambda^{\leq}_{{\rm
p},\varepsilon}}a_{l}^{*}\frac{1}{(T+k^{2}+l^{2}-\lambda)^{2}}a_{k}\geq 0,$
(3.31)

which is verified by writing

$\displaystyle\frac{1}{(T+k^{2}+l^{2}-\lambda)^{2}}\,=\,\int_{0}^{\infty}\exp\left(-t(T-\lambda+k^{2}+l^{2})^{2}\right)\,\textnormal{d}t$
(3.32)

and estimating

$\displaystyle\exp\left(-t(T-\lambda+k^{2}+l^{2})^{2}\right)\geq\exp\left(-t4l^{4}\right)\exp\left(-t2(T-\lambda)^{2}\right)\exp\left(-t4k^{4}\right).$
(3.33)

With $T\geq E_{0}(\mu)-\mu$ on $\mathscr{H}_{N(\mu)-1}$ and
$E_{0}(\mu)-\lambda\geq|e_{P}(\mu,E_{B})|$ we next get

$\displaystyle\eqref{eq: bound for Pi P Pi line
3}\leq\frac{1}{L^{2}}\sum_{k^{2}>\mu}\frac{1}{(k^{2}-\mu+|e_{\rm
P}(\mu,E_{B})|)^{2}}.$ (3.34)

By Lemma A.1 and the estimate

$\displaystyle\int\limits_{\sqrt{\mu}}^{\infty}\frac{\textnormal{d}t}{(t^{2}-\mu+|e_{\rm
P}(\mu,E_{B})|)^{2}}$
$\displaystyle\,\leq\,\int\limits_{\sqrt{\mu}}^{\infty}\frac{\textnormal{d}t}{((t-\sqrt{\mu})^{2}+|e_{\rm
P}(\mu,E_{B})|)^{2}}\,=\,\frac{\pi}{4|e_{\rm P}(\mu,E_{B})|^{3/2}},$ (3.35)

one finds the upper bound

$\displaystyle\eqref{eq: bound for Pi P Pi line 3}$
$\displaystyle\leq\frac{1}{4\pi|e_{\rm P}(\mu,E_{B})|}+\frac{1}{L|e_{\rm
P}(\mu,E_{B})|^{3/2}}+\bigg{(}\frac{4\sqrt{\mu}}{L}+\frac{6}{L^{2}}\bigg{)}\frac{1}{|e_{\rm
P}(\mu,E_{B})|^{2}}\leq\frac{C\log\widetilde{\mu}}{\mu}.$ (3.36)

Hence,

$\displaystyle\Pi_{\varepsilon}\,P(\lambda)\,\Pi_{\varepsilon}\geq-C\left(\frac{f(\mu,E_{B})\mu}{\varepsilon\log\widetilde{\mu}}+\frac{\log\widetilde{\mu}}{f(\mu,E_{B})\mu}\right)\Pi_{\varepsilon}\geq-\frac{C}{\sqrt{\varepsilon}}\,\Pi_{\varepsilon},$
(3.37)

if we choose $f(\mu,E_{B})=(\sqrt{\varepsilon}\log\widetilde{\mu})/\mu$.

Line (3.21). Since states of the form
$\widetilde{w}=a^{*}(\eta)w\in\mathscr{H}_{N(\mu)}$ with
$w\in\text{Ran}(\Pi^{\perp}_{\varepsilon})$ have at least one momentum mode
occupied in $\Lambda_{{\rm p},\varepsilon}^{\geq}$, it follows that

$\displaystyle\Pi^{\perp}_{\varepsilon}\,H_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}\,\leq\,\Pi^{\perp}_{\varepsilon}\left(\frac{a(\eta)a^{*}(\eta)}{|e_{\rm
P}(\mu,E_{B})|+\mu/(\varepsilon\log\widetilde{\mu})}\right)\Pi^{\perp}_{\varepsilon}.$
(3.38)

The remaining expression is estimated using

$\displaystyle a(\eta)a^{*}(\eta)\leq\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}1\leq
C\mu$ (3.39)

which follows from the CAR in combination with (3.7).

Lines (3.22) and (3.23). It is straightforward to verify that for any two
orthogonal projectors $Q,\widetilde{Q}$ acting on $\mathscr{H}_{N(\mu)-1}$ and
for any $f(\mu,E_{B}),g(\mu,E_{B})>0$,

$\displaystyle QX_{p}(\lambda)\widetilde{Q}\,+\,\text{h.c.}$
$\displaystyle\,\leq\,Q\left(f(\mu,E_{B})\,A_{p}(\lambda)A_{p}^{*}(\lambda)+\frac{a(\eta)a^{*}(\eta)}{g(\mu,E_{B})}\right)Q\,$
$\displaystyle\hskip
42.67912pt+\,\widetilde{Q}\left(g(\mu,E_{B})\,A_{p}(\lambda)A_{p}^{*}(\lambda)+\frac{a(\eta)a^{*}(\eta)}{f(\mu,E_{B})}\right)\widetilde{Q}.$
(3.40)

Similar as in the analysis of (3.25), one further shows

$\displaystyle
A_{p}(\lambda)A^{*}_{p}(\lambda)\restriction\mathscr{H}_{N(\mu)-1}\,\leq\,C\frac{\log\widetilde{\mu}}{\mu}.$
(3.41)

Together with (3.39) and (3.40) this leads to

$\displaystyle Q\,X_{p}(\lambda)\,\widetilde{Q}\,+\,\text{h.c.}\,$
$\displaystyle\quad\leq\,C\left(\frac{f(\mu,E_{B})\log\widetilde{\mu}}{\mu}+\frac{\mu}{g(\mu,E_{B})}\right)Q\,+\,C\left(\frac{g(\mu,E_{B})\log\widetilde{\mu}}{\mu}+\frac{\mu}{f(\mu,E_{B})}\right)\widetilde{Q}.$
(3.42)

For $f(\mu,E_{B})=g(\mu,E_{B})=\mu/\sqrt{\log\widetilde{\mu}}$, this shows
(3.22), whereas the inequality in (3.23) follows from
$f(\mu,E_{B})=\sqrt{\varepsilon}\mu$ and
$g(\mu,E_{B})=\mu/(\sqrt{\varepsilon}\log\widetilde{\mu})$. (In the latter
case we set $Q=\Pi^{\perp}_{\varepsilon}$ and
$\widetilde{Q}=\Pi_{\varepsilon}$.)

Line (3.24). Using $a_{l}\Pi_{\varepsilon}=0$ for $l\in\Lambda_{\rm
p}\setminus\Lambda_{{\rm p},\varepsilon}^{\leq}$ and the pull-through formula
together with the CAR, we find

$\displaystyle\Pi_{\varepsilon}\,P_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}=\Pi_{\varepsilon}\Bigg{(}-\frac{1}{L^{2}}\sum_{\begin{subarray}{c}l\in\Lambda^{\leq}_{{\rm
p},\varepsilon}\\\ k^{2}>\mu\end{subarray}}a_{k}\frac{1}{\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda}a_{l}^{*}\Bigg{)}\Pi^{\perp}_{\varepsilon},$ (3.43)

where we made use of

$\displaystyle\Pi_{\varepsilon}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-l)^{2}+T+l^{2}-\lambda}\Pi^{\perp}_{\varepsilon}=\frac{1}{\frac{1}{M}(p-P_{\rm
f}-l)^{2}+T+l^{2}-\lambda}\Pi_{\varepsilon}\Pi^{\perp}_{\varepsilon}=0.$
(3.44)

From here the proof works the same way as for (3.28) (with the difference that
we end up with an identity on the right side). We obtain

$\displaystyle\Pi_{\varepsilon}\,P_{p}(\lambda)\,\Pi^{\perp}_{\varepsilon}+\text{h.c.}\geq-\frac{C}{\sqrt{\varepsilon}},$
(3.45)

which completes the proof of the lemma and thus also the proof of Lemma 3.1. ∎

##  4 Analysis of $\Phi_{p}(\lambda,\varepsilon)$: perturbed polaron equation

In this section we show that the condition $\Phi_{p}(\lambda,\varepsilon)\geq
0$ leads to a perturbed polaron equation for $\lambda$ and then provide a
suitable estimate for the solution of this equation.

In order to obtain the presumably optimal asymptotics of the error in (1.14),
we introduce another orthogonal projector $\Pi_{\varepsilon,1}$ with
$\text{Ran}(\Pi_{\varepsilon,1})\subseteq\text{Ran}(\Pi_{\varepsilon})$
defined as the closed subspace of all states containing exactly one unoccupied
momentum mode (a hole) in the lattice region

$\displaystyle\Lambda_{{\rm
h},\varepsilon}^{\leq}=\left\\{k\in\kappa\mathbb{Z}^{2}\,:\,k^{2}\leq\left(1-\frac{1}{\varepsilon\log\widetilde{\mu}}\right)\mu\right\\}\subset\Lambda_{\rm
h}.$ (4.1)

More precisely we set for $n\geq 0$,
$\text{Ran}(\Pi_{\varepsilon,n})=\overline{\text{lin}}V_{\varepsilon,n}$ with
$V_{\varepsilon,n}\subset\mathscr{H}_{N(\mu)-1}$ the subset

$\displaystyle
V_{\varepsilon,n}=\Big{\\{}w=a^{*}_{l_{1}}...a_{l_{m-1}}^{*}a_{k_{1}}...a_{k_{m}}\left|\rm{FS}_{\mu}\right\rangle\big{|}\,$
$\displaystyle m\geq 1,\ k_{1},...,k_{m}\in\Lambda_{\rm h},\
l_{1},....,l_{m-1}\in\Lambda_{{\rm p},\varepsilon}^{\leq},$
$\displaystyle\hskip 56.9055pt\text{and}\ \sum_{k\in\Lambda_{{\rm
h},\varepsilon}^{\leq}}a_{k}a_{k}^{*}w\,=\,nw\Big{\\}}.$ (4.2)

Clearly $\text{Ran}(\Pi_{\varepsilon})=\bigoplus_{n\geq
0}\text{Ran}(\Pi_{\varepsilon,n})$. (Note that the operator
$\sum_{k\in\Lambda_{{\rm h},\varepsilon}^{\leq}}a_{k}a_{k}^{*}$ counts the
number of holes in $\Lambda_{{\rm h},\varepsilon}^{\leq}$.)

The next lemma provides a more accurate localization of the polaron energy
inside the subspace $\text{Ran}(\Pi_{\varepsilon,1})$.

######  Lemma 4.1.

There are constants $c_{0},\varepsilon_{0}>0$ such that for all
$p\in\kappa\mathbb{Z}^{2}$, $L^{2}|E_{B}|\geq 1$, $\mu/|E_{B}|\geq c_{0}$ and
$\varepsilon\in(0,\varepsilon_{0})$, we have

$\displaystyle\Phi_{p}(\lambda,\varepsilon)\,$
$\displaystyle\geq\,\Pi_{\varepsilon,1}\big{(}pol(\lambda)-\varepsilon^{-3}\big{)}\Pi_{\varepsilon,1}$
(4.3)

where $pol(\lambda)=G(0,T-\lambda)-a(\eta)(T-\lambda)^{-1}a^{*}(\eta)$.

###### Proof.

We write
$\Pi_{\varepsilon}=\Pi_{\varepsilon,0}+\Pi_{\varepsilon,1}+\Pi_{\varepsilon,2+}$
with $\Pi_{\varepsilon,2+}=\sum_{n\geq 2}\Pi_{\varepsilon,n}$. Below we prove
the inequality

$\displaystyle\Phi_{p}(\lambda,\varepsilon)\,$
$\displaystyle\geq\,\Pi_{\varepsilon,1}\big{(}G(p-P_{\rm
f},T-\lambda)-H_{p}(\lambda)-C\varepsilon^{-2}\big{)}\Pi_{\varepsilon,1}$
$\displaystyle\hskip
8.5359pt\,+\,(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+})\big{(}G(p-P_{\rm
f},T-\lambda)-C\varepsilon\log\widetilde{\mu}\big{)}(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+})$
(4.4)

from which the statement of the lemma follows by

$\displaystyle G(p-P_{\rm f},T-\lambda)-H_{p}(\lambda)\geq
G(0,T-\lambda)-a(\eta)(T-\lambda)^{-1}a^{*}(\eta)-C$ (4.5)

together with inequality (3.18). By choosing $\varepsilon_{0}$ small enough
the second line in (4.4) is positive for all $\widetilde{\mu}\geq c_{0}$. The
bound in (4.5) is a direct consequence of (3.17) and the fact that
$T-\lambda\geq|e_{\rm P}(\mu,E_{B})|$ on $\mathscr{H}_{N(\mu)}$.

