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∗∗≈0.36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m^{**}}\approx 0.36$$\end{document}, 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 [4,5,12,28,29], but later they were also successfully applied to many other areas of physics, like polarons (see [20] and references there) or cold atomic gases [30].
It was already known to Thomas [28] 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 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 [25], 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,Theorem 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 short x 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., G μ (k 0 , k) := 1 2m We denote by F α,N the quadratic form used in [6,25] 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 (1.5) 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 ⊂ C N +1 L , where C L = (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 − 1 2 N i=1 Δ i for wavefunctions in H 1 as (R 3N ) with Dirichlet boundary conditions on ∂C L . It equals the sum of the N lowest eigenvalues of the Dirichlet Laplacian on C L , and it is easy to see that (2.2) (The additional factor (m + 1)/(2m) compared to [25, Theorem 2.1] results from the separation of the center-of-mass motion used in [25].) It was also shown in [25] that Λ(m) < 1 if m > m * * ≈ 0.36. For particles confined to the cube C L 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). Theorem 2.1 shows that the presence of the impurity affects the groundstate energy by a term that is bounded independently of N . 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. This is due to our method of proof; in particular, the utilized localization technique introduces an error which is independent of the interaction strength.
Using various types of trial states, the ground-state energy of pointinteracting systems is extensively discussed in the physics literature (see [20] and references there). We note that with this method it is only possible to derive upper bounds, while Theorem 2.1 gives a lower bound on the groundstate energy.
From a physics perspective, it would of course be interesting to extend Theorem 2.1 to the case of several (or even many) impurities [30]. However, even the basic question of stability, i.e., boundedness of the energy from below, is open for more than one impurity.

Proof Outline
For the proof of Theorem 2.1, we first prove in Sect. 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 Sect. 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 non-sharp 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 the simpler form of the Greens function for the Laplacian allows to perform explicit computations in momentum space.
Because the ground-state energy of the confined non-interacting Nparticle system is strictly positive, we are allowed to choose μ negative in the definition of F per α,N . Applying the method of [25] 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 Sect. 5, we give the proof of Theorem 2.1. Using the IMS formula of Proposition 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. When optimizing over the distribution of particles, we shall conclude from these bounds that only O(ρ 3 ) = O(1) particles will be in the small box. This allows us to improve the rough bound of Theorem 4.1 and show Theorem 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. Afterward, 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 C L = (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 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 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.

Lemma 3.1.
For J ∈ C ∞ (R 3(N +1) ) bounded and with bounded derivatives, we (3.11) 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), i.e., as multiplication by the function restricted to the relevant plane {x i = x 0 }. 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, . 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 which indeed equals (3.14). This completes the proof of the lemma.
Using Lemma 3.1 we get that 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.
In particular, and we have 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 asF α,N (ψ 1 , ψ 2 ). It suffices to prove that for smooth functions J, since theñ (3.28) Recall the definition H μ = H N 0 + μ. The left side of (3.27) equals 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 representation (3.21), which together with (3.11) implies that as shown in the proof of Lemma 3.1, we can rewrite the terms in the integrand as 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 as (3.14) 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 C N +1 with C = (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 [25] 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 C 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 of F 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. This latter property would also be true for Dirichlet boundary conditions; the simpler form of the Greens function of the Laplacian makes periodic ones more convenient, however. In particular, it allows to apply the method of [25], 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 between Λ(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 ⊂ C N +1 , we extend ψ to a periodic function ψ per , defined as (4.5) 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 Greens function of the Laplacian 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 center-of-mass and relative coordinates for the pair (k 0 , k 1 ), i.e., More generally, we have (4.10) Then, (4.12) 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 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 cutoff in momentum space would not be suitable.
We shall now defineF per α,N with domain where H 1 per (C N +1 ) and H 1/2 per (C 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 (3.8), and the singular parts of the quadratic form are given bỹ 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 definition (3.9). In the sense of distributions, we can apply H μ to φ μ , and in particular ). In this sense, we can write the regular part ofF α,N as We use the identity χφ μ = χφ per is supported on C \ C − , 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 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 For the terms with i = j, we can use dominated convergence in momentum space to conclude that For the terms with i = j, we can further write (4.26) 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 equalsT per,μ,N dia ( ξ per ). Combining the above, we obtain This completes the proof of the lemma.  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 , . . . , k N Using the resolvent identity, we see that the regular part of the quadratic form satisfies A straightforward computation using definitions (4.13)-(4.16) shows that Combining both statements yields the desired identity (4.32)

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 C 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.

Lemma 4.4. Given μ and q such that
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 q1 = 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.
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 [25], 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. We are only interested in values of q = (q 1 , . . . , q N ) where ξ per ( q) is nonzero, which requires all the q i for i ≥ 2 to be different since ξ per is antisymmetric in these variables. (We note that in comparison with [25] 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 (4.45) In particular, Lemma 4.4 yields the bound (4.46) on the diagonal term of the singular part of F per α,N . Following the same steps as in [25], we can obtain the following lower bound for the off-diagonal term.

Proposition 4.1. Assume that μ ≥ −κN 5/3 −2 for some κ < c T . Then for all
Proof. The proof works in almost the exact same way as in [25]; 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 [25, Eq. (4.12)]) differently, namely as For comparison, δ = 0 was used in [25]. Following the proof in [25,Sect. 4], this choice gives a lower bound to the off-diagonal term of the form with a prefactorΛ δ,μ (m) equal to sup s,K∈R 3 ,Q 2 >cT N 5/3 −2 Since (4.45) holds under our assumption on μ, we can use Q 2 ≤ Q 2 μ (1 − κ/c T ) −1 in the first numerator in (4.52) to conclude that inf δ>0Λδ,μ (m) ≤ (1 − κ/c T ) −1Λ (m, κ), which yields the desired result.

