Energy contribution of a point interacting impurity in a Fermi gas

We give a bound on the ground state energy of a system of $N$ non-interacting fermions in a three dimensional cubic box interacting with an impurity particle via point interactions. We show that the change in energy compared to the system in the absence of the impurity is bounded in terms of the gas density and the scattering length of the interaction, independently of $N$. Our bound holds as long as the ratio of the mass of the impurity to the one of the gas particles is larger than a critical value $m_c\approx 0.36$, which is the same regime for which we recently showed stability of the system.


Introduction
Quantum systems of particles interacting with forces of very short range allow for an idealized description in terms of point interactions. The latter are characterized by a single number, the scattering length. Originally point interactions were introduced in the 1930s to model nuclear interactions [4,5,12,27,28], but later they were also successfully applied to many other areas of physics, like polarons (see [19] and references there) or cold atomic gases [29].
It was already known to Thomas [27] that the spectrum of a bosonic many-particle system depends strongly on the range of the interactions, and that an idealized point-interacting system with more than two particles is inherently unstable, i.e., the energy is not bounded from below. This collapse can be counteracted by the Pauli principle for fermions with two species (e.g., spin states). In this paper we are interested in the impurity problem where there is only one particle for one of the species.
Given N ≥ 1 fermions of one type with mass 1 and one particle of another type with mass m > 0, a model of point interactions gives a meaning to the formal expression for γ ∈ R. We note that because of the antisymmetry constraint on the wavefunctions there are only interactions between particles of different species. The expression (1.1) is ill-defined in d ≥ 2 dimensions since H 1 (R d ), the form domain of the Laplacian, contains discontinuous functions for which the meaning of the δ-function as a potential is unclear. In the following we restrict our attention to the case d = 3, but we note that also two-dimensional systems exhibit interesting behavior [10,11,15,16,18]. For d ≥ 4 there are no point interactions as the Laplacian restricted to functions supported away from the hyperplanes of interactions is essentially selfadjoint.
A mathematically precise meaning to (1.1) in three dimensions was given in [10,13,20] and we will work with the model introduced there. Our analysis will start from this well-defined model, but we note that the question whether the model can be obtained as a limit of Schrödinger operators with genuine interaction potentials of shrinking support is still open. (See, however, [1] for the case N = 1, and [2] for models in one dimension.) In this paper we study the energy contribution of the point-interacting impurity. We confine the N + 1 particles to a box (0, L) 3 and investigate the ground state energy of the system. In particular, our goal is the show that at given mean particle densityρ = N/L 3 , the difference between the ground state energies of the interacting and the non-interacting system is bounded independently of the system size.
Previous work on this model was mostly concerned with stability and hence studied the model without confinement. For example, it is possible to analyze the 2 + 1 model, i.e,. two fermions of one kind and one impurity of another kind, in great detail [3, 6-8, 10, 20-23]. It turns out that the mass of the impurity plays an important role for stability. It was shown in [6] that for the 2 + 1 system there is a critical mass m * ≈ 0.0735 such that the system is stable for m ≥ m * and unstable otherwise. This critical mass does not depend on the strength of the interaction, i.e., the scattering length.
Building on these results it was shown in [24] that a similar statement holds for the N + 1 system. In particular, it was proven that there is a critical mass m * * ≈ 0.36 such that the system is stable for all m ≥ m * * , independently of N. This bound is presumably not sharp and stability is still open for m ∈ [m * , m * * ). Recently also the stability of the 2 + 2 system was proved in a suitable mass range [25]. The general case with N + M particles still poses an open problem, however.
In all cases where stability of the system was established, the ground state energy in infinite volume is actually zero in case the scattering length is negative, and there are no bound states. For positive scattering length there are bound states, but one still expects that only a finite number of particles can bind to the impurity. In particular, the ground state energy of the N + 1 system is bounded from below independently of N [24]. Intuitively one would expect that if one confines the system to a box in order to have a non-zero mean particle density, the interaction with the impurity should again only affect a finite number of particles, and hence the energy change compared to the non-interacting system should be O(1), independently of N. This is what we prove here. We note that it is sufficient to derive a lower bound on the ground state energy, as point interactions are always attractive, i.e., they lower the energy.
Even for regular interaction potentials, it is highly non-trivial to show that an impurity causes only an O(1) change to the energy of a non-interacting Fermi gas. For fixed, i.e., non-dynamical impurities, this was established in [14] as a consequence of a positive density version of the Lieb-Thirring inequality. The result in [14] applies to systems in infinite volume, as well as to systems in a box with periodic boundary conditions. In the appendix we provide an extension to Dirichlet boundary conditions, since this result will be an essential ingredient in our proof.
Compared to [14] we face here two additional difficulties: the impurity is dynamic and has a finite mass, and the interaction with the gas particles is through singular point interactions.
Besides the methods of [14] and [24], a key ingredient in our analysis is a proof of an IMS type formula for the quadratic form defining the model, which allows for a localization of the particles into regions close and far away from the impurity. It has the same form as the IMS formula for regular Schrödinger operators (see [9,Thm. 3.2]), but is much harder to prove.