The derivation of (4.4) occupies the remainder of this proof. To this end,
note

$\displaystyle\Pi_{\varepsilon}G(p-P_{\rm f},T-\lambda)\Pi_{\varepsilon}$
$\displaystyle=\,\Pi_{\varepsilon,1}G(p-P_{\rm
f},T-\lambda)\Pi_{\varepsilon,1}$
$\displaystyle\quad+(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+})G(p-P_{\rm
f},T-\lambda)(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+}).$ (4.6)

$\bullet$ Introducing $\Lambda_{{\rm h},\varepsilon}^{>}=\Lambda_{\rm
h}\setminus\Lambda_{{\rm h},\varepsilon}^{\leq}$ and
$a(\eta^{>}_{\varepsilon})=\sum_{k\in\Lambda_{{\rm h},\varepsilon}^{>}}a_{k}$,
we can start with

$\displaystyle\Pi_{\varepsilon,0}\,H_{p}(\lambda)\,\Pi_{\varepsilon,0}\leq\Pi_{\varepsilon,0}\left(\frac{a(\eta^{>}_{\varepsilon})a^{*}(\eta^{>}_{\varepsilon})}{|e_{\rm
P}(\mu,E_{B})|}\right)\Pi_{\varepsilon,0}\leq
C\varepsilon^{-1}\,\Pi_{\varepsilon,0}$ (4.7)

which follows from

$\displaystyle\left(\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda\right)\\!\upharpoonright\\!\mathscr{H}_{N(\mu)}\geq|e_{\rm
P}(\mu,E_{B})|\geq\frac{C\mu}{\log\widetilde{\mu}},$ (4.8)

$a^{*}(\eta)\Pi_{0}(\varepsilon)=a^{*}(\eta^{>}_{\varepsilon})\Pi_{0}(\varepsilon)$,
and

$\displaystyle a(\eta^{>}_{\varepsilon})a^{*}(\eta^{>}_{\varepsilon})\ \leq\
\frac{1}{L^{2}}\sum_{k\in\Lambda_{h,\varepsilon}^{>}}1\ \leq\
\frac{C\mu}{\varepsilon\log\widetilde{\mu}}.$ (4.9)

The latter is obtained via (3.7).

$\bullet$ Next we consider

$\displaystyle\Pi_{\varepsilon,0}\,$ $\displaystyle
H_{p}(\lambda)\,\Pi_{\varepsilon,1}\,+\,\text{h.c.}$ $\displaystyle=\
\Pi_{\varepsilon,0}\,a(\eta^{>}_{\varepsilon})\frac{1}{\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda}a^{*}(\eta)\,\Pi_{\varepsilon,1}\,+\,\text{h.c.}$
$\displaystyle\leq\frac{(\varepsilon\log\widetilde{\mu})^{2}}{\mu}\,\Pi_{\varepsilon,0}\,a(\eta^{>}_{\varepsilon})a^{*}(\eta^{>}_{\varepsilon})\,\Pi_{\varepsilon,0}+\frac{\mu}{(\varepsilon\log\widetilde{\mu})^{2}}\,\Pi_{\varepsilon,1}\,a(\eta)\frac{1}{(T-\lambda)^{2}}a^{*}(\eta)\
\Pi_{\varepsilon,1}$ $\displaystyle\leq
C\big{(}\varepsilon\log\widetilde{\mu}\,\Pi_{\varepsilon,0}+\varepsilon^{-2}\,\Pi_{\varepsilon,1}\big{)},$
(4.10)

where we made use of (3.39), (4.8) and (4.9).

$\bullet$ The contribution

$\displaystyle\Pi_{\varepsilon,0}\,H_{p}(\lambda)\,\Pi_{\varepsilon,2+}\,+\,\text{h.c.}=0$
(4.11)

vanishes identically since

$\displaystyle a(\eta)\frac{1}{\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda}a^{*}(\eta)\Pi_{\varepsilon,2+}w\in\text{Ran}(\Pi_{\varepsilon,1})\oplus\text{Ran}(\Pi_{\varepsilon,2+})$
(4.12)

for any $w\in\mathscr{H}_{N(\mu)-1}$ and
$\Pi_{\varepsilon,0}\Pi_{\varepsilon,1}=\Pi_{\varepsilon,0}\Pi_{\varepsilon,2+}=0$.
(The operator $a^{*}(\eta)$ can reduce the number of unoccupied modes at most
by one.)

$\bullet$ We proceed with

$\displaystyle\Pi_{\varepsilon,1}\,H_{p}(\lambda)\,\Pi_{\varepsilon,2+}\,+\,\text{h.c.}\,=\,\Pi_{\varepsilon,1}\,a(\eta^{>}_{\varepsilon})\frac{1}{\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda}a^{*}(\eta)\,\Pi_{\varepsilon,2+}\,+\,\text{h.c.}$ (4.13)

which holds because of

$\displaystyle(a(\eta)-a(\eta^{>}_{\varepsilon}))\frac{1}{\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda}a^{*}(\eta)\,\Pi_{\varepsilon,2+}w\in\text{Ran}(\Pi_{\varepsilon,2+})$
(4.14)

and $\Pi_{\varepsilon,1}\Pi_{\varepsilon,2+}=0$. (Note that
$a(\eta)-a(\eta^{>}_{\varepsilon})$ adds an unoccupied mode in
$\Lambda^{\leq}_{{\rm h},\varepsilon}$.) We estimate the r.h.s. of (4.13) from
above by

$\displaystyle\frac{\varepsilon\log\widetilde{\mu}}{\mu}\,\Pi_{\varepsilon,1}a(\eta^{>}_{\varepsilon})a^{*}(\eta^{>}_{\varepsilon})\,\Pi_{\varepsilon,1}+\frac{\mu}{\varepsilon\log\widetilde{\mu}}\,\Pi_{\varepsilon,2+}\,a(\eta)\frac{1}{(\frac{1}{M}(p-P_{\rm
f})^{2}+T-\lambda)^{2}}a^{*}(\eta)\,\Pi_{\varepsilon,2+}$ $\displaystyle\hskip
227.62204pt\leq
C\big{(}\Pi_{\varepsilon,1}+\varepsilon\log\widetilde{\mu}\,\Pi_{\varepsilon,2+}\big{)},$
(4.15)

where we used another time that states of the form
$\psi=a^{*}(\eta)\Pi_{\varepsilon,2+}w\in\mathscr{H}_{N(\mu)}$ are either zero
or have at least one unoccupied mode in $\Lambda_{h,\varepsilon}^{\leq}$. The
latter implies

$\displaystyle\left\langle\psi,(T-\lambda)^{-s}\psi\right\rangle\leq\left\langle\psi,\psi\right\rangle\frac{(\varepsilon\log\widetilde{\mu})^{s}}{\mu^{s}}\quad(s>0).$
(4.16)

$\bullet$ In the bound for
$\Pi_{\varepsilon,2+}\,H_{p}(\lambda)\,\Pi_{\varepsilon,2+}$ we use (4.16)
with $s=1$ to get

$\displaystyle\Pi_{\varepsilon,2+}\,H_{p}(\lambda)\,\Pi_{\varepsilon,2+}\leq\frac{C\varepsilon\log\widetilde{\mu}}{\mu}\Pi_{\varepsilon,2+}a(\eta)a^{*}(\eta)\Pi_{\varepsilon,2+}\leq\
C\varepsilon\log\widetilde{\mu}\,\Pi_{\varepsilon,2+}$ (4.17)

by means of (3.39).

So far we have shown

$\displaystyle\Pi_{\varepsilon}\,H_{p}(\lambda)\,\Pi_{\varepsilon}-\Pi_{\varepsilon,1}\,H_{p}(\lambda)\,\Pi_{\varepsilon,1}\,\geq\,C\varepsilon^{-2}\,\Pi_{\varepsilon,1}+C\varepsilon\log\widetilde{\mu}\,\big{(}\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+}\big{)}.$
(4.18)

For the bounds involving $X_{p}(\lambda)$, we recall (3.42).

$\bullet$ With $f(\mu,E_{B})=g(\mu,E_{B})=\mu/\sqrt{\log\widetilde{\mu}}$, we
obtain

$\displaystyle(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+})\,X_{p}(\lambda)\,(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+})\,\leq\,C\sqrt{\log\widetilde{\mu}}\,\big{(}\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+}\big{)}.$
(4.19)

$\bullet$ Choosing $f(\mu,E_{B})=\mu/(\varepsilon\log\widetilde{\mu})$ and
$g(\mu,E_{B})=\varepsilon\mu$ leads to

$\displaystyle\Pi_{\varepsilon,1}\,X(\lambda)\,(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+})\,+\,\text{h.c.}\,\leq\,C\varepsilon^{-1}\
\Pi_{\varepsilon,1}\,+\,C\varepsilon\log\widetilde{\mu}\
(\Pi_{\varepsilon,0}+\Pi_{\varepsilon,2+}).$ (4.20)

$\bullet$ For the term with $\Pi_{\varepsilon,1}$ on both sides, the estimate
in (3.42) is not good enough (for obtaining an error of order one w.r.t.
$\widetilde{\mu}$). A possible improvement, however, is readily obtained from

$\displaystyle\Pi_{\varepsilon,1}\,X_{p}(\lambda)\,\Pi_{\varepsilon,1}\,=\,\Pi_{\varepsilon,1}\,\left(A_{p}(\lambda)a^{*}(\eta^{>}_{\varepsilon})+\text{h.c.}\right)\,\Pi_{\varepsilon,1}$
(4.21)

which is true since
$A_{p}(\lambda)(a^{*}(\eta)-a^{*}(\eta^{>}_{\varepsilon}))\Pi_{\varepsilon,1}w\in\text{Ran}(\Pi_{\varepsilon,0})$
and $\Pi_{\varepsilon,1}\Pi_{\varepsilon,0}=0$. Following now the same steps
that led to (3.42) and using in addition (4.9), we obtain

$\displaystyle\Pi_{\varepsilon,1}\,X_{p}(\lambda)\,\Pi_{\varepsilon,1}\,$
$\displaystyle\leq\,\Pi_{\varepsilon,1}\left(f(\mu,E_{B})\,A_{p}(\lambda)A_{p}^{*}(\lambda)\,+\,\frac{a(\eta^{>}_{\varepsilon})a^{*}(\eta^{>}_{\varepsilon})}{g(\mu,E_{B})}\right)\Pi_{\varepsilon,1}\,$
$\displaystyle\leq\,C\left(\frac{f(\mu,E_{B})\log\widetilde{\mu}}{\mu}\,+\,\frac{\mu}{g(\mu,E_{B})\varepsilon\log\widetilde{\mu}}\right)\,\Pi_{\varepsilon,1}.$
(4.22)

With $f(\mu,E_{B})=\mu/(\sqrt{\varepsilon}\log\widetilde{\mu})$ and
$g(\mu,E_{B})=\sqrt{\varepsilon}\mu/\log\widetilde{\mu}$, this provides
$\Pi_{\varepsilon,1}X_{p}(\lambda)\,\Pi_{\varepsilon,1}\leq
C\varepsilon^{-1/2}\,\Pi_{\varepsilon,1}$.

Bringing the above bounds together proves (4.4). ∎

The goal of the next lemma is to analyze the condition $pol(\lambda)-r\geq 0$
for a given number $r\geq 0$. To see for which $\lambda$ such a bound may
hold, we use the fact that this operator is given by an expression of the form
$K-V^{*}V$ with $K=G-r$ and $V=(T-\lambda)^{-1/2}a^{*}(\eta)$. If $K$ is self-
adjoint and $K\geq c$ for some number $c>0$, it follows easily that

$\displaystyle K-V^{*}V\,$
$\displaystyle=\,(K-V^{*}V)K^{-1}(K-V^{*}V)+V^{*}(1-VK^{-1}V^{*})V$
$\displaystyle\geq\,V^{*}(1-VK^{-1}V^{*})V.$ (4.23)

This is a key argument in the proof of the following proposition.

######  Proposition 4.2.

For any fixed $r>0$ there exists a constant $c_{0}>0$ such that for all
$L^{2}|E_{B}|\geq 1$ and $\mu/|E_{B}|\geq c_{0}$ the following implication
holds: $pol(\lambda)\geq r$ if $\lambda$ satisfies

$\displaystyle
E_{0}(\mu)-\lambda-\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\frac{1}{G(0,E_{0}(\mu)-\lambda-k^{2})-r}=0.$
(4.24)

Remark 4.1. We call (4.24) the perturbed polaron equation.