A Bound onΛ(m, κ)
We will not evaluateΛ(m, κ) directly, but we will compare it with Λ(m), which is defined in [25, Eq. (2.8)] and which was already referred to in (2.2) above.

The expression Λ(m) can be written as Λ(m) := sup
The additional constraint on Q μ in the latter supremum has no effect because of the scaling properties of λs ,Qμ,K,m,0 , specifically λ νs,νQμ,νK,m,0 (νt) = ν −3 λs ,Qμ,K,m,0 (t) for any ν > 0, which allows to fix one of the parameters when taking the supremum. Expression (4.48) differs from (4.53) by the nonzero 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 ,Qμ,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., Proof. For the pointwise bound (4.55), we will proceed similarly to [25,Sect. 6].
Using the Cauchy-Schwarz inequality, we have and also (4.58) By minimizing over K, we find that from which (4.55) readily follows.
We denote the right side of (4.55) by λ > (t) = λ > s,Qμ,K,m,δ (t), and we will write λ(t) = λs ,Qμ,K,m,δ (t) in the following. That is, (4.55) reads λ(t) λ > (t). First, we treat the termt = AK in (4.56). Using (4.61), we can bound for anyt and hence, in particular, fort ∈ C (AK). For the case 0 = t ∈ L, we note that for τ 1 , τ 2 ∈ C (t) the bound |τ 1 | ≤ √ 11|τ 2 | holds, and hence In particular, the maximal value of λ > in C (τ ) is dominated by the average value, and therefore As a last step, we explicitly evaluate the integral, which results in the bound This completes the proof of the lemma. (4.66) Proof. As in the proof of the previous Lemma, we denote λ(t)=λs ,Qμ,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 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 For the second term, we obtain 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 bound (4.56) holds also with λs ,Qμ,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 Proof. We first note that Λ(m) ≤ 1 implies m 1. Moreover, from definition (4.49) we have (4.76) Combining this with Lemma 4.6 and taking the supremum overs, K and where we also used that Λ(m) For an upper bound, we shall choose δ ∼ N 4/9 , which yields the desired bound

Proof of Theorem 4.1
Using Proposition 4.1, Eq. (4.46) and Lemma 4.7, we get the lower bound 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 nonnegative, 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 (C 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.
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 nonnegative, with η(0) > 0, 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 ∈ C L , define Note that since η is assumed to be strictly positive in a small ball around the origin, the denominator in (5.1) is strictly positive for x ∈C L , hence (5.1) is well defined. We have supp J i ⊆ C 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 ∈ C L 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 ⊂ C N +1 L and ψ 2 = 1. We use the IMS formula, Proposition 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 (5.4) 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) Figure 1 visualizes this setup.
We localize all the remaining particles using the IMS formula in Proposition 3.1, with the localization functions . . , N}\A. For short we define A straightforward calculation using Proposition 3.1 and the fact 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 ℓ L 1 ℓ ℓ ℓ 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.
Proof. We define ξ j and φ μ for some μ > 0 using the unique decomposition (5.10) 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 where we used the fact that . 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 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 (5.14) and (5.15) 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 C L = (0, L) 3 was defined in the beginning of Sect. 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, 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 ϕ p0, pA i,A is an antisymmetric function supported in C 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 C L = (0, L) 3 .
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 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 −Δ 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 −Δ 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 ; 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, t = t + − t − .
There exist positive con-stantsK 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 [27]. With the aid of the spectral projections P e := 1 1(|Δ L + μ| ≥ e), we have the layer cake representation Let us assume that γ is a smooth enough finite rank operator with 0 ≤ γ ≤ 1. Then, tr |Δ 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 · S2 . Because γ ≤ 1, we further get where B(R) denotes the centered open ball with radius R andB(R) its closure.
Here, we used (A.14) where we bounded the eigenfunction φ p of −Δ 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 where we used (1+x) 3/2 − (1− x) 3 Using the explicit form of g, one readily checks that where we have also used that ( In combination with (A.16), this shows 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. 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 use Lemma A.1. For the second term, we need to use the specific form of the eigenfunctions of 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, The sum of (A.31) is included in the second line of the previous calculation by extending the sum over p, q ∈ N 3 to p, q ∈ (Z \ {0}) 3 , and we have again used (A. 19) in the last step.
p,q∈(πZ 3 /L) 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 −Δ L − μ > 0, and therefore tr(−Δ 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 This equality can be extended to allow A = −Δ − μ and B = V (see [14,Theorem 4.1]). Using this and applying Theorem A.1, we get where the infimum in the first line is over functions ρ : R 3 → R, while in the second we can restrict to nonnegative functions ρ. We can pull the infimum inside the integral for a lower bound. Clearly, we can assume that ρ ≥ ηL −1 μ. (A.49) Using that We apply the above theorem for a potential V ∈ L 5/2 (C L ) with V ≤ 0, choosing μ as e N , the N th eigenvalue of the Dirichlet Laplacian −Δ L . In particular, μ ≥ e 1 = 3π 2 L −2 which allows us to use Theorem A.2. The groundstate energy E D N for N non-interacting particles confined to C L was defined in the beginning of Sect. We used that since V ≤ 0 the operator −Δ L + V − e N has at least N nonpositive 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.