The point interaction model
We consider a system of N fermions of mass 1, interacting with another particle of mass m > 0. Let be the non-interacting part of the Hamiltonian, acting on L 2 (R 3 )⊗ L 2 as (R 3N ), where L 2 as denotes the totally antisymmetric functions in ⊗ N L 2 (R 3 ). The N +1 coordinates we denote by x 0 , x 1 , . . . , x N ∈ R 3 and throughout this paper we will use the notation x = (x 1 , . . . , x N ). If we want to exclude a set of coordinates labeled by A ⊆ {1, . . . , N} we usex A = (x i ) i A and for shortx i =x {i} . If we want to restrict to certain coordinates we write x A = (x i ) i∈A .
For µ > 0, we define G µ as the resolvent of H N 0 in momentum space, i.e., We denote by F α,N the quadratic form used in [6,24] describing point interactions between N fermions and the impurity. Its domain is given by where G µ ξ is defined via its Fourier transform (denoted by a· ) as The space H 1 as (R 3N ) contains all totally antisymmetric functions in H 1 (R 3N ). For a given ψ ∈ D(F α,N ) and µ > 0, the splitting ψ = φ µ + G µ ξ is unique. We point out that while φ µ depends on the choice of µ, ξ is independent of µ. We will call φ µ the regular part and ξ the singular part of ψ. Note that D(F α,N ) is independent of the choice of µ, and so is the quadratic form F α,N defined as (1.10) The quadratic form F α,N describes N fermions interacting with an impurity particle via point interactions with scattering length a = −2π 2 /α, with α ∈ R. The non-interacting system is recovered in the limit α → +∞. Notation. Throughout the paper we will use the following notation. We define the relation by x y ⇐⇒ ∃C > 0 : x ≤ Cy (1.11) where C is independent of x and y. In the obvious way we define . In case that x y and y x we write x ∼ y.

Main result for confined wavefunctions
Let us assume that supp ψ ⊆ B N+1 , where B = (0, L) 3 for some L > 0. The mean particle density will be denoted byρ = N/L 3 . Let E D N be the ground state energy of − (2.2) (The additional factor (m + 1)/(2m) compared to [24,Thm. 2.1] results from the separation of the center-of-mass motion used in [24].) It was also shown in [24] that Λ(m) < 1 if m > m * * ≈ 0.36.
For particles confined to the box B with mean densityρ we can show that under the condition Λ(m) < 1 the correction to E D N is small, i.e., it is O(1) independently of N. Our main result is the following.
where the constant is independent of ψ, m, N, L and α, and α − denotes the negative part of α, i.e., α − = 1 2 (|α| − α). Thm. 2.1 shows that the presence of the impurity affects the ground state energy by a term that is bounded independently of N. The bound (2.3) is an extension of (2.2) in the sense that if we take L → ∞ in (2.3) we recover (2.2) up to the value of the constant.
Remark. For α → ∞ one would expect that the optimal lower bound converges to the ground state energy of the non-interacting Hamiltonian H N 0 with Dirichlet boundary conditions. This is not the case for (2.3) which is independent of α for α ≥ 0.
Using various types of trial states the ground state energy of point-interacting systems is extensively discussed in the physics literature (see [19] and references there). We note that with this method it is only possible to derive upper bounds, while Thm. 2.1 gives a lower bound on the ground state energy.

Proof outline
For the proof of Theorem 2.1 we first prove in Section 3 an IMS type formula, which allows to localize the impurity in a small box, of side length independent of L. In a second step we localize all of the remaining particles to be either close to the impurity or separated from it. Doing this we partly violate the antisymmetry constraint on the wavefunctions, which makes it necessary to first extend the quadratic form F α,N toF α,N . The latter does not require the antisymmetry, but coincides with F α,N on D(F α,N ).
In Section 4 we give a rough lower bound on the energy in case the wavefunction is compactly supported in a box (0, ) 3 . This lower bound is of the order N 5/3 / 2 , as expected, but with a nonsharp prefactor. We shall introduce a quadratic form F per α,N with periodic boundary conditions and show that it is equivalent to F α,N for confined wavefunctions. The reason we work with periodic boundary conditions instead of Dirichlet ones is that it allows to perform explicit computations in momentum space.
Because the ground state energy of the confined non-interacting N-particle system is strictly positive, we are allowed to choose µ negative in the definition of F per α,N . Applying the method of [24] then leads to the lower bound on F per α,N in Theorem 4.1. The downside of working with F per α,N will be that because of the discrete nature of momentum space for periodic functions, we have to work with sums instead of integrals, and the difference between the sum and the integral versions will have to be carefully controlled.
In Section 5 we give the proof of Theorem 2.1. Using the IMS formula of Prop. 3.1, we localize the particles either in a small box with side length ∼ρ −1/3 containing the impurity, or in the large complement. In the small box we use Theorem 4.1 for a lower bound, whereas in the large complement we use Theorem A.2, which is a version of the positive density Lieb-Thirring inequality in [14] adapted to our setting of Dirichlet boundary conditions, and which is proved in the appendix. This allows us to improve the rough bound of Thm. 4.1 and show Thm. 2.1.

Properties of the quadratic form
In this section we will first extend the quadratic form F α,N to functions that are not required to be antisymmetric in the last N variables. Afterwards we shall discuss how the splitting ψ = φ µ + G µ ξ is affected when multiplying ψ by a smooth function (which need not be symmetric under permutations). This will be utilized in the last part of this section where an IMS formula for the (extended) quadratic form is shown.