###### Proof.

Since $T\geq E_{0}(\mu)-\mu$ on $\mathscr{H}_{N(\mu)-1}$ and $\lambda\leq
E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$, it follows by (3.18) that $G(0,T-\lambda)$
exceeds the value of $r$ for $\widetilde{\mu}$ large enough. For such
$\widetilde{\mu}$ we can use (4.23) to find

$\displaystyle
pol(\lambda)-r\,\geq\,a(\eta)\frac{1}{T-\lambda}\,\mathcal{F}(T-\lambda,r)\,\frac{1}{T-\lambda}a^{*}(\eta)\\!\upharpoonright\\!\mathscr{H}_{N(\mu)-1}$
(4.25)

where

$\displaystyle\mathcal{F}(T-\lambda,r)=\left(T-\lambda-a^{*}(\eta)\frac{1}{G(0,T-\lambda)-r}a(\eta)\right)\\!\upharpoonright\\!\mathscr{H}_{N(\mu)}.$
(4.26)

From here it follows similarly as in the proof of [20, Lemma 4.2] that
$\mathcal{F}(T-\lambda,r)\geq 0$ if $\lambda$ satisfies the inequality

$\displaystyle
E_{0}(\mu)-\lambda-\frac{1}{L^{2}}\sum_{k^{2}}\frac{1}{G(0,E_{0}(\mu)-k^{2}-\lambda)-r}\geq
0.$ (4.27)

For convenience of the reader we provide the proof of the last statement in
Appendix B. ∎

Next we prove the existence of a unique solution to the perturbed polaron
equation (4.24) in the interval $(-\infty,E_{0}(\mu)+e_{\rm P}(\mu,E_{B})]$
and provide a suitable estimate for the difference of this solution and
$E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$.

######  Lemma 4.3.

For any fixed $r>0$ there exists a constant $c_{0}>0$ such that for all
$L^{2}|E_{B}|\geq 1$ and $\mu/|E_{B}|\geq c_{0}$, the perturbed polaron
equation (4.24) admits a unique solution in the interval
$(-\infty,E_{0}(\mu)+e_{\rm P}(\mu,E_{B})]$. We call this solution
$\lambda(\mu,E_{B})$.222The omission of the $r$ dependence of
$\lambda(\mu,E_{B})$ is justified by (4.28). Moreover there exists a constant
$C>0$ such that

$\displaystyle E_{0}(\mu)+e_{\rm
P}(\mu,E_{B})-\lambda(\mu,E_{B})\,\leq\,C(1+r)\,\frac{|e_{P}(\mu,E_{B})|}{\log(\mu/|E_{B}|)}$
(4.28)

for all $L^{2}|E_{B}|\geq 1$ and $\mu/|E_{B}|\geq c_{0}$.

###### Proof.

To prove the existence of a solution we write (4.24) as
$E_{0}(\mu)-\lambda=f(\lambda)$ with

$\displaystyle
f(\lambda)=\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\frac{1}{G(0,E_{0}(\mu)-\lambda-k^{2})-r}$
(4.29)

a continuous monotonically increasing function
$f(\lambda):(-\infty,E_{0}(\mu)+e_{\rm P}(\mu,E_{B})]\to\mathbb{R}$. By
definition of $G(q,\tau)$ we have $f(\lambda)\to 0$ as $\lambda\to-\infty$.
Next consider $f(\Lambda(\mu,E_{B}))$ with
$\Lambda(\mu,E_{B})=E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$. With the help of (1.10),

$\displaystyle f(\Lambda(\mu,E_{B}))$ $\displaystyle=-e_{\rm P}(\mu,E_{B})$
$\displaystyle\quad+\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\frac{r+G(k,-e_{\rm
P}(\mu,E_{B})-k^{2})-G(0,-e_{\rm P}(\mu,E_{B})-k^{2})}{(G(0,-e_{\rm
P}(\mu,E_{B})-k^{2})-r)\,G(k,-e_{\rm P}(\mu,E_{B})-k^{2})},$ (4.30)

and by way of Lemma 3.2, $G(k,-e_{\rm P}(\mu,E_{B})-k^{2})-G(0,-e_{\rm
P}(\mu,E_{B})-k^{2})\geq-2C$, we see that the second line in (4.30) is bounded
from below by a constant times $-\mu/(\log\widetilde{\mu})^{2}$. Hence for all
$\widetilde{\mu}$ large enough, we infer
$f(\Lambda(\mu,E_{B}))\geq\frac{1}{2}|e_{\rm P}(\mu,E_{B})|>0$. These
observations imply that there is a unique
$\lambda(\mu,E_{B})\in(-\infty,\Lambda(\mu,E_{B}))$ such that
$f(\lambda(\mu,E_{B}))=E_{0}(\mu)-\lambda(\mu,E_{B})$.

The difference between $\Lambda(\mu,E_{B})$ and $\lambda(\mu,E_{B})$ is
estimated by

$\displaystyle\Lambda(\mu,E_{B})-\lambda(\mu,E_{B})$
$\displaystyle=\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\bigg{(}\frac{r+G(k,-e_{\rm
P}(\mu,E_{B})-k^{2})-G(0,E_{0}(\mu)-k^{2}-\lambda(\mu,E_{B}))}{G(k,-e_{\rm
P}(\mu,E_{B})-k^{2})(G(0,E_{0}(\mu)-k^{2}-\lambda(\mu,E_{B}))-r)}\bigg{)}$
$\displaystyle\leq\frac{C(1+r)}{(\log\widetilde{\mu})^{2}}\bigg{(}\mu+\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\big{(}G(k^{2},-e_{\rm
P}(\mu,E_{B})-k^{2})-G(0,E_{0}(\mu)-k^{2}-\lambda(\mu,E_{B}))\big{)}\bigg{)},$
(4.31)

where we used Lemma 3.2 to estimate the denominator from below by a constant
times $(\log\widetilde{\mu})^{2}$. In the remainder we show

$\displaystyle\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}\big{(}G(k,-e_{\rm
P}(\mu,E_{B})-k^{2})-G(0,E_{0}(\mu)-k^{2}-\lambda(\mu,E_{B}))\big{)}\leq C\mu$
(4.32)

which proves that the left side of (4.28) is bounded from above by
$C(1+r)\mu/(\log(\mu/|E_{B}|))^{2}$.

To verify (4.32) we use $\lambda(\mu,E_{B})\leq E_{0}(\mu)+e_{\rm
P}(\mu,E_{B})$ and again Lemma 3.2 to estimate the expression inside the
brackets from above by

$\displaystyle\frac{1}{4\pi m}\log\Bigg{(}\frac{\frac{k^{2}}{M+1}-e_{\rm
P}(\mu,E_{B})+m\mu-k^{2}}{-e_{\rm P}(\mu,E_{B})+m\mu-k^{2}}\Bigg{)}+2C.$
(4.33)

With $0\leq k^{2}\leq\mu$, $0\leq-e_{\rm P}(\mu,E_{B})\leq\mu$ and $|e_{\rm
P}(\mu,E_{B})|=O(\mu/\log\widetilde{\mu})$, one further verifies that the
logarithm is bounded from above by $\log((2M^{2}+4M+1)/(M+1))\leq C$. ∎

Let us summarize the result of this section.

######  Corollary 4.4.

For any fixed $\varepsilon>0$ there exist constants $c_{0},C>0$ such that
$\lambda(\mu,E_{B})\leq E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$, the unique solution
of the perturbed polaron equation (4.24) with $r=\varepsilon^{-3}$, satisfies
the following two properties: $\Phi_{p}(\lambda(\mu,E_{B}))\geq 0$ and

$\displaystyle\lambda(\mu,E_{B})-E_{0}(\mu)-e_{\rm
P}(\mu,E_{B})\,\geq\,-C(1+\varepsilon^{-3})\,\frac{|e_{P}(\mu,E_{B})|}{\log(\mu/|E_{B}|)}$
(4.34)

for all $p\in\kappa\mathbb{Z}^{2}$, $L^{2}|E_{B}|\geq 1$ and $\mu/|E_{B}|\geq
c_{0}$.

In the next section we show that $\Psi_{p}(\lambda,\varepsilon)\geq 0$ for all
$\lambda\leq E_{0}(\mu)+e_{\rm P}(\mu,E_{B})$ provided that $M>1.225$ and
$\varepsilon$ is sufficiently small.

##  5 Analysis of $\Psi_{p}(\lambda,\varepsilon)$: stability condition

On the subspace $\text{Ran}(\Pi^{\perp}_{\varepsilon})$ it is not clear how to
obtain a suitable $L$-independent bound for the operator $P_{p}(\lambda)$. A
possible solution to this difficulty is to estimate its negative part in terms
of $G(p-P_{\rm f},T-\lambda)$. Such a bound was derived in [19, 13] in the
context of the 2D Fermi polaron at zero density (there the model is defined on
$\mathbb{R}^{2}$ instead of the box $\Omega$ and the kinetic energy
$E_{0}(\mu=0)$ is zero). The strategy of our proof follows the one developed
there, but several new obstacles need to be dealt with in the present case.
The new obstacles are due to $\mu>0$ and the fact that we have to work with
momentum sums instead of integrals.

We write
$P_{p}(\lambda)=P_{p}(\lambda)-\widetilde{P}_{p}(\lambda,\varepsilon)+\widetilde{P}_{p}(\lambda,\varepsilon)$
where

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)=\frac{1}{L^{2}}\sum_{k^{2},l^{2}>\mu/\varepsilon}a_{l}^{*}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-k-l)^{2}+T+k^{2}+l^{2}-\lambda}a_{k}.$ (5.1)

The operator $P_{p}(\lambda)\,-\,\widetilde{P}_{p}(\lambda,\varepsilon)$ is
the easy part and can be estimated by the following lemma.

######  Lemma 5.1.

There are constants $c_{0},\varepsilon_{0},C>0$ such that

$\displaystyle
P_{p}(\lambda)\,-\,\widetilde{P}_{p}(\lambda,\varepsilon)\,\geq-C\,\sqrt{\varepsilon^{-1}\log(\mu/|E_{B}|)}$
(5.2)

on $\mathscr{H}_{N(\mu)-1}$ for all $p\in\kappa\mathbb{Z}^{2}$,
$L^{2}|E_{B}|\geq 1$, $\mu/|E_{B}|\geq c_{0}$ and
$\varepsilon\in(0,\varepsilon_{0})$.

###### Proof.

Write

$\displaystyle P_{p}(\lambda)\,-\,\widetilde{P}_{p}(\lambda,\varepsilon)$
$\displaystyle=\frac{1}{L^{2}}\sum_{\mu<k^{2},l^{2}\leq\mu/\varepsilon}a_{l}^{*}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-k-l)^{2}+T+k^{2}+l^{2}-\lambda}a_{k}$ (5.3)
$\displaystyle\quad+\frac{1}{L^{2}}\sum_{\begin{subarray}{c}\mu<k^{2}\leq\mu/\varepsilon\\\
l^{2}>\mu/\varepsilon\end{subarray}}a_{l}^{*}\frac{1}{\frac{1}{M}(p-P_{\rm
f}-k-l)^{2}+T+k^{2}+l^{2}-\lambda}a_{k}+\text{h.c}.$ (5.4)

We proceed as in the proof of (3.20) to obtain

$\displaystyle\eqref{eq: first line of P minus tilde P}$
$\displaystyle\geq-C\bigg{(}f(\mu,E_{B})\frac{\mu}{\varepsilon}+\frac{1}{f(\mu,E_{B})|e_{\rm
P}(\mu,E_{B})|}\bigg{)}.$ (5.5)

Choosing $f(\mu,E_{B})=\sqrt{\varepsilon^{-1}\log\widetilde{\mu}}/\mu$ we get
the desired bound for this line. The proof for the second line works in
complete analogy. ∎

To a large extent this section is about the derivation of a lower bound for
$\widetilde{P}_{p}(\lambda,\varepsilon)$. For that purpose we need to
introduce some further notation. For $\varepsilon$ small enough, let the
function $\beta(\cdot,\varepsilon):[0,1]\to(0,1]$ be given by

$\displaystyle\beta(u,\varepsilon)=\min\Bigg{\\{}1,\frac{(M+1-u)(M+2)\big{(}1-(1+\frac{M+1-u}{M(M+2)})\sqrt{\varepsilon}\big{)}}{(M+1-u)(1-2\sqrt{\varepsilon})+M(M+2)(1-\sqrt{\varepsilon})}\Bigg{\\}}$
(5.6)

and set

$\displaystyle\alpha(M,\varepsilon)=\frac{1}{2}\left(\frac{1}{M(1-\sqrt{\varepsilon})+1}+\int_{0}^{1}\frac{1}{\beta(u,\varepsilon)(M(1-\sqrt{\varepsilon})+1-u)}\textnormal{d}u\right).$
(5.7)

######  Proposition 5.2.

There are constants $c_{0},\varepsilon_{0},C>0$ such that

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)\,\geq\,-\frac{1}{1-\varepsilon}\,\bigg{(}\frac{\alpha(M,\varepsilon)}{4\pi}\,\log\left(1+\frac{T-\lambda+2\mu}{\mu}\right)+\frac{C}{\sqrt{\mu/|E_{B}|}}\bigg{)}$
(5.8)

on $\mathscr{H}_{N(\mu)-1}$ for all $p\in\kappa\mathbb{Z}^{2}$,
$L^{2}|E_{B}|\geq 1$, $\mu/|E_{B}|\geq c_{0}$ and
$\varepsilon\in(0,\varepsilon_{0})$.

