The Excitation Spectrum of Two-Dimensional Bose Gases in the Gross–Pitaevskii Regime

We consider a system of N bosons, in the two-dimensional unit torus. We assume particles to interact through a repulsive two-body potential, with a scattering length that is exponentially small in N (Gross–Pitaevskii regime). In this setting, we establish the validity of the predictions of Bogoliubov theory, determining the ground state energy of the Hamilton operator and its low-energy excitation spectrum, up to errors that vanish in the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rightarrow \infty $$\end{document}N→∞.


Introduction
In the past decades, Bose-Einstein condensates (BEC) have emerged as important quantum systems, in view of the precision and flexibility with which they can be manipulated. Experiments on thin films [5] or in highly elongated magnetic and pancake-shaped optical traps (see e.g., [11,Sect. 1.6]) have also pushed forward the study of BEC in low-dimensional systems. As a matter of fact, dimensionality plays a crucial role in situations where spontaneous symmetry breaking of continuous symmetries occurs [23,30]. Hence, it is not surprising that equilibrium properties of the two-dimensional Bose gas exhibit significant differences compared with the three-dimensional case (see e.g., [27,Chapter 3], [35,Chapter 23], [17]).
In this paper, we are interested in the low energy spectrum of two dimensional dilute Bose gases, describing systems where both the quantum and thermal motions are frozen in one direction (see [10,36] for a discussion of regimes where the confined system has rather a three-dimensional character). In particular, we consider N bosons moving in the two-dimensional box Λ = [−1/2; 1/2] 2 , with periodic boundary conditions (the two-dimensional unit torus) and described by the Hamilton operator acting on the Hilbert space L 2 s (Λ N ), the subspace of L 2 (Λ N ) consisting of functions that are symmetric with respect to permutations of the N particles.
Here V is a non-negative, compactly supported and spherically symmetric two body potential. The form of the scaled interaction V N (x) = e 2N V (e N x) is chosen so that the scattering length of V N is equal to e −N a, with a the scattering length of V . Indeed in two dimensions and for a potential V with finite range R 0 the scattering length is defined by where R > R 0 , B R is the disk of radius R centered at the origin and the infimum is taken over functions φ ∈ H 1 (B R ) with φ(x) = 1 for all x with |x| = R (see for example [27,Sect. 6.2]). In the scaling limit defined by (1.1), known as the two-dimensional Gross-Pitaevskii regime, we provide an expression for the ground state energy and the low-energy excitation spectrum, up to errors vanishing as N → ∞, validating the predictions of Bogoliubov theory [9]. In particular we exhibit a proof of the linear dependence of the dispersion of low-lying excitation at low-momenta, a fact which is interpreted in the physics literature as a signature for superfluidity. Remark that, rescaling lengths, the two-dimensional Gross-Pitaevskii regime can be interpreted as describing an extended Bose gas (of particles interacting through the unscaled potential V ) at a density that is exponentially small in N . While the exponential smallness of the density (or, equivalently, of the scattering length) makes it difficult to directly apply our results to physically relevant situations, it should be stressed that the Gross-Pitaevskii regime provides a first example of scaling limit in which peculiarities of twodimensional systems can be observed.
The following theorem is our main result.
Remarks. (i) To keep our analysis as simple as possible, we restrict our attention to bosons moving in the two-dimensional unit torus. Our results could be extended to more general trapping potentials, combining the proof of Theorem 1.1 with ideas from [14,15,32,33], recently developed in the three-dimensional setting. (ii) To leading order, the first rigorous computation of the ground state energy of a dilute two-dimensional Bose gas has been obtained in [28].