Extension to functions without symmetry
To prove our main theorem, we want to localize the particles in different subsets of the cube B = (0, L) 3 . Hence it is necessary to extend the quadratic form F α,N by removing the antisymmetry constraint. To this aim we define The quadratic formF α,N is defined as (3.6) Each ξ i in (3.2) corresponds to a function supported on the hyperplane x 0 = x i . The only overlap between hyperplanes for i j is on the set x i = x 0 = x j , which implies that N i=1ξ i (k 0 + k i ,k i ) has a unique decomposition into (ξ i ) N i=1 , and thus the splitting ψ = φ µ + N i=1 G µ ξ i is unique. To stress the dependence on ψ, we will sometimes use the notation φ ψ µ and ξ ψ i below. In the case that ψ is antisymmetric in the last N coordinates, the uniqueness of the decomposition ψ = φ µ + N i=1 G µ ξ i shows that there exists a function ξ ∈ H 1/2 (R 3 ) ⊗ H 1/2 as (R 3(N−1) ) such that ξ i = (−1) i+1 ξ, and hence N i=1 G µ ξ i = G µ ξ, defined in (1.5). Furthermore we havẽ in this case, which shows thatF α,N (ψ) = F α,N (ψ) for ψ antisymmetric in the last N coordinates. In particular,F α,N is an extension of F α,N , and for a lower bound it therefore suffices to work withF α,N . In the following, it will be convenient to introduce the notatioñ as well as

Localization of wavefunctions
An important ingredient in the proof of Theorem 2.1 will be to localize the particles. For this purpose we will study in this subsection how the splitting ψ = φ ψ µ + N i=1 G µ ξ ψ i is affected when multiplying ψ by a smooth function. Then and the regular part φ Jψ µ of Jψ is given by Remark. We clarify that J acts on functions on R 3(N+1) , and in particular on φ ψ µ and G µ ξ ψ i , as a multiplication operator, whereas on functions in L 2 (R 3N ) it acts as in (3.10). Hence the commutator [J, G µ ] has no meaning here independently of its application on ξ, and is only used as a convenient notation.
It remains to show that [J, G µ ]ξ i ∈ H 1 (R 3(N+1) ) for ξ i ∈ L 2 (R 3N . In order to do so, we shall in fact show that where we used the notation introduced in (3.8) and (3.9). From (3.14) the H 1 property readily follows, using that In the last step we did an explicit integration over 1 In order to show (3.14), we note that since J is smooth, H −1 µ JH µ is a bounded operator. In the sense of distributions, we have and hence H −1 µ JH µ G µ ξ i = G µ Jξ i . In particular, which indeed equals (3.14). This completes the proof of the lemma.
Since this holds for all J with the above property, the claim follows.

Alternative representation of the singular part
The following Lemma gives an alternative representation of the singular part of the quadratic form, defined in (3.4). It will turn out to be useful in the proof of the IMS formula in the next subsection.
is integrable on [µ, ∞) for any µ > 0, and we havẽ In particular, For the terms i j, on the other hand, we have Here the exchange of the order of integration is justified by Fubini's theorem, since the integrand in the first line on the right is absolutely integrable for ξ i ∈ H 1/2 . This completes the proof.

IMS formula
In this subsection we will prove the following Lemma.
Proof. By using the polarization identity, we can extendF α,N to a sesquilinear form, denoted as F α,N (ψ 1 , ψ 2 ). It suffices to prove that for smooth functions J, since theñ where we introduced the sesquilinear formT α,µ,N ( ξ 1 , ξ 2 ) corresponding to the quadratic form (3.4). We use Lemma 3.1 to identify the regular and singular parts of the various wavefunctions. For the quadratic formT α,µ,N , we utilize the representation (3.21), which together with (3.11) implies thatT as shown in the proof of Lemma 3.1, we can rewrite the terms in the integrand as Using that (∂/∂ν)G ν ξ ψ i = −H −1 ν G ν ξ ψ i as well as [J, [H ν , J]] = 2|∇J| 2 , one readily checks that this further equals For the regular part, we use (3.12) to rewrite the first line in (3.29) as The second term on the right side equals shows. Also the last line in (3.34) can be evaluated with the aid of (3.14), with the result that In combination, (3.33), (3.34) and (3.35) imply the desired identity (3.27). This completes the proof of the lemma.