To prove Proposition 5.2 we combine the next two lemmas.

######  Lemma 5.3.

Let $T_{>\mu/\varepsilon}=\sum_{k^{2}>\mu/\varepsilon}k^{2}\,a_{k}^{*}a_{k}$.
For any $\varepsilon\in(0,1)$, it holds that

$\displaystyle\frac{T_{>\mu/\varepsilon}}{T-\lambda+2\mu}\restriction\mathscr{H}_{N(\mu)-1}\,\leq\,\frac{1}{1-\varepsilon}.$
(5.9)

######  Lemma 5.4.

There are constant $c_{0},\varepsilon_{0},C>0$ such that

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)\geq-\frac{T_{>\mu/\varepsilon}}{T-\lambda+2\mu}\,\bigg{(}\frac{\alpha(M,\varepsilon)}{4\pi}\,\log\left(1+\frac{T-\lambda+2\mu}{\mu}\right)+\,\frac{C}{\sqrt{\mu/|E_{B}|}}\bigg{)}$
(5.10)

on $\mathscr{H}_{N(\mu)-1}$ for all $p\in\kappa\mathbb{Z}^{2}$,
$L^{2}|E_{B}|\geq 1$, $\mu/|E_{B}|\geq c_{0}$ and
$\varepsilon\in(0,\varepsilon_{0})$.

###### Proof of Proposition 5.2.

Since $T-\lambda\restriction\mathscr{H}_{N(\mu)-1}\geq-\mu$, the operator

$\displaystyle\log\left(1+\frac{T-\lambda+2\mu}{\mu}\right)\restriction\mathscr{H}_{N(\mu)-1}\geq\log(2)$
(5.11)

is positive. Since $T$ and $T_{>\mu/\varepsilon}$ commute, we can use Lemma
5.3 to prove Proposition 5.2 with the aid of (5.10). ∎

###### Proof of Lemma 5.3.

Using again $T-\lambda\restriction\mathscr{H}_{N(\mu)-1}\geq-\mu$ in
combination with $0\leq T_{>\mu/\varepsilon}\leq T$, the operator

$\displaystyle\frac{T_{>\mu/\varepsilon}}{T-\lambda+2\mu}\leq
1+\frac{\lambda-2\mu}{T-\lambda+2\mu}\leq 1+\frac{E_{0}(\mu)-2\mu}{\mu}$
(5.12)

is bounded when restricted to $\mathscr{H}_{N(\mu)-1}$. Hence it is sufficient
to show (5.9) on the dense subspace
$\text{lin}D\subseteq\mathscr{H}_{N(\mu)-1}$ given by all finite linear
combinations of anti-symmetric products of plane waves, see (3.4). Since the
states in $\text{lin}D$ are linear combinations of simultaneous eigenstates of
$T\\!\upharpoonright\\!\mathscr{H}_{N(\mu)-1}$ and
$T_{>\mu/\varepsilon}\\!\upharpoonright\\!\mathscr{H}_{N(\mu)-1}$, we can
restrict the argument further to the set $D$ itself. This becomes particularly
useful when writing $D=\bigcup_{n\geq 0}W_{n}(\varepsilon)$ with

$\displaystyle W_{n}(\varepsilon)=\Big{\\{}w\in
D\,:\,\Big{(}\sum_{k^{2}>\mu/\varepsilon}a_{k}^{*}a_{k}\Big{)}w=nw\Big{\\}}$
(5.13)

the set of anti-symmetric products of plane waves with exactly $n$ momentum
modes occupied in $\\{k\in\kappa\mathbb{Z}^{2}:k^{2}>\mu/\varepsilon\\}$.

Since $T_{>\mu/\varepsilon}w=0$ for $w\in W_{0}(\varepsilon)$, we consider
$w\in W_{n}(\varepsilon)$, $n\geq 1$, with $||w||^{2}=1$. We call $\beta(w)$
the eigenvalue of $T$ and $\gamma(w)$ the eigenvalue of
$T_{>\mu/\varepsilon}$. It follows that

$\displaystyle\gamma(w)=\mbox{$\langle
w,T_{>\mu/\varepsilon}w\rangle$}>n\,\mu\varepsilon^{-1}$ (5.14)

as well as

$\displaystyle\beta(w)-\gamma(w)=\mbox{$\langle
w,(T-T_{>\mu/\varepsilon})w\rangle$}\geq E_{0}(\mu)-(n+1)\,\mu.$ (5.15)

To derive the last inequality we denote the eigenvalues of $-\Delta$ by
$\lambda_{i}(-\Delta)$ ($i\geq 1$, numbered with increasing order and counting
multiplicities) and use

$\displaystyle\mbox{$\langle
w,(T-T_{>\mu/\varepsilon})w\rangle$}=\mbox{$\langle
w,(\sum_{k^{2}\leq\mu/\varepsilon}k^{2}a_{k}^{*}a_{k})w\rangle$}$
$\displaystyle\geq E_{0}(\mu)-\sum_{i=N(\mu)-n}^{N(\mu)}\lambda_{i}(-\Delta).$
(5.16)

From $\lambda_{i}(-\Delta)\leq\mu$ for $i\leq N(\mu)$, we obtain (5.15). The
latter together with $\lambda\leq E_{0}(\mu)$ implies

$\displaystyle\beta(w)-\lambda\,\geq\,\gamma(w)-(n+1)\,\mu,$ (5.17)

and combining this with (5.14), we get

$\displaystyle\mbox{$\langle
w,\frac{T_{>\mu/\varepsilon}}{T-\lambda+2\mu}w\rangle$}=\frac{\gamma(w)}{\beta(w)-\lambda+2\mu}\leq\frac{\gamma(w)}{\gamma(w)-(n-1)\,\mu}\leq\frac{n\,\mu\varepsilon^{-1}}{n\,\mu\varepsilon^{-1}-(n-1)\,\mu}.$
(5.18)

Since the expression on the right does not exceed the value
$\frac{1}{1-\varepsilon}$, we have proven the statement. ∎

###### Proof of Proposition 5.2.

We start by introducing the abbreviations

$\displaystyle\widehat{k}=k+\frac{1}{M+2}(p-P_{\rm
f}),\quad\quad\widehat{l}=l+\frac{1}{M+2}(p-P_{\rm f})$ (5.19)

by which one writes the denominator in the expression defining
$\widetilde{P}(\lambda,\varepsilon)$ as

$\displaystyle
m(\widehat{k}^{2}+\widehat{l}^{2})+\frac{2}{M}\widehat{k}\cdot\widehat{l}+\frac{1}{M+2}(p-P_{\rm
f})^{2}+T-\lambda.$ (5.20)

For $w\in\mathscr{H}_{N(\mu)-1}$ we define $\widetilde{w}\in
L^{2}(\kappa\mathbb{Z}^{2};\mathscr{H}_{N(\mu)-2})$ by
$\widetilde{w}(k)=a_{k}w$. Moreover we define the unitary operator
$U\in\mathscr{L}(L^{2}(\kappa\mathbb{Z}^{2};\mathscr{H}_{N(\mu)-2}))$
by333Note that we omit the $p$-dependence of the unitary operator $U$.

$\displaystyle(U\varphi)(k;k_{1},...,k_{N(\mu)-2})=\varphi\big{(}k+\frac{1}{M+2}\big{(}p-\sum_{i=1}^{N(\mu)-2}k_{i}\big{)};k_{1},...,k_{N(\mu)-2}\big{)},$
(5.21)

where we use the notation $(U\varphi)(k;k_{1},...,k_{N(\mu)-2})$ for the
Fourier space representation of $(U\varphi)(k)\in\mathscr{H}_{N(\mu)-2}$. With
these definitions at hand, it is not difficult to compute

$\langle w,\widetilde{P}_{p}(\lambda,\varepsilon)w\rangle$
$\displaystyle=\,\frac{1}{L^{2}}\sum_{k,l}\mbox{$\langle(\chi_{\mu/\varepsilon}\widetilde{w})(k),U\sigma(k,l)U^{*}(\chi_{\mu/\varepsilon}\widetilde{w})(l)\rangle$}$
(5.22)

where $\chi_{(\mu/\varepsilon,\infty)}$ stands for the characteristic function
$k\mapsto\chi_{(\mu/\varepsilon,\infty)}(k^{2})$ and where

$\displaystyle\sigma(k,l)=\frac{1}{L^{2}}\,\frac{1}{m(k^{2}+l^{2})+\frac{2}{M}k\cdot
l+\frac{1}{M+2}(p-P_{\rm f})^{2}+T-\lambda}.$ (5.23)

Denoting the scalar product on
$L^{2}(\kappa\mathbb{Z}^{2};\mathscr{H}_{N(\mu)-2})$ by
$\langle\langle\cdot,\cdot\rangle\rangle$, (5.22) is rewritten as

$\displaystyle\mbox{$\langle
w,\widetilde{P}_{p}(\lambda,\varepsilon)w\rangle$}\,=\,\langle\langle\chi_{(\mu/\varepsilon,\infty)}\widetilde{w},U\,\sigma\,U^{*}\chi_{(\mu/\varepsilon,\infty)}\widetilde{w}\rangle\rangle$
(5.24)

where $\sigma$ is the operator on
$L^{2}(\kappa\mathbb{Z}^{2};\mathscr{H}_{N(\mu)-2})$ with operator-valued
kernel $\sigma(k,l)$. Next, we show that the negative part of $\sigma$ has the
kernel $\sigma^{-}(k,l)=\frac{1}{2}\left(\sigma(-k,l)-\sigma(k,l)\right)$. To
this end, consider the reflection operator $R$ defined by
$(R\widetilde{w})(k)=\widetilde{w}(-k)$ for any $\widetilde{w}\in
L^{2}(\kappa\mathbb{Z}^{2};\mathscr{H}_{N(\mu)-2})$. It is straightforward to
verify $R\sigma=\sigma R$. Moreover, $R\sigma$ is a positive operator, which
can be seen as follows. The integral kernel of $R\sigma$ is given by
$(R\sigma)(k,l)=\sigma(-k,l)$ and has the integral representation

$\displaystyle\sigma(-k,l)$
$\displaystyle\,=\,\frac{1}{L^{2}}\,\int_{0}^{\infty}\,e^{-tk^{2}}\left(e^{-t(k-l)^{2}/M}\
e^{-t(\frac{1}{M+2}(p-P_{\rm
f})^{2}+T-\lambda)}\right)e^{-tl^{2}}\,\textnormal{d}t.$ (5.25)

Then use the following identity for $\psi\in L^{2}(\Omega)$ and its Fourier
transform $\widehat{\psi}\in\ell^{2}(\kappa\mathbb{Z}^{2})$,

$\displaystyle\frac{1}{L^{2}}\sum_{k,l}\overline{\widehat{\psi}(k)}\
e^{-t(k-l)^{2}/M}\ \widehat{\psi}(l)\ =\
\int_{\Omega}|\psi(x)|^{2}\sum_{k}e^{ikx}e^{-tk^{2}/M}\,\textnormal{d}x.$
(5.26)

This together with Poisson’s summation formula (see e.g. [12, Section 3.2])
and the fact that the Fourier transform of a Gaussian is a positive function
implies that $R\sigma$ is a positive operator. Consequently, we have
$R\sigma=|\sigma|$ since $R\sigma$ is positive and
$\sigma^{2}=(R\sigma)(R\sigma)$. The positive and negative parts of $\sigma$
are thus given by $\sigma^{\pm}=\pm(\sigma\pm R\sigma)/2$ and the
corresponding kernels by
$\sigma^{\pm}(k,l)=\pm(\sigma(k,l)\pm\sigma(-k,l))/2$.