In this paper, the authors considered a system of N particles, moving in a box with side length L and interacting through a two-body potential with scattering length a. In the thermodynamic limit N, L → ∞ at fixed density ρ = N/L 3 , they considered the ground state energy per particle, e(ρ), and they proved that Translating to the Gross-Pitaevskii regime (where ρ = N and the scattering length is given by e −N a), this bound implies that which is consistent with the leading order term in (1.3). The estimate (1.5) has been extended to general trapping potentials in [25]. Recently, also the free energy of a two-dimensional dilute Bose gas at positive temperature has been computed to leading order in [18,29] (thermodynamic limit) and in [19] (Gross-Pitaevskii regime). (iii) It is interesting to compare our bound (1.3) with the second order approximation of the energy per particle in the thermodynamic limit, given by with b = | log(ρa 2 )| −1 and where γ = 0.577.. is Euler's constant. This expression, first predicted in [2,31,34], has been recently proved, for all positive potentials with finite scattering length, in [20] (partial results have been previously obtained in [21], restricting the analysis to quasifree states). In the Gross-Pitaevskii limit, where ρ = N and b = (2N − log N − log a 2 ) −1 , one can check that (1.6) is consistent with (1.3) (in the thermodynamic limit, the lattice spacing in Λ * tends to zero, and the sum over p ∈ Λ * is replaced by an integral, which is convergent because lim r→0 J 0 (r) = 1).
(iv) It is interesting to observe that (1.3) and (1.4) only depend on the interaction potential through the term π 2 a 2 and the argument of the Bessel function J 0 in the expression for the ground state energy. Observing that the quantity −2π log( /a) + π 2 and the dispersion of the low-energy excitations would be given by ε (R) (p) = p 4 + 8πR p 2 . Approximating the sum with an integral (in the limit of large R, after replacing the variable p with p/ √ R), this leads to which is perfectly consistent with the formula (1.6) obtained in the thermodynamic limit. (v) The assumptions on V are technical; the result is expected to hold true for any positive interaction with finite scattering length (in particular bounds compatible with (1.3) and upper bounds matching (1.4) for hard core interactions can be obtained following [3,20]) and also, more generally, for a certain class of (not necessarily non-negative) potentials having positive scattering length. The condition V ∈ L 3 (R 2 ) is used to show some properties of the solution of the scattering equation, in Lemma 2.1. The restriction to V ≥ 0 is used to discard certain error terms, when proving lower bounds for the eigenvalues of (1.1).
The proof of Theorem 1.1 is based on Fock space methods, recently developed in the three-dimensional setting, to study the dynamics of Bose-Einstein condensates [4,13] and to investigate the equilibrium properties of dilute gases Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2881 in the Gross-Pitaevskii regime. In particular, these techniques led to the verification of the predictions of Bogoliubov theory for the ground state energy and the excitation spectrum of three-dimensional Bose gas in the Gross-Pitaevskii regime, confined on the unit torus [8,22] or by more general trapping potentials [15,33]. The starting observation is that, in order to investigate the low-energy properties of Bose gases, it is convenient to factor out the Bose-Einstein condensate and to focus on its orthogonal excitations. This suggests to introduce a unitary transformation U N , mapping the N -particle Hilbert space L 2 s (Λ N ) into the truncated bosonic Fock space constructed over the orthogonal complement L 2 ⊥ (Λ) of the condensate wave function ϕ 0 (defined by ϕ 0 (x) = 1, for all x ∈ Λ). On the Hilbert space F ≤N + , we introduce the excitation Hamilton L N = U N H N U * N , given by the sum of a constant and of terms that are quadratic, cubic and quartic in (appropriately defined) modified creation and annihilation operators (see (2.3)). In the very spirit of the Bogoliubov approximation, we aim at reducing L N to a quadratic (and therefore diagonalizable) Hamiltonian, up to error terms vanishing in the limit of large N . To achieve this goal, we conjugate L N with suitable unitary operators, modeling the correlation structure created by the singular two-body interaction.
The main input for our analysis are the recent results of [16], proving a bound of the form for the ground state energy and, most importantly, showing that the ground state and low-energy states of (1.1) exhibit complete Bose-Einstein condensation, with at most order log N excitations. This estimate is used here to show that several error terms, emerging from the unitary conjugations can be neglected. While this strategy is similar to the one used in the three-dimensional setting (see, for example, [6,8,12,15,22,33]), the choice of the appropriate unitary transformations and their action strongly depend on the specific problem under consideration.
Compared with the three-dimensional setting, a first important difference we have to face to prove Theorem 1.1 is the fact that, in the two-dimensional Gross-Pitaevskii regime, correlations among particles are much stronger. This can already be seen by noticing that the expectation of (1.1) on factorized states is of the order N 2 , in the limit of large N . Hence, correlations among particles are responsible for reducing the ground state energy of (1.1) to a quantity of order