A rough bound
In this section we give a rough lower bound on the ground state energy of F α,N when restricted to wavefunctions ψ ∈ D(F α,N ) that are supported in B N+1 with B = (0, ) 3 for some > 0. This lower bound has the desired scaling in N and , i.e., it is proportional to N 5/3 −2 , but with a non-sharp prefactor. For its proof, we will first reformulate the problem using periodic boundary conditions, and then apply the methods previously introduced in [24] to show stability in infinite space.
The statement of the following theorem involves three positive constants c T , c L and c Λ , which are independent of m, N, and α and which will be defined later. In particular, c T is defined in Eq. (4.44), c L in Eq. (4.84) and c Λ in Lemma 4.7.
We note that this result gives a lower bound only for particle numbers N > N 0 (m, κ). In the case that N ≤ N 0 , we can still use (2.2), however.
The remainder of this section contains the proof of Theorem 4.1. An important role will be played by a reformulation using periodic boundary conditions. We will start by introducing the functionalF per α,N which is defined for periodic functions. In Lemma 4.2 we will show that it is in fact equivalent to the original quadratic formF α,N when applied to wavefunctions with compact support in B N+1 . Working with periodic boundary conditions comes with the inconvenience of having to work with sums, rather than with integrals, in momentum space. In particular, this makes the explicit form of the singular part ofF per α,N rather complicated; we shall compare it with the singular part ofF α,N in Lemma 4.4 and bound the difference. It comes with the big advantage of allowing us to choose µ negative, however, which will be essential to show a positive lower bound to the energy. We shall use the method of [24] which gives positivity of the singular part of F per α,N for µ ≥ −κN 5/3 −2 for small enough κ, under a condition of the formΛ(m, κ) < 1. In Lemmas 4.5-4.7, we investigate the difference betweenΛ(m, κ) and Λ(m). In the last subsection we combine these results to prove Theorem 4.1.

Periodic boundary conditions
Given ψ ∈ D(F α,N ) such that supp ψ ⊆ B N+1 , we extend ψ to a periodic function ψ per , defined as In the following we shall rewrite the functionalF α,N (ψ) in terms of ψ per . Compared to Dirichlet boundary conditions, periodic ones have the advantage that one can work easily in the associated momentum space, similar to the unconfined case. For this purpose, we define the lattice in momentum space as The function ψ per is then determined by its Fourier coefficientsψ per (k 0 , k), which can be viewed as a function L N+1 → C.
Hence we can extend it in a similar way as ψ to a periodic function ξ per . In momentum space we can write it asξ per : L N → C. For periodic functions, G µ ψ per does not make sense anymore, but instead choosing G per µ as the resolvent of the non-interacting Hamiltonian with periodic boundary conditions allows us to define G per µ ξ per i by the Fourier coefficients In order to motivate the quadratic form introduced below, we note that the expression L µ,N ( k) in (1.10) originates from the limit whereH 0 is the non-interacting Hamiltonian in momentum space, expressed in terms of centerof-mass and relative coordinates for the pair (k 0 , k 1 ), i.e., More generally, we have (4.10) Sinceτ(0) = 1 andτ(t) ≤ 1 for all other t, the result follows from dominated convergence.
When replacing integrals by sums, we have to keep in mind that a change of coordinates from (k 0 , k 1 ) to s = k 0 + k 1 and t = m m+1 k 1 − 1 m+1 k 0 changes the domain over which we have to take the sums. Whereas s ∈ L we have to sum for a fixed s the variable t over L s L + ms m+1 . Let τ be chosen as in Lemma 4.1, and define (4.13) We shall see below that this definition is actually independent of τ. For us it will be important that τ has compact support, hence a sharp cut-off in momentum space would not be suitable.
We shall now defineF per α,N with domain (4.14) where H 1 per (B N+1 ) and H 1/2 per (B N ) denotes the spaces of functions defined by Fourier coefficients in 2 (L, (1 + p 2 )) ⊗(N+1) and 2 (L, (1 + p 2 ) 1/2 ) ⊗N respectively. The quadratic form is given bỹ , and the singular parts of the quadratic form are given byT We also define F per α,N as the restriction ofF per α,N to functions antisymmetric in the last N coordinates. Further we define T per,µ,N dia , T per,µ,N off and T per α,µ,N in the natural way similar to T µ,N dia , T µ,N off and T α,µ,N originating fromT µ,N dia ,T µ,N off andT α,µ,N , respectively (compare with (1.7) and (3.7)).
Proof. Recall the splitting of ψ into its regular and singular parts, and similarly for ψ per : Recall also the definition (3.9). In the sense of distributions we can apply H µ to φ µ , and in particular H µ φ µ ∈ H −1 (R 3(N+1) ) as φ µ ∈ H 1 (R 3(N+1) ). In this sense we can write the regular part ofF α,N as φ µ |H µ φ µ . Because supp ψ ⊆ B N+1 we have ε dist(supp ψ, ∂B) > 0. Let χ be a smooth cutoff function such that χ(x) = 1 if x ∈ B 0 = [ε/2, − ε/2] 3 and χ(x) = 0 if x ∈ B c . As supp(H µ G µ ξ) ⊆ B N+1 0 and supp ψ ⊆ B N+1 0 also supp(H µ φ µ ) ⊆ B N+1 0 , and therefore φ µ |H µ φ µ = χφ µ |H µ φ µ . (4.21) We use the identity χφ µ = χφ per Note that H µ χ(G per µ ξ per i − G µ ξ i ) is supported on B \ B 0 , and ψ per vanishes on this set. Hence (4.23) We claim that (4.23) is equal to the differenceT per α,µ,N ( ξ per ) −T α,µ,N ( ξ). Let τ be given as in Lemma 4.1. We approximate the distribution (χH µ G per We assume that R is large enough such that τ R is supported in a ball of radius ε/2, and hence ξ j τ R is supported in B N+1 . Because G per µ ξ per i − G µ ξ i is actually a smooth function, as H µ (G per µ ξ per i − G µ ξ i ) = 0 on B N+1 , we conclude that (4.23) is equal to For the terms with i j, we can use dominated convergence in momentum space to conclude that lim For the terms with i = j, we can further write Lemma 4.1 implies that the limit of the last two terms exists, is independent of the choice of τ and is equal toT µ,N dia ( ξ). Because also (4.23) does not depend on τ we conclude that exists and is independent of τ. Comparing with (4.13) and (4.17), we see that it actually equals T per,µ,N dia ( ξ per ). Combining the above, we obtain This completes the proof of the lemma.
For fermions, described by wavefunctions ψ per that are antisymmetric in the last N variables, the expression G per µ ξ per in (4.7) is also well defined for negative µ as long as µ > −E per N−1 , where E per N−1 denotes the ground state energy of the non-interacting Hamiltonian for N − 1 fermions with periodic boundary conditions on ∂B. (Note than G µ ξ, on the other hand, is only defined for µ > 0.) The following lemma shows that for such µ the quadratic form F per α,N is actually independent of µ. Proof. We first note that G per µ ξ per is well defined for µ > −E per N−1 , because of the antisymmetry of ξ per in the last N − 1 variables, which implies that N − 1 of the variables (k 1 , . . . , Using the resolvent identity, we see that the regular part of the quadratic form satisfies A straightforward computation using the definitions (4.13)-(4.16) shows that Combining both statements yields the desired identity |∇φ per ν | 2 + ν|φ per ν | 2 − ν ψ per 2 + T per α,ν,N (ξ per ) .

Approximation by integrals
In the previous subsection we have shown that the original and the periodic formulations of the energy functionals,F α,N andF per α,N , agree if applied to functions ψ compactly supported in B N+1 . One complication in the periodic form is that L per µ,N is not given as explicitly as L µ,N . The following lemma gives a bound on the difference.
where the constant c L is independent of N, q, m, and µ.
Proof. We recall the definitions of L µ,N and L per µ,N for some arbitrary τ fulfilling the requirements of Lemma 4.1: withH 0 defined in (4.9). For simplicity we assume that q 1 is such that L q 1 = L but all other cases work analogously as a shift in momentum space only introduces a phase factor in configuration space, which vanishes when taking absolute values. In the following we denote f ∞ (s) = (H 0 (q 1 , s,q 1 ) + µ) −1 and f R (s) = f ∞ (s)τ(s/R) and suppress the dependence on q for simplicity. We can express the difference between the Riemann sum and the integral using Poisson's summation formula For short we write γ 1 2(1+m) q 2 1 + 1 2q 2 1 + µ, which is bounded from below by Q 2 µ and hence is positive, by our assumption (4.33). The function f ∞ and its Fourier transform are given by We will show thatf R (s) is summable over Z 3 \ {0}. In fact for |z| , where we assumed that R is large enough such that τ(Rw) = 0 for |w| > |z|/2, and used that τ = 1, which was required by Lemma 4.1. Asf ∞ is summable over Z 3 \ {0} we get by dominated convergence that We bound the sum overf ∞ (|z|) by for η = (2m/(m + 1)) 1/2 √ γ . Combining (4.36), (4.40) and (4.41) and using that γ ≥ Q 2 µ , we conclude that for some constant c L > 0. This completes the proof of the lemma.

Bound on the singular parts
The strategy for obtaining a lower bound on F per α,N is to find a µ such that T per α,µ,N ≥ 0, in which case we obtain the lower bound F per α,N (ψ per ) ≥ −µ ψ per 2 . Hence we want to choose µ as negative as possible. We shall use the method of [24], which yields the desired positivity of T per α,µ,N (for large enough m) as long as µ ≥ −κN 5/3 −2 for κ small enough. (More precisely, −µ will be equal to the right side of (4.3).) If we define Q 2 = 1 2 N i=2 q 2 i for N > 2, we observe that there exists a constant c T > 0 such that if all q i ∈ L are different, as required by the antisymmetry constraint. (We note that in comparison with [24] Q 2 is defined with an additional factor 1/2 here.) From now on we restrict µ to satisfy µ ≥ −κN 5/3 −2 for some κ < c T . This implies that In particular, Lemma 4.4 yields the bound on the diagonal term of the singular part of F per α,N . Following the same steps as in [24] we can obtain the following lower bound for the off-diagonal term.
Proof. The proof works in almost the exact same way as in [24], hence we will not spell out the details. The main difference is that we now have to write sums instead of integrals, and in particular this implies that we have to choose the weight function h(s,q 1 ) (see [24, Eq. (4.12)]) differently, namely as For comparison δ = 0 was used in [24]. Following the proof in [