We proceed by writing the kernel of the negative part as
$\sigma^{-}(k,l)=\frac{1}{2}\int_{-1}^{1}\frac{d}{du}\sigma(-uk,l)\textnormal{d}u$,
and hence

$\displaystyle\sigma^{-}(k,l)$ $\displaystyle=\frac{Mk\cdot
l}{L^{2}}\int\limits_{-1}^{1}\frac{1}{[(M+1)(k^{2}+l^{2})-2uk\cdot
l+B]^{2}}\textnormal{d}u,$ (5.27)

where $B=\frac{M}{M+2}(p-P_{\rm f})^{2}+M(T-\lambda)$. Using this in
combination with (5.24), we get

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)\,\geq\,-\frac{M}{L^{2}}\sum_{k^{2},l^{2}>\mu/\varepsilon}a_{k}^{*}\left(\
\int\limits_{-1}^{1}\frac{\widehat{k}\cdot\widehat{l}}{\big{[}(M+1)({\widehat{k}}^{2}+{\widehat{l}}^{2})-2u\widehat{k}\cdot\widehat{l}+B\big{]}^{2}}\textnormal{d}u\right)a_{l}.$
(5.28)

To the expression on the right we apply the following inequality which is a
version of the Schur test and is easily proven by applying the Cauchy-Schwarz
inequality two times,

$\displaystyle\sum_{k^{2},l^{2}>\mu/\varepsilon}a_{k}^{*}\,J(k,l)\,a_{l}\,\leq\,\sum_{k^{2}>\mu/\varepsilon}k^{2}\,a_{k}^{*}\left(\sum_{l^{2}>\mu/\varepsilon}\frac{|J(k,l)|}{l^{2}}\right)a_{k}$
(5.29)

for any family of bounded operators $(J(k,l))_{k,l\in\kappa\mathbb{Z}^{2}}$ on
$\mathcal{F}$ satisfying $J(k,l)^{*}=J(k,l)$. This provides

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)\,\geq\,-M\sum_{k^{2}>\mu/\varepsilon}k^{2}\,a_{k}^{*}\underbrace{\left(\frac{1}{L^{2}}\sum_{l^{2}>\mu/\varepsilon}\
\int\limits_{-1}^{1}\frac{|\widehat{k}\cdot\widehat{l}|}{l^{2}\big{[}(M+1)(\widehat{k}^{2}+\widehat{l}^{2})-2u\widehat{k}\cdot\widehat{l}+B\big{]}^{2}}\textnormal{d}u\right)}_{=f(k^{2},p-P_{\rm
f},T)}a_{k}$ (5.30)

as an operator inequality on $\mathscr{H}_{N(\mu)-1}$. Our next goal is to
find a suitable function $g$ such that for $k^{2}>\mu/\varepsilon$, we have
$f(k^{2},p-P_{\rm f},T)\leq g(T+k^{2})$ on $\mathscr{H}_{N(\mu)-2}$. For such
a function we have

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)$
$\displaystyle\,\geq\,-M\sum_{k^{2}>\mu/\varepsilon}k^{2}\,a^{*}_{k}\,g(T+k^{2})\,a_{k}\,\geq\,-MT_{>\mu/\varepsilon}\
g(T)\,$ (5.31)

since $g(T+k^{2})a_{k}=a_{k}g(T)$ and
$T_{>\mu/\varepsilon}=\sum_{k^{2}>\mu/\varepsilon}k^{2}a_{k}^{*}a_{k}$.

To find a suitable function $g$, it is helpful to check that the expression
inside the square brackets in the denominator in (5.30) is positive for
$\varepsilon$ small enough. To see this we use

$\displaystyle\widehat{k}^{2}\geq\sqrt{\varepsilon}k^{2}-\frac{\sqrt{\varepsilon}}{1-\sqrt{\varepsilon}}\,\frac{(p-P_{\rm
f})^{2}}{(M+2)^{2}},$ (5.32)

and similarly for $\widehat{l}^{2}$, together with
$k^{2},l^{2}>\mu/\varepsilon$ and $T-\lambda\geq-2\mu$ on
$\mathscr{H}_{N(\mu)-2}$ to find

$\displaystyle(M+1)(\widehat{k}^{2}+{\widehat{l}}^{2})-2u\widehat{k}\cdot\widehat{l}+B$
$\displaystyle\qquad\geq
2M(\varepsilon^{-1/2}-1)\mu+\bigg{(}1-\frac{\sqrt{\varepsilon}}{1-\sqrt{\varepsilon}}\,\frac{1}{M+2}\bigg{)}\frac{M}{M+2}(p-P_{\rm
f})^{2}.$ (5.33)

Next we use that $-2u\widehat{k}\cdot\widehat{l}\geq 0$ either for
$u\in[-1,0]$ or for $u\in[0,1]$. This makes the quotient in the definition of
$f$ larger and also independent of $u$ on the respective interval. On the
other interval, we employ
$0\geq-2u\widehat{k}\cdot\widehat{l}\geq-|u|(\widehat{k}^{2}+\widehat{l}^{2})$.
In both cases this leads to

$\displaystyle\int\limits_{-1}^{1}\frac{|\widehat{k}\cdot\widehat{l}|}{l^{2}\big{[}(M+1)(\widehat{k}^{2}+\widehat{l}^{2})-2u\widehat{k}\cdot\widehat{l}+B\big{]}^{2}}\textnormal{d}u$
$\displaystyle\leq\frac{|\widehat{k}\cdot\widehat{l}|}{l^{2}\big{[}(M+1)(\widehat{k}^{2}+\widehat{l}^{2})+B\big{]}^{2}}$
(5.34a)
$\displaystyle+\int\limits_{0}^{1}\frac{|\widehat{k}\cdot\widehat{l}|}{l^{2}\big{[}(M+1-u)(\widehat{k}^{2}+\widehat{l}^{2})+B\big{]}^{2}}\textnormal{d}u.$
(5.34b)

In the denominators we proceed with the bound

$\displaystyle(M+1-u)(\widehat{k}^{2}+\widehat{l}^{2})+B\geq
2\,|\widehat{k}\cdot\widehat{l}|\big{(}M(1-\sqrt{\varepsilon})+1-u\big{)}.$
(5.35)

The latter is verified by

$\displaystyle(M+1-u)(\widehat{k}^{2}+\widehat{l}^{2})+B$
$\displaystyle\quad\geq\bigg{(}\widehat{k}^{2}+\widehat{l}^{2}+\frac{(p-P_{\rm
f})^{2}M}{(M+1)(M+2)}\bigg{)}\bigg{(}M+1-u-\frac{2\mu
M}{\widehat{k}^{2}+\widehat{l}^{2}+\frac{(p-P_{\rm
f})^{2}}{(M+1)(M+2)}}\bigg{)}$ (5.36)

on $\mathscr{H}_{N(\mu)-2}$ in combination with

$\displaystyle\widehat{k}^{2}+\widehat{l}^{2}+\frac{(p-P_{\rm
f})^{2}M}{(M+1)(M+2)}>\frac{2\mu}{\sqrt{\varepsilon}}$ (5.37)

which, in turn, follows from (5.32) and $k^{2}+l^{2}\geq 2\mu/\varepsilon$.
Putting the different steps together, one obtains

$\displaystyle f(k^{2},p-P_{\rm f},T)$
$\displaystyle\leq\widetilde{f}(k^{2},p-P_{\rm
f},T,0)+\int_{0}^{1}\widetilde{f}(k^{2},p-P_{\rm f},T,u)\textnormal{d}u$
(5.38)

with

$\displaystyle\widetilde{f}(k^{2},p-P_{\rm
f},T,u)=\frac{1}{L^{2}}\sum_{l^{2}>\mu/\varepsilon}\frac{1}{2l^{2}(M(1-\sqrt{\varepsilon})+1-u)\big{[}(M+1-u)(\widehat{k}^{2}+\widehat{l}^{2})+B\big{]}}.$
(5.39)

In the expression inside the square brackets we estimate $\widehat{k}^{2}$ and
$\widehat{l}^{2}$ by (5.32) to get the lower bound

$\displaystyle\big{[}...\big{]}\geq(M+1-u)\bigg{(}\sqrt{\varepsilon}l^{2}+\sqrt{\delta}k^{2}-\frac{2\mu
M}{(M+1-u)}\bigg{)}+M(T-\lambda+2\mu)$ $\displaystyle\hskip
42.67912pt+\bigg{(}M(M+2)-\frac{\sqrt{\varepsilon}}{1-\sqrt{\varepsilon}}(M+1-u)-\frac{\sqrt{\delta}}{1-\sqrt{\delta}}(M+1-u)\bigg{)}\frac{(p-P_{\rm
f})^{2}}{(M+2)^{2}}.$ (5.40)

Requiring that the second line vanishes implies

$\displaystyle\sqrt{\delta}=\frac{M(M+2)(1-\sqrt{\varepsilon})-\sqrt{\varepsilon}(M+1-u)}{M(M+2)(1-\sqrt{\varepsilon})+(M+1-u)(1-2\sqrt{\varepsilon})}.$
(5.41)

Hence we can bound the expression in square brackets by

$\displaystyle(M+1-u)(\widehat{k}^{2}+\widehat{l}^{2})+B$
$\displaystyle\geq\frac{\sqrt{\varepsilon}}{2}l^{2}(M+1-u)+M\beta(u,\varepsilon)(T+k^{2}-\lambda+2\mu)$
(5.42)

with

$\displaystyle\beta(u,\varepsilon)=\min\\{1,\sqrt{\delta}(M+1-u)/M\\}.$ (5.43)

Note that for $\varepsilon$ small enough $\beta(0,\varepsilon)=1$. Applying
this to (5.39), we obtain

$\displaystyle\widetilde{f}(k^{2},p-P_{\rm f},T,u)$
$\displaystyle\leq\frac{1}{2(M(1-\sqrt{\varepsilon})+1-u)}\,\frac{1}{L^{2}}\sum_{l^{2}>\mu/\varepsilon}\frac{l^{-2}}{\frac{\sqrt{\varepsilon}}{2}l^{2}(M+1-u)+M\beta(u,\varepsilon)(T+k^{2}-\lambda+2\mu)}.$
(5.44)

Here we sum a non-negative and monotonically decreasing function of $l^{2}$ so
that we can apply Lemma A.1. To follow the next steps with more ease, let us
write

$\displaystyle\eqref{eq: f
tilde}=\frac{1}{X}\,\bigg{(}\frac{1}{L^{2}}\sum_{l^{2}>\mu/\varepsilon}\frac{1}{l^{2}(1+Yl^{2})}\bigg{)}$
(5.45)

with (all understood as operator on $\mathscr{H}_{N(\mu)-2}$)

$\displaystyle X=2(M(1-\sqrt{\varepsilon})+1-u)Z,\quad
Y=\frac{\sqrt{\varepsilon}(M+1-u)}{2Z},$ (5.46)

and $Z=M\beta(u,\varepsilon)(T+k^{2}-\lambda+2\mu)$. Since for $b>0$

$\displaystyle\int\limits_{\sqrt{\mu/\varepsilon}}^{\infty}\frac{1}{s(1+bs^{2})}\textnormal{d}s=\frac{1}{2}\log\bigg{(}1+\frac{\varepsilon}{\mu\,b}\bigg{)},\quad\int\limits_{\sqrt{\mu/\varepsilon}}^{\infty}\frac{1}{s^{2}(1+bs^{2})}\textnormal{d}s\leq\frac{1}{\sqrt{\mu/\varepsilon}}+\frac{\pi\sqrt{b}}{2},$
(5.47)

we obtain the bound

$\displaystyle\frac{1}{L^{2}}\sum_{l^{2}>\mu/\varepsilon}\frac{1}{l^{2}(1+Yl^{2})}$
$\displaystyle\leq\frac{1}{4\pi}\log\bigg{(}1+\frac{\varepsilon}{\mu\,Y}\bigg{)}$
$\displaystyle\quad+\frac{2}{\pi
L}\bigg{(}\frac{1}{\sqrt{\mu/\varepsilon}}+\frac{\pi}{2}\sqrt{Y}\bigg{)}+\bigg{(}\frac{4\sqrt{\mu/\varepsilon}}{\pi
L}+\frac{6}{L^{2}}\bigg{)}\frac{1}{\frac{\mu}{\varepsilon}(1+\frac{\mu}{\varepsilon}Y)}.$
(5.48)

Using $T-\lambda\geq-2\mu$ on $\mathscr{H}_{N(\mu)-2}$,
$k^{2}\geq\mu/\varepsilon$ and $L^{2}|E_{B}|\geq 1$, the second line is easily
seen to be bounded by a constant times $\widetilde{\mu}^{-1/2}$. In the first
line, we estimate

$\displaystyle\log\bigg{(}1+\frac{2\sqrt{\varepsilon}M\beta(u,\varepsilon)(T+k^{2}-\lambda+2\mu)}{(M+1-u)\mu}\bigg{)}\leq\log\bigg{(}1+\frac{T+k^{2}-\lambda+2\mu}{\mu}\bigg{)}$
(5.49)

by choosing $\varepsilon$ sufficiently small. This together with (5.45) leads
to

$\displaystyle\widetilde{f}(k^{2},p-P_{\rm f},T,u)$
$\displaystyle\leq\frac{1}{T+k^{2}-\lambda+2\mu}\bigg{(}\frac{\log\big{(}1+\frac{T+k^{2}-\lambda+2\mu}{\mu}\big{)}}{8\pi
M(M(1-\sqrt{\varepsilon})+1-u)\beta(u,\varepsilon)}+\frac{C}{\sqrt{\widetilde{\mu}}}\bigg{)}.$
(5.50)

Recalling definition (5.7) for $\alpha(M,\varepsilon)$, we set

$\displaystyle
g(T)=\frac{1}{T-\lambda+2\mu}\bigg{(}\frac{\alpha(M,\varepsilon)}{4\pi
M}\log\left(1+\frac{T-\lambda+2\mu}{\mu}\right)+\frac{C}{\sqrt{\widetilde{\mu}}}\bigg{)}$
(5.51)

for some suitable constant $C$. In view of (5.38) and (5.50), it follows that
$f(k^{2},p-P_{\rm f},T)\leq g(T+k^{2})$ as desired. With the aid of (5.31)
this leads to

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)\,$
$\displaystyle\geq\,-\frac{T_{>\mu/\varepsilon}}{T-\lambda+2\mu}\left(\frac{\alpha(M,\varepsilon)}{4\pi}\,\log\left(1+\frac{T-\lambda-2\mu}{\mu}\right)+\frac{C}{\sqrt{\widetilde{\mu}}}\right)$
(5.52)

for some constant $C>0$ and thus the proof of the lemma is complete. ∎

The next statement is the main result of this section. Let us mention that the
condition $M>1.225$ enters as a technical assumption and is not expected to be
optimal.

######  Corollary 5.5.

Let $M>1.225$ and $\lambda\leq E_{0}(\mu)+e_{\rm{P}}(\mu,E_{B})$. There exist
constants $c_{0},\varepsilon_{0}>0$ such that
$\Psi_{p}(\lambda,\varepsilon)\geq 0$ for all $p\in\kappa\mathbb{Z}^{2}$,
$L^{2}|E_{B}|\geq 1$, $\mu/|E_{B}|\geq c_{0}$ and
$\varepsilon\in(0,\varepsilon_{0})$.

###### Proof.

Recalling the definition of $\Psi_{p}(\lambda,\varepsilon)$ in (3.6), we write

$\displaystyle\Psi_{p}(\lambda,\varepsilon)=\Pi^{\perp}(\varepsilon)(\Psi_{p,1}(\lambda,\varepsilon)+\Psi_{p,2}(\lambda,\varepsilon))\Pi^{\perp}(\varepsilon)$
(5.53)

with

$\displaystyle\Psi_{p,1}(\lambda,\varepsilon)$
$\displaystyle=\varepsilon^{1/3}\,G(p-P_{\rm
f},T-\lambda)+P_{p}(\lambda)-\widetilde{P}_{p}(\lambda,\varepsilon)-K(\varepsilon,\widetilde{\mu})-d,$
(5.54) $\displaystyle\Psi_{p,2}(\lambda,\varepsilon)$
$\displaystyle\,=\,(1-\varepsilon^{1/3})G(p-P_{\rm
f},T-\lambda)\,+\,\widetilde{P}_{p}(\lambda,\varepsilon)+d,$ (5.55)

where
$K(\varepsilon,\widetilde{\mu})=\varepsilon^{-1}+\varepsilon^{-1/2}\sqrt{\log\widetilde{\mu}}+\varepsilon^{1/2}\log\widetilde{\mu}$
and $d>0$ is a constant that we choose large enough but fixed w.r.t. all
parameters.

By means of inequality (3.18) and Lemma 5.1, we estimate

$\displaystyle\Psi_{p,1}(\lambda,\varepsilon)\geq\frac{\varepsilon^{1/3}}{4\pi
m}\log\widetilde{\mu}-C\big{(}\varepsilon^{1/3}+d+\sqrt{\varepsilon^{-1}\,\log\widetilde{\mu}}+K(\varepsilon,\widetilde{\mu})\big{)}$
(5.56)

for some ($\varepsilon$-independent) $C>0$. With $\varepsilon_{0}>0$ small
enough, the right side is non-negative for all $\widetilde{\mu}$ large enough.

In line (5.55) we apply (3.17) to get

$\displaystyle(1-\varepsilon^{1/3})G(p-P_{\rm f},T-\lambda)+\frac{d}{2}$
$\displaystyle\geq\frac{1-\varepsilon^{1/3}}{4\pi
m}\log\left(\frac{T-\lambda+m\mu}{|E_{B}|}\right)$ (5.57)

on $\mathscr{H}_{N(\mu)-1}$. Since $m=1+\frac{1}{M}$, $\mu/|E_{B}|\geq
c_{0}\geq 2M$ as well as
$(T-\lambda+\mu)\restriction\mathscr{H}_{N(\mu)-1}\geq 0$, we can estimate the
logarithm further by

$\displaystyle\log\left(\frac{T-\lambda+m\mu}{|E_{B}|}\right)\geq\log\left(\frac{T-\lambda+\mu+c_{0}\mu/M}{\mu}\right)\geq\log\left(\frac{T-\lambda+3\mu}{\mu}\right).$
(5.58)

Proposition 5.2 gives a bound for the second term in (5.55),

$\displaystyle\widetilde{P}_{p}(\lambda,\varepsilon)+\frac{d}{2}$
$\displaystyle\,\geq\,-\frac{1}{1-\varepsilon}\,\frac{\alpha(M,\varepsilon)}{4\pi}\log\bigg{(}1+\frac{T-\lambda+2\mu}{\mu}\bigg{)}.$
(5.59)

Adding both estimates together leads to

$\displaystyle\Psi_{p,2}(\lambda,\varepsilon)$
$\displaystyle\,\geq\,\bigg{(}\frac{1-\varepsilon^{1/3}}{4\pi
m}-\frac{1}{1-\varepsilon}\,\frac{\alpha(M,\varepsilon)}{4\pi}\bigg{)}\log\bigg{(}1+\frac{T-\lambda+2\mu}{\mu}\bigg{)}$
(5.60)

on $\mathscr{H}_{N(\mu)-1}$.

The condition $\Psi_{p}(\lambda,\varepsilon)\geq 0$ is thus satisfied if

$\displaystyle(1-\varepsilon^{1/3})\,\frac{M}{M+1}-\frac{1}{1-\varepsilon}\,\alpha(M,\varepsilon)\,\geq\,0.$
(5.61)

This is similar to the stability condition at zero density that was derived in
[13]. There it was shown that the Fermi polaron defined on $\mathbb{R}^{2}$ is
stable if

$\displaystyle\frac{M}{M+1}-\alpha(M,0)\geq 0$ (5.62)

which was proven to hold for all $M>1.225$ [13, Theorem 1]. Since
$\alpha(M,\varepsilon)$ depends continuously on $\varepsilon$, we can conclude
that (5.61) holds for any given $M>1.225$ if we choose $\varepsilon$
sufficiently small. This completes the proof of the corollary. ∎

###  5.1 Proof of Theorem 1.2

The lower bound in (1.14) is a direct consequence of the Birman–Schwinger
principle (2.2) together with Corollaries 4.4 and 5.5. As the upper bound was
already discussed in Section 2.1, we have completed the proof of Theorem 1.2.

##  6 Proof of Lemma 3.2

As a first step we replace $G(q,\tau)$ by

$\displaystyle\widetilde{G}(q,\tau)=\frac{1}{L^{2}}\sum_{k}\left(\frac{1}{mk^{2}-E_{B}}-\frac{\xi_{\mu}(k^{2})}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau}\right),$
(6.1)

where

$\displaystyle\xi_{\mu}(s)=\begin{cases}&\quad\quad\quad\quad 0\hskip
81.09052pt(s\leq\mu)\\\
&\frac{1}{2}\cos\Big{(}\frac{\pi(s-\mu)\log\widetilde{\mu}}{\mu}\Big{)}+\frac{1}{2}\quad\quad(\mu\leq
s\leq\mu+\mu/\log\widetilde{\mu})\\\ &\quad\quad\quad\quad 1\hskip
82.22844pt(s\geq\mu+\mu/\log\widetilde{\mu}).\end{cases}$ (6.2)

Compared to $G(q,\tau)$ we have replaced the characteristic function
$\chi_{(\mu,\infty)}(k^{2})$ with a smoother cutoff described by
$\xi_{\mu}(k^{2})$. The error for this can be controlled by a crude estimate
like

$\displaystyle|G(q,\tau)-\widetilde{G}(q,\tau)|$
$\displaystyle\leq\frac{1}{L^{2}}\sum_{k^{2}\geq\mu}\frac{\chi_{(\mu,\mu+\mu/\log\widetilde{\mu})}(k^{2})}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau}\leq
C\frac{\mu}{(\mu+\tau)\log\widetilde{\mu}},$ (6.3)

which is easily justified by means of (3.7). Next we write
$\widetilde{G}(q,\tau)=L^{-2}\sum_{k}g(k)$ with

$\displaystyle
g(k)=\frac{1}{mk^{2}+|E_{B}|}-\frac{\xi_{\mu}(k^{2})}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau},$
(6.4)

and apply Poisson’s summation formula (see e.g. [12, Section 3.2]) to find

$\displaystyle\widetilde{G}(q,\tau)-\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{2}}g(k)\textnormal{d}^{2}k=\frac{1}{4\pi^{2}}\bigg{(}\left(\frac{2\pi}{L}\right)^{2}\sum_{k\in\kappa\mathbb{Z}^{2}}g(k)-(2\pi)\widehat{g}(0)\bigg{)}=\frac{1}{2\pi}\sum_{\begin{subarray}{c}z\in
L\mathbb{Z}^{2}\\\ z\neq 0\end{subarray}}\widehat{g}(z).$ (6.5)

To compute $\widehat{g}(0)$ we replace $\xi_{\mu}(s)$ again with
$\chi_{(\mu,\infty)}(k^{2})$ and estimate the difference

$\displaystyle\Bigg{|}\,(2\pi)\widehat{g}(0)-\int_{\mathbb{R}^{2}}\left(\frac{1}{mk^{2}+|E_{B}|}-\frac{\chi_{(\mu,\infty)}(k^{2})}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau}\right)\textnormal{d}^{2}k\,\Bigg{|}\leq
C\frac{\mu}{(\mu+\tau)\log\widetilde{\mu}}.$ (6.6)

The integral can be evaluated explicitly,

$\displaystyle\int_{\mathbb{R}^{2}}\left(\frac{1}{mk^{2}+|E_{B}|}-\frac{\chi_{(\mu,\infty)}(k^{2})}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau}\right)\textnormal{d}^{2}k$
$\displaystyle\hskip
99.58464pt=\frac{\pi}{m}\log\left(\frac{\frac{1}{M+1}q^{2}+\tau+m\mu}{|E_{B}|}\right)+\frac{\pi}{m}\log\left(1-\frac{F(q^{2},\tau)}{2}\right)$
(6.7)

where

$\displaystyle
F(s,\tau)=\frac{\frac{1}{M}s+\tau+m\mu}{\frac{1}{M+1}s+\tau+m\mu}\left(1-\sqrt{1-\frac{4s\frac{\mu}{M^{2}}}{\big{(}\frac{1}{M}s+\tau+m\mu\big{)}^{2}}}\right).$
(6.8)

The last expression is not larger than $1+1/M$ such that for $M>1$, we have

$\displaystyle\bigg{|}\log\left(1-\frac{F(q^{2},\tau)}{2}\right)\bigg{|}\leq
C.$ (6.9)

It follows that

$\displaystyle\Bigg{|}\,(2\pi)\widehat{g}(0)-\frac{\pi}{m}\log\left(\frac{\frac{1}{M+1}q^{2}+\tau+m\mu}{|E_{B}|}\right)\Bigg{|}\leq
C\bigg{(}1+\frac{\mu}{(\mu+\tau)\log\widetilde{\mu}}\bigg{)}.$ (6.10)

Next we need to estimate the right side in (6.5). To this end, write
$g(k)=g_{1}(k)+g_{2}(k)$ with