A bound onΛ(m, κ)
We will not evaluateΛ(m, κ) directly but we will compare it with Λ(m), which is defined in [24, Eq. (2.8)] and which was already referred to in (2.2) above. The expression Λ(m) can be written as The additional constraint on Q µ in the latter supremum has no effect because of the scaling properties of λs ,Q µ ,K,m,0 , specifically λ νs,νQ µ ,νK,m,0 (νt) = ν −3 λs ,Q µ ,K,m,0 (t) for any ν > 0, which allows to fix one of the parameters when taking the supremum. The expression (4.48) differs from (4.53) by the non-zero value of δ, as well as the sum instead of an integral. In the following lemmas we will compare the two. The next Lemma gives a pointwise bound on λs ,Q µ ,K,m,δ . For its statement it will be convenient to define C (s) as the cube with side length 2π/ centered at s ∈ R 3 , i.e., C (s) = − π , π 3 + s. (4.56) Proof. For the pointwise bound (4.55) we will proceed similarly to [24,Sect. 6]. Using the Cauchy-Schwarz inequality we have and also (4.58) By minimizing over K we find that By combining these bounds we get for (4.49) the pointwise bound from which (4.55) readily follows.
Lemma 4.6. For m 1 we have (4.66) Proof. As in the proof of the previous Lemma, we denote λ(t) = λs ,Q µ ,K,m,δ (t), and write it as with appropriate coefficients c 1 , c 2 , c 3 , c 4 , c 5 depending on m. Its gradient equals (4.68) We can quantify the difference between the Riemann sum and the integral by With the aid of the triangle inequality we can treat the terms I − IV separately. We can bound I as |I| ≤ 2 For the second term we obtain |II| ≤ 1 2 For III, we use similar estimates as in Lemma 4.5 to get |III| |t| + |s| Finally, for IV we have to proceed slightly differently. If we use instead of (4.57), we see that we can bound |III| from above by Q −1 µ times the right side of (4.61). Using Lemma 4.5 we conclude that (4.74) Here we have used that the bound (4.56) holds also with λs ,Q µ ,K,m,δ replaced by the right side of (4.61), as shown in the proof of Lemma 4.5. This completes the proof.
Lemma 4.7. There exists a c Λ > 0 such that whenever κ < c T and Λ(m) ≤ 1, where c T is defined in (4.44).
Proof. We first note that Λ(m) ≤ 1 implies m 1. Moreover, from the definition (4.49) we have λs ,Q µ ,K,m,δ (t) ≤ 1 + Nδ 2 2 Q 2 µ λs ,Q µ ,K,m,0 (t) . (4.76) Combining this with Lemma 4.6 and taking the supremum overs, K and Q 2 µ ≥ (c T − κ)N 5/3 −2 , we obtaiñ (4.77) where we also used that Λ(m) m −1 for m 1. The supremum over Q µ is clearly achieved for For an upper bound, we shall choose δ ∼ N 4/9 , which yields the desired boundΛ for any 0 < κ < c T and µ ≥ −κN 5/3 / 2 . Note that the coefficient in front of the last sum is positive for all N > N 0 (κ, m), defined in (4.2). If α is large enough such that also the first term on the right side of (4.79) is non-negative, we conclude that T per α,µ,N (ξ per ) ≥ 0. In case 2mα < (m + 1)c L (c T − κ) −1 N −5/3 −1 , on the other hand, we need to dominate the first term on the right side of (4.79) by the second. We use (4.44) to obtain the lower bound In particular, if we choose we again conclude that T per α,µ,N (ξ per ) ≥ 0. Note that for our choice of µ, satisfying in particular µ ≥ −c T N 5/3 −2 , we have for all φ per µ ∈ H 1 per (B N+1 ) that are antisymmetric in the last N variables. Hence the positivity of T per α,µ,N (ξ per ) implies that F per α,N (ψ per ) ≥ −µ ψ per 2 . In combination with Lemmas 4.2 and 4.3, this completes the proof of Theorem 4.1. To simplify its statement, we have additionally used that

Proof of Theorem 2.1
In this section we will give the proof of our main result, Theorem 2.1. Let B = (0, L) 3 andB = M iB i a disjoint decomposition into cubes B i = (0, ) 3 + z i with z i ∈ R 3 . We will choose such that L/ ∈ N in which case M = (L/ ) 3 . Let 1/4 > ε > 0 and let η ∈ C ∞ 0 (B ε (0)) be non-negative, where we denote by B ε (0) the centered ball of radius ε. In the following we will assume that ε is a fixed constant independent of all parameters (for example ε = 1/8 works). For x ∈ B, define Then supp J i ⊆ B i + B ε (0) and J i (x) = 1 for x ∈ (ε, 1 − ε) 3 + z i . Moreover, M i=1 J 2 i (x) = 1 for x ∈ B by construction. The derivative of J i can be bounded uniformly in i and M by a constant c η depending only on η (and hence ε) as Let ψ ∈ D(F α,N ) be such that supp ψ ⊆ B N+1 and ψ 2 = 1. We use the IMS formula, Prop. 3.1, for the quadratic form F α,N to localize the impurity particle (with coordinate x 0 ). With J i ψ denoting the function ( We note that the last term is bounded by Recall the definition of the mean density,ρ = NL −3 . We will choose ∼ρ −1/3 which means that (5.3) is of the orderρ 2/3 . In the next step we want to localize the other particles, to be able to distinguish whether they are close to the impurity or far from it. Because we violate the antisymmetry constraint by doing so, we will work with the extended quadratic formF α,N defined in (3.4). Let V ∈ C ∞ 0 (R 3 ) satisfy 0 ≤ V ≤ 1, with supp V ⊆ [−2ε, 1 + 2ε] 3 and V(x) = 1 for x ∈ [−ε, 1 + ε] 3 . We define Figure 1 visualizes this setup. We localize all the remaining particles using the IMS formula in Prop. 3.1, with the localization functions (x 1 , . . . , A straightforward calculation using Prop. 3.1 and the fact that V 2 i +Ṽ 2 i = 1 shows that Here it is necessary to introduce the extended quadratic formF α,N since the functions ϕ i,A are not antisymmetric in all N variables (x 1 , . . . , x N ). They are still separately antisymmetric in the coordinates in A and in the ones in A c , however.
In the next lemma we will show that the energyF α,N (ϕ i,A ) splits up into a non-interacting energy for the particles in A c that are localized away from the impurity, and in a point interacting quadratic form for particles in A. Lemma 5.1. We define the functions ϕ p A c i,A ∈ L 2 (R 3(|A|+1) ) and ϕ p 0 , p A i,A ∈ L 2 (R 3|A c | ) via their Fourier transforms asφ Proof. We define ξ j and φ µ for some µ > 0 using the unique decomposition ϕ i,A = φ µ + N j=1 G µ ξ j . Corollary 3.1 implies that ξ j = 0 for j ∈ A c . Hencẽ Following the argumentation in the proof of Lemma 4.3 we see that the expression inside the integral over p A c is independent of µ. In particular this allows us to shift µ → µ − p 2 A c /2 for fixed p A c , which gives where we used the fact that L µ− p 2 A c /2,N (p i ,p i ) = L µ,|A| (p i , p A\{i} ). The result then follows by noting that the Fourier transform of the regular part of ϕ p A c i,A for fixed p A c is equal toφ µ− p 2 A c ( · , p A c ), and using the the antisymmetry of ϕ p A c i,A . We can apply a similar decomposition also to the second term in (5.7). For simplicity, let Then (5.7) and (5.9) imply that we can write and To obtain a lower bound on A i,A we can use Theorem 4.1, and for the non-interacting part B i,A we use the following proposition. We recall that the energy E D n on the box B = (0, L) 3 was defined in the beginning of Section 2 as the ground state energy of the non-interacting Hamiltonian H n 0 with Dirichlet boundary conditions. Proposition 5.1. For n ∈ N, let φ ∈ H 1 as (R 3n ) be supported in (0, L) 3n , with φ 2 = 1, and let 1 ≤ i ≤ M. Then n j=1 1 2 |∇ j φ| 2 − W i (x j )|φ| 2 ≥ E D n − const.
Proof. The result follows in a straightforward way from Corollary A.1, which is an adaptation of the Lieb-Thirring inequality at positive density derived in [14]. We use that | supp(W i )| 3 and W i ∞ −2 . This allows us to bound the right side of (A.54) as from which the statement readily follows. Since is an antisymmetric function supported in B |A c | , Prop. 5.1 implies that where we used |A c | ≤ N in the error term. To minimize the error we choose ∼ρ −1/3 . The factor on the right side of (5.16) then equals E D N−|A| − const.ρ 2/3 . Because of the condition that L/ ∈ N we cannot choose without restriction but it is always possible to choose a value such that ∼ρ −1/3 . We define e N to be the N-th eigenvalue of the one-particle Dirichlet Laplacian on B = (0, L) 3 . Then E D N−|A| ≥ E D N − |A|e N . Moreover, we can bound e N ρ 2/3 . In particular, We proceed with a lower bound on A i,A . Theorem 4.1 can be used for a lower bound on F α,|A| only if |A| > N 0 , with N 0 defined in (4.2). In case that |A| ≤ 2N 0 we use the bound (2.2) originating form [24] instead, which implies that using m 1. In combination with W i ∞ ρ 2/3 this gives the lower bound and hence For |A| ≥ 2N 0 , we use the bound in Theorem 4.1 on F α,|A| (ϕ p A c i,A ). Since ϕ p A c i,A is an |A|+1-particle wavefunction supported in a cube of side length (1 + 2ε), Theorem 4.1 implies that In combination with (5.19) and W i ∞ ρ 2/3 this yields the bound where we have minimized over |A| in the last step, and used that ε 1 and ∼ρ −1/3 . We are still free to choose κ in such a way as to minimize the error terms. We shall choose κ = c T ν(1 − Λ(m)) for some 0 < ν < 1 (e.g., ν = 1/2). Then N 0 (1 − Λ(m)) −9/2 , and hence (5.22) and (5.25) together yield the bound

A Lieb-Thirring inequality in a box
In this appendix we will follow the analysis of [14] to show a positive density Lieb-Thirring inequality for a system of non-interacting fermions in a box with Dirichlet boundary conditions. When reformulated via a Legendre transformation as a bound on the difference between the ground state energies with and without an external potential, we will see that this inequality in particular implies Prop. 5.1. Let C L = [−L/2, L/2] 3 be the cube in R 3 and let Π − L,µ 1 1(−∆ L ≤ µ), where ∆ L denotes the Dirichlet Laplacian on C L . For short we will just write Π − for Π − L,µ , and Π + = 1 − Π − . For a density matrix γ we denote the corresponding density by ρ γ . Of particular relevance for us is the density corresponding to Π − , which we denote by ρ 0 . Differently to the case of periodic boundary conditions (discussed in [14]), ρ 0 is not a constant and is given by where φ p are the eigenvectors of −∆ L to the eigenvalues p 2 , i.e., Since the absolute value of each eigenvector is pointwise bounded by (2/L) 3/2 we have Remark. Since the lowest eigenvalue of −∆ L equals 3π 2 L −2 , the problem simplifies for µ < 3π 2 L −2 since the projections Π ± L,µ become trivial. In this case we can simply apply the original Lieb-Thirring inequality [17] to obtain the desired bound. For our application we shall need µ L −2 , however, hence we shall restrict our attention to µ ≥ 3π 2 L −2 in the following theorem.
For a real number t we denote its positive part by t + and its negative part by t − . In particular, L,µ , with density ρ Q . There exist positive constantsK and η independent of µ, L and Q such that Remark. In [14] a similar result was proven for the Laplacian with periodic boundary conditions and we mostly follow that proof.
Remark. The crucial properties of the function S are its positivity and the fact that S (ρ) behaves like µ −1/2 ρ 2 for small ρ and like ρ 5/3 for large ρ. For technical reasons it will also be convenient that S is convex.
Essential for the proof will be to separate a given Q into Q = (Π + + Π − )Q(Π + + Π − ) Q ++ + Q +− + Q −+ + Q −− . The densities associated to Q ±± will be denoted by ρ ±± . Before we proceed with the proof of the theorem we show the following Lemma.
Proof. We claim that Q 2 ≤ Q ++ − Q −− , which follows from the condition on Q. In fact, Expanding the last inequality proves the claim. Hence Proof of Theorem A.1. We shall treat Q ±± separately and combine the various terms at the end using the convexity of S .
We shall follow the method introduced by Rumin in [26]. With the aid of the spectral projections P e 1 1(|∆ L + µ| ≥ e) we have the layer cake representation (A.9) Let us assume that γ is a smooth enough finite rank operator with 0 ≤ γ ≤ 1. Then tr |∆ L + µ|γ = where ρ e denotes the density of the finite rank operator P e γP e . For a bounded measurable set A we estimate A ρ e (x) dx = tr(1 1 A P e γP e ) = 1 1 A P e γ 1/2 2 where ρ γ denotes the density of γ and we used the triangle inequality for the Hilbert-Schmidt norm · S 2 . Because γ ≤ 1 we further get where B(R) denotes the centered open ball with radius R andB(R) its closure. Here we used where we bounded the eigenfunction φ p of −∆ L to the eigenvalue p 2 by |φ p (x)| ≤ (2/L) 3/2 . Taking A = B(R) + x with R → 0 we obtain the pointwise bound Hence we get To obtain the desired result we have to analyze R(ρ) in more detail. In the following we will use C to denote a generic constant, whose value can change throughout the computation. Obviously and the same statement holds if one takes the closureB(R) instead of B(R). For 0 < x < 1 and M > 0, (A.18) allows us to bound using that µ L −2 by assumption. For e ≥ µ we get Combining both statements we have thus shown that f (e) ≤ C µ L + µ 1/2 e1 1(e ≤ µ) + e 3/2 1 1(e > µ) = u + g(e) (A. 22) with g(e) Ce max(µ 1/2 , e 1/2 ) , u C µ L . (A.23) Using the explicit form of g, one readily checks that where we have also used that ( In the next step we want to prove bounds for Q +− and Q −+ . We introduce The following three parts of the proof will treat these terms. We start with Q ± 00 . Part 3. Q +−

00
The density of Q +− 00 is equal to Using Q ≤ 1, we can bound this as where we applied (A. 19) in the last step.
Part 4. Q +− 10 , Q +− 01 Next we will bound ρ +− 10 . For a general function W (viewed as a multiplication operator), we have To bound the first factor, we can used Lemma A.1. For the second term we need to use the specific form of the eigenfunctions for the Dirichlet Laplacian. Using (A.2) we get where (A j p j ) j and (B j q j ) j denote the vectors obtained by component-wise multiplication. Hence By taking a Legendre transform, the result above implies that following potential version of the Lieb-Thirring inequality. Remark. In case that µ < 3π 2 L −2 we have −∆ L − µ > 0, and therefore tr(−∆ L − µ) − = 0 and also ρ 0 = 0. One can thus obtain a lower bound using the standard Lieb-Thirring inequality [17] applied to a potential V − µ in this case.
Proof. We start with the identity where the infimum in the first line is over functions ρ : R 3 → R, while in the second we can restrict to non-negative functions ρ. We can pull the infimum inside the integral for a lower bound. Clearly we can assume that ρ ≥ ηL −1 µ. Introducing γ = ρ − ηL −1 µ we have We apply the above theorem for a potential V ∈ L 5/2 (C L ) with V ≤ 0, choosing µ as e N , the Nth eigenvalue of the Dirichlet Laplacian −∆ L . In particular, µ ≥ e 1 = 3π 2 L −2 which allows us to use Theorem A.2. The ground state energy E D N for N non-interacting particles confined to C L was defined in the beginning of Section 2 and can be written as E D N = N i=1 e i . We used that since V ≤ 0 the operator −∆ L + V − e N has at least N non-positive eigenvalues, and therefore we can get a lower bound on the trace of its negative part by summing only the first N of them. From the above calculation, together with ρ 0 µ 3/2 and µ = e N N 2/3 /L 2 , we deduce the following corollary.