$\displaystyle g_{1}(k)=\frac{1}{mk^{2}+|E_{B}|},\quad\quad
g_{2}(k)=\frac{\xi_{\mu}(k^{2})}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau},$ (6.11)

and use rotational symmetry, i.e.
$\widehat{g}_{i}(z)=\widehat{g}_{i}(|z|e_{u})$, $i=1,2$, where $e_{u}$ denotes
the first unit vector in the $(k_{u},k_{v})$ plane. We can then use
integration by parts to compute the Fourier transform for $z\neq 0$,

$\displaystyle\widehat{g}_{1}(z)$
$\displaystyle=m^{-1}\int\limits_{-\infty}^{\infty}\textnormal{d}k_{u}e^{ik_{u}|z|}\int\limits_{-\infty}^{\infty}\textnormal{d}k_{v}\frac{1}{k^{2}+|E_{B}|/m}$
$\displaystyle=\frac{1}{m(i|z|)^{3}}\int\limits_{-\infty}^{\infty}\textnormal{d}k_{u}\left(\frac{\partial^{3}}{\partial
k_{u}^{3}}e^{ik_{u}|z|}\right)\int\limits_{-\infty}^{\infty}\textnormal{d}k_{v}\frac{1}{k^{2}+|E_{B}|/m}$
$\displaystyle=\frac{1}{im|z|^{3}}\int\limits_{-\infty}^{\infty}\textnormal{d}k_{u}e^{ik_{u}|z|}\int\limits_{-\infty}^{\infty}\textnormal{d}k_{v}\frac{18k_{u}(k_{v}^{2}+|E_{B}|/m)-24k_{u}^{3}}{(k^{2}+|E_{B}|/m)^{4}}.$
(6.12)

Of the last expression we estimate the absolute value to get

$\displaystyle|\widehat{g}_{1}(z)|$
$\displaystyle\leq\frac{1}{m|z|^{3}}\int\textnormal{d}^{2}k\left(\frac{18|k|}{(k^{2}+|E_{B}|/m)^{3}}+\frac{24|k|^{3}}{(k^{2}+|E_{B}|/m)^{4}}\right)\leq\frac{C}{|z|^{3}|E_{B}|^{3/2}}.$
(6.13)

The bound for $|\widehat{g}_{2}(z)|$ works similarly but is slightly more
cumbersome. We start again with

$\displaystyle\widehat{g}_{2}(z)$
$\displaystyle=\frac{1}{im|z|^{3}|E_{B}|^{3/2}}\int\limits_{-\infty}^{\infty}\textnormal{d}k_{u}e^{ik_{u}|z|}\int\limits_{-\infty}^{\infty}\textnormal{d}k_{v}\frac{\partial^{3}}{\partial
k_{u}^{3}}\left(\frac{\xi_{\mu}(k^{2})|E_{B}|^{3/2}}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau}\right)$
(6.14)

for which we need to compute the different derivatives. A straightforward
computation shows

$\displaystyle\Big{|}\frac{\partial^{n}}{\partial
k_{u}^{n}}\xi_{\mu}(k^{2})\Big{|}$ $\displaystyle\leq
C\frac{(\log\widetilde{\mu})^{n}}{\mu^{n/2}}\chi_{(\mu,\mu+\mu/\log\widetilde{\mu})}(k^{2}),\quad
n\in\\{1,2,3\\}.$ (6.15)

Abbreviating the denominator as $D(k)=\frac{1}{M}(q-k)^{2}+k^{2}+\tau$ it is
not difficult to verify

$\displaystyle\chi_{(\mu,\mu+\mu/\log\widetilde{\mu})}(k^{2})\bigg{|}\frac{\partial}{\partial
k_{u}}\frac{1}{D(k)}\bigg{|}$ $\displaystyle\leq
C\bigg{(}\frac{1}{\mu^{1/2}(k^{2}+\tau)}+\frac{\mu^{1/2}}{(k^{2}+\tau)^{2}}\bigg{)},$
(6.16)
$\displaystyle\chi_{(\mu,\mu+\mu/\log\widetilde{\mu})}(k^{2})\bigg{|}\frac{\partial^{2}}{\partial
k_{u}^{2}}\frac{1}{D(k)}\bigg{|}$ $\displaystyle\leq
C\bigg{(}\frac{1}{(k^{2}+\tau)^{2}}+\frac{\mu}{(k^{2}+\tau)^{3}}\bigg{)},$
(6.17)
$\displaystyle\chi_{(\mu,\mu+\mu/\log\widetilde{\mu})}(k^{2})\bigg{|}\frac{\partial^{3}}{\partial
k_{u}^{3}}\frac{1}{D(k)}\bigg{|}$ $\displaystyle\leq
C\bigg{(}\frac{1}{\mu^{1/2}(k^{2}+\tau)^{2}}+\frac{\mu^{1/2}}{(k^{2}+\tau)^{3}}+\frac{\mu^{3/2}}{(k^{2}+\tau)^{4}}\bigg{)}.$
(6.18)

To illustrate this for the first line, we compute

$\displaystyle\bigg{|}\frac{\partial}{\partial
k_{u}}\frac{1}{D(k)}\bigg{|}=\bigg{|}\frac{1}{D(k)^{2}}\bigg{(}\frac{2}{M}(k_{u}-q_{u})+2k_{u}\bigg{)}\bigg{|}\leq
C\bigg{(}\frac{|k-q|}{D(k)^{2}}+\frac{|k|}{D(k)^{2}}\bigg{)}$ (6.19)

and use $D(k)\geq\frac{1}{M}(k-q)^{2}$ and $k^{2}\leq 2\mu$ in combination
with a balanced Cauchy-Schwarz estimate. This leads to

$\displaystyle\bigg{|}\frac{\partial}{\partial
k_{u}}\frac{1}{D(k)}\bigg{|}\leq
C\bigg{(}\frac{1}{D(k)^{3/2}}+\frac{\sqrt{\mu}}{D(k)^{2}}\bigg{)}\leq
C\bigg{(}\frac{1}{\sqrt{\mu}D(k)}+\frac{\sqrt{\mu}}{D(k)^{2}}\bigg{)}$ (6.20)

from which the bound in (6.16) follows by $D(k)\geq k^{2}+\tau$. The other two
lines are obtained in close analogy.

Summing up the different combinations we obtain

$\displaystyle\bigg{|}\frac{\partial^{3}}{\partial
k_{u}^{3}}\bigg{(}\frac{\xi_{\mu}(k^{2})|E_{B}|^{3/2}}{\frac{1}{M}(p-k)^{2}+k^{2}+\tau}\bigg{)}\bigg{|}$
$\displaystyle\quad\quad\leq\frac{C(\log\widetilde{\mu})^{3}}{\widetilde{\mu}^{3/2}}\bigg{(}\chi_{(\mu,\mu+\mu/\log\widetilde{\mu})}(k^{2})\frac{1}{k^{2}+\tau}+\chi_{(\mu,\infty)}(k^{2})\sum_{j=2}^{4}\frac{\mu^{j-1}}{(k^{2}+\tau)^{j}}\bigg{)}$
(6.21)

by which we can estimate the integral

$\displaystyle\int_{\mathbb{R}^{2}}\bigg{|}\frac{\partial^{3}}{\partial
k_{u}^{3}}\bigg{(}\frac{\xi_{\mu}(k^{2})|E_{B}|^{3/2}}{\frac{1}{M}(q-k)^{2}+k^{2}+\tau}\bigg{)}\bigg{|}\textnormal{d}^{2}k$
$\displaystyle\leq
C\bigg{(}1+\frac{\mu}{(\mu+\tau)\log\widetilde{\mu}}\bigg{)}^{3}.$ (6.22)

Together with (6.13) and (6.14), this gives

$\displaystyle|\widehat{g}(z)|\leq\frac{C}{|z|^{3}|E_{B}|^{3/2}}\bigg{(}1+\frac{\mu}{(\mu+\tau)\log\widetilde{\mu}}\bigg{)}^{3}.$
(6.23)

The remaining series can be bounded as

$\displaystyle\sum_{\begin{subarray}{c}z\in L\mathbb{Z}^{2}\\\ z\neq
0\end{subarray}}|z|^{-3}|E_{B}|^{-3/2}=(L|E_{B}|^{1/2})^{-3}\sum_{\begin{subarray}{c}z\in\mathbb{Z}^{2}\\\
z\neq 0\end{subarray}}|z|^{-3}\leq C$ (6.24)

because of $L^{2}|E_{B}|\geq 1$ and

$\displaystyle\sum_{\begin{subarray}{c}z\in\mathbb{Z}^{2}\\\ z\neq
0\end{subarray}}\frac{1}{|z|^{3}}$ $\displaystyle=\sum_{n,m\geq
1}\frac{4}{(n^{2}+m^{2})^{3/2}}+\sum_{n\geq
1}\frac{4}{n^{3}}\leq\int\limits_{1}^{\infty}\bigg{(}\frac{8\pi}{s^{2}}+\frac{4}{s^{3}}\bigg{)}\textnormal{d}s\leq
C$ (6.25)

by the integral test of convergence.

We conclude that the absolute value of the right side in (6.5) is bounded from
above by

$\displaystyle\frac{1}{2\pi}\sum_{\begin{subarray}{c}z\in L\mathbb{Z}^{2}\\\
z\neq 0\end{subarray}}|\widehat{g}(z)|\leq
C\bigg{(}1+\frac{\mu}{(\mu+\tau)\log\widetilde{\mu}}\bigg{)}^{3}.$ (6.26)

Hence the proof of the lemma is complete.

##  Appendix A Replacing sums by integrals

For a short proof of the following lemma, see [20, Appendix B].

######  Lemma A.1.

(a) Let $f:[0,\infty)\to[0,\infty)$ be monotonically decreasing. Then,

$\displaystyle\Bigg{|}\frac{1}{L^{2}}\sum_{k}f(k^{2})-\frac{1}{2\pi}\int\limits_{0}^{\infty}f(t^{2})t\
\textnormal{d}t\Bigg{|}\leq\frac{2}{\pi
L}\int\limits_{0}^{\infty}f(t^{2})\textnormal{d}t+\frac{3f(0)}{L^{2}}.$ (A.1)

(b) Let $m\geq 0$ and $f:[m,\infty)\to[0,\infty)$ be monotonically decreasing.
Then,

$\displaystyle\Bigg{|}\frac{1}{L^{2}}\sum_{k^{2}\geq
m}f(k^{2})-\frac{1}{2\pi}\int\limits_{\sqrt{m}}^{\infty}f(t^{2})t\
\textnormal{d}t\Bigg{|}\leq\frac{2}{\pi
L}\int\limits_{\sqrt{m}}^{\infty}f(t^{2})\textnormal{d}t+\Big{(}\frac{4\sqrt{m}}{\pi
L}+\frac{6}{L^{2}}\Big{)}f(m).$ (A.2)

##  Appendix B Completing the proof of Lemma 4.2

It remains to analyze the condition $\mathcal{F}(T-\lambda,r)\geq 0$. For that
we approximate $\mathcal{F}(T-\lambda,r)$ by $\mathcal{F}^{(n)}(T-\lambda,r)$
where the operator $\mathcal{F}^{(n)}(T-\lambda,r)$ arises by replacing the
function $G(0,\tau)$ in (4.26) by

$\displaystyle G^{(n)}(0,\tau)=\frac{1}{L^{2}}\sum_{k^{2}\leq
n}\left(\frac{1}{k^{2}-E_{B}}-\frac{\chi_{(\mu,\infty)}(k^{2})}{k^{2}+\tau}\right).$
(B.1)

Note that $G^{(n)}(0,\tau)\to G(0,\tau)$ as $n\to\infty$ for every
$\tau>-\mu$. Thus, $G^{(n)}(0,T-\lambda)\psi\to G_{\mu}(0,T-\lambda)\psi$ as
$n\to\infty$ for every $\psi\in D$ (recall that
$D\subset\mathscr{H}_{N(\mu)-1}$ is the set of all anti-symmetric product
states, see (3.4)). Since $D$ forms a total set of eigenstates of
$G(0,T-\lambda)$ on
$\mathscr{H}_{N\mspace{-1.0mu}(\mspace{-1.0mu}\mu\mspace{-1.0mu})-1}$, its
linear hull
$\text{lin}D\subseteq\mathscr{H}_{N\mspace{-1.0mu}(\mspace{-1.0mu}\mu\mspace{-1.0mu})-1}$
is a domain of essential self-adjointness for this operator. Furthermore,
$G^{(n)}(0,T-\lambda)\geq G^{(n)}(0,-\mu-e_{\rm P}(\mu,E_{B}))$ and thus, by
the convergence of $G^{(n)}(0,\tau)$ and the fact that $G(0,-\mu-e_{\rm
P}(\mu))\geq C\log\widetilde{\mu}$, it follows that there is a $c>0$ such that
$G^{(n)}(0,T-\lambda)-r>c$ for $n$ large enough. Hence as $n\to\infty$,
$(G^{(n)}(0,T-\lambda)-r)^{-1}\to(G(0,T-\lambda)-r)^{-1}$ and
$\mathcal{F}^{(n)}(T-\lambda,r)\to\mathcal{F}(T-\lambda,r)$ strongly.

Using the pull-through formula (2.6), we can write

$\displaystyle\mathcal{F}^{(n)}(T-\lambda,r)$
$\displaystyle=T-\lambda-\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}(G^{(n)}(0,T-k^{2}-\lambda)-r)^{-1}$
$\displaystyle\hskip
28.45274pt+\frac{1}{L^{2}}\sum_{k^{2},l^{2}\leq\mu}\\!\\!a_{k}(G^{(n)}(0,T-k^{2}-l^{2}-\lambda)-r)^{-1}a_{l}^{*}$
(B.2)

on $\mathscr{H}_{N\mspace{-1.0mu}(\mspace{-1.0mu}\mu\mspace{-1.0mu})}$.
Assuming that the last term in (B.2), which we call
$\mathcal{P}^{(n)}(T-\lambda,r)$ in the following, is a positive operator on
$\mathscr{H}_{N\mspace{-1.0mu}(\mspace{-1.0mu}\mu\mspace{-1.0mu})}$, we obtain

$\displaystyle\mathcal{F}^{(n)}(T-\lambda,r)\geq
E_{0}(\mu)-\lambda-\frac{1}{L^{2}}\sum_{k^{2}\leq\mu}(G^{(n)}(0,E_{0}(\mu)-k^{2}-\lambda)-r)^{-1},$
(B.3)

since $T\geq E_{0}(\mu)$ on
$\mathscr{H}_{N\mspace{-1.0mu}(\mspace{-1.0mu}\mu\mspace{-1.0mu})}$. In view
of (4.25), this completes the proof of Lemma 4.2.

It remains to show $\mathcal{P}^{(n)}(T-\lambda,r)\geq 0$. For
$\psi\in\mathscr{H}_{N\mspace{-1.0mu}(\mspace{-1.0mu}\mu\mspace{-1.0mu})}$,

$\displaystyle
L^{2}\,\mbox{$\langle\psi,\mathcal{P}^{(n)}(T-\lambda,r)\psi\rangle$}$
$\displaystyle=\int\limits_{0}^{\infty}\sum_{k^{2},l^{2}\leq\mu}\mbox{$\langle\psi,a_{k}\>\exp(-t\,[G^{(n)}(0,T-k^{2}-l^{2}-\lambda)-r)\>a_{l}^{*}\psi\rangle$}\,\textnormal{d}t$
$\displaystyle=\int\limits_{0}^{\infty}\>\exp(-t[L^{-2}\sum_{p^{2}\leq
n}\tfrac{1}{p^{2}-E_{B}}\\!-\\!r])\,\mathcal{I}^{(n)}(t)\,\textnormal{d}t$
(B.4)

with

$\displaystyle\mathcal{I}^{(n)}(t)=\sum_{k^{2},l^{2}\leq\mu}\mbox{$\langle\psi,a_{k}\prod\limits_{\mu<q^{2}\leq
n}\\!\\!\exp(t(q^{2}+T-k^{2}-l^{2}-\lambda)^{-1})\>\,a_{l}^{*}\psi\rangle$}.$
(B.5)

We show that $\mathcal{I}^{(n)}(t)\geq 0$ for all $t\in[0,\infty)$ and
$n\in\mathbb{N}$. Note that the product in the definition of
$\mathcal{I}^{(n)}(t)$ has only finitely many factors, because
$A_{n}=\\{q\in\frac{2\pi}{L}\mathbb{Z}^{2}\>|\>\mu<q^{2}\leq n\\}$ is a finite
set. We consider the exponential series and obtain

$\displaystyle\mathcal{I}^{(n)}(t)=\sum_{k^{2},l^{2}\leq\mu}\mbox{$\langle\psi,a_{k}\prod\limits_{q\in
A_{n}}\\!\\!\left(\sum\limits_{m=0}^{\infty}\frac{t^{m}}{m!}\,\frac{1}{(q^{2}+T-k^{2}-l^{2}-\lambda)^{m}}\right)a_{l}^{*}\psi\rangle$}.$
(B.6)

By the absolute convergence of the exponential series, we can rearrange the
product of series to get

$\displaystyle\mathcal{I}^{(n)}(t)=\sum_{m:A_{n}\to\mathbb{N}_{0}}\>\>\sum_{k^{2},l^{2}\leq\mu}\mbox{$\langle\psi,a_{k}\prod\limits_{q\in
A_{n}}\\!\\!\left(\frac{t^{m(q)}}{m(q)!}\,\frac{1}{(q^{2}+T-k^{2}-l^{2}-\lambda)^{m(q)}}\right)a_{l}^{*}\psi\rangle$},$
(B.7)

where we sum over all $\mathbb{N}_{0}$-valued functions $m$ on the finite set
$A_{n}$. Note that the factor in parentheses indexed by $q$ is equal to $1$ if
$m(q)=0$. For all factors with $m(q)\neq 0$, we use the identity

$\displaystyle\frac{1}{a^{\tau}}=\frac{1}{c_{\tau}}\int\limits_{0}^{\infty}\\!\textnormal{d}s\>e^{-as^{1/\tau}}\qquad\text{with}\qquad
c_{\tau}=\int\limits_{0}^{\infty}\\!\textnormal{d}s\>e^{-s^{1/\tau}}$ (B.8)

for $a,\tau>0$ to rewrite each of the summands in the $m$-sum as

$\displaystyle\sum_{k^{2},l^{2}\leq\mu}\>\>\prod\limits_{\begin{subarray}{c}q\in
A_{n}\\\ m(q)\neq
0\end{subarray}}\>\left(\frac{t^{m(q)}}{m(q)!\,c_{m(q)}}\int\limits_{0}^{\infty}\\!\textnormal{d}s_{q}\right)\mbox{$\langle\psi,a_{k}\prod\limits_{\begin{subarray}{c}p\in
A_{n}\\\ m(p)\neq
0\end{subarray}}\\!\\!e^{-(p^{2}+T-k^{2}-l^{2}-\lambda)s_{p}^{1/m(p)}}a_{l}^{*}\psi\rangle$}$
$\displaystyle=\prod\limits_{{\begin{subarray}{c}q\in A_{n}\\\ m(q)\neq
0\end{subarray}}}\left(\frac{t^{m(q)}}{m(q)!\,c_{m(q)}}\int\limits_{0}^{\infty}\\!\textnormal{d}s_{q}\right)\big{|}\big{|}\sum\limits_{k^{2}\leq\mu}\prod\limits_{{\begin{subarray}{c}p\in
A_{n}\\\ m(p)\neq
0\end{subarray}}}e^{-\frac{1}{2}(p^{2}+T-2k^{2}-\lambda)s_{p}^{1/m(p)}}a_{k}^{*}\psi\big{|}\big{|}^{2}.$
(B.9)

This yields $\mathcal{I}^{(n)}(t)\geq 0$ and thus
$\mbox{$\langle\psi,\mathcal{P}^{(n)}(T-\lambda,r)\psi\rangle$}\geq 0$ for all
$n\in\mathbb{N}$.

Acknowledgements.

I am very grateful to Ulrich Linden for introducing me to the Fermi polaron as
well as for his contributions to this project in its early stage.

## References

  * [1]
  * [2]
  * AGHH [05] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden.  Solvable models in quantum mechanics. AMS Chelsea Publishing, 2nd edition. (2005)
  * BM [10] G.M. Bruun and P. Massignan.  Decay of polarons and molecules in a strongly polarized Fermi gas.  Phys. Rev. Lett. 105:020403. (2010)
  * Che [06] F. Chevy. Universal phase diagram of a strongly interacting Fermi gas with unbalanced spin populations. Phys. Rev. A 74, 063628. (2006)
  * CM [09] F. Chevy and C. Mora. Ground state of a tightly bound composite dimer immersed in a Fermi sea. Phys. Rev. A 80, 033607. (2009)
  * CG [08] R. Combescot and S. Giraud. Normal state of highly polarized Fermi gases: Full many-body treatment. Phys. Rev. Lett. 101, 050404. (2008)
  * CDFMT [12] M. Correggi, G.F. Dell’Antonio, D. Finco, A. Michelangeli and A. Teta. Stability for a system of $N$ fermions plus a different particle with zero-range interactions. Rev. Math. Phys. 24(7), 1250017, 32. (2012)
  * DFT [94] G.F. Dell’Antonio, R. Figari and A. Teta.  Hamiltonians for systems of $N$ particles interacting through point interactions  Ann. Inst. H. Poincaré Phys. Théor. 60(3), 253–290. (1994)
  * DR [04] J. Dimock and S.G. Rajeev. Multi-particle Schrödinger operators with point interactions in the plane. J. Phys. A 37(39), 9157–9173. (2004)
  * Frö [54] H. Fröhlich. Electrons in lattice fields. Adv. Phys 3(11):325. (1954)
  * Gra [14] L. Grafakos. Classical Fourier Analysis. Graduate texts in mathematics. Springer, 3rd edition. (2014)
  * GL [18] M. Griesemer and U. Linden.  Stability of the two-dimensional Fermi polaron.  Lett. Math. Phys. 108(8), 1837–1849. (2018)
  * GL [19] M. Griesemer and U. Linden. Spectral Theory of the Fermi Polaron. Ann. Henri Poincaré 20(6), 1931–1967. (2019)
  * GD [15] F. Grusdt and E. Demler. New theoretical approaches to Bose polarons. Preprint. (2015) arXiv:1510.04934
  * Lan [33] L.D. Landau. Über die Bewegung der Elektronen in Kristallgitter. Phys. Z. Sowjetunion. 3, 644–645. (1933)
  * LP [48] L.D. Landau and S.I. Pekar. Zh. Eksp. Teor. Fiz. 18, 419. (1948)
  * LLP [53] T.D. Lee, F.E. Low and D. Pines. The motion of slow electrons in a polar crystal.  Phys. Rev. 90(2), 297–302. (1953)
  * Lin [17] U. Linden. Energy estimates for the two-dimensional Fermi polaron. PhD thesis. Universität Stuttgart. (2017)
  * LM [19] U. Linden and D. Mitrouskas.  High density limit of the Fermi polaron with infinite mass.  Lett. Math. Phys. 109, 1805–1825. (2019)
  * MO [18] A. Michelangeli and A. Ottolini. Multiplicity of self-adjoint realisations of the (2 + 1)-fermionic model of Ter-Martirosyan-Skornyakov type. Rep. Math. Phys. 81(1):1–38. (2018)
  * Min [11] R. Minlos. On point-like interaction between $n$ fermions and another particle. Mosc. Math. J. 11(1), 113–127, 182. (2011)
  * MS [17] T. Moser and R. Seiringer. Stability of a fermionic $N+1$ particle system with point interactions. Commun. Math. Phys. 356(1), 329–355. (2017)
  * MS [19] T. Moser and R. Seiringer.  Energy contribution of a point interacting impurity in a Fermi gas. Ann. Henri Poincaré 20, 1325–1365. (2019)
  * Par [11] M.M. Parish. Polaron-molecule transitions in a two-dimensional Fermi gas. Phys. Rev. A 83, 051603. (2011)
  * PL [13] M.M. Parish and J. Levinsen. Highly polarized Fermi gases in two dimensions. Phys. Rev. A 87, 033616. (2013)
  * Pek [54] S.I. Pekar. Untersuchung über die Elektronentheorie der Kristalle. Berlin, Akad. Verlag. (1954)
  * PS [08] N. Prokof’ev and B. Svistunov. Fermi-polaron problem: Diagrammatic monte carlo method for divergent sign-alternating series. Phys. Rev. B 77, 020408. (2008)
  * PDZ [09] M. Punk, P.T. Dumitrescu and W. Zwerger. Polaron-to-molecule transition in a strongly imbalanced fermi gas. Phys. Rev. A 80, 053605. (2009)
  * SEPD [12] R. Schmidt, T. Enss, V. Pietilä and E. Demler. Fermi polarons in two dimensions. Phys. Rev. A 85, 0216029. (2012)
  * SL [15] R. Schmidt and M. Lemeshko. Rotation of quantum impurities in the presence of a many-body environment. Phys. Rev. Lett. 114, 20300. (2015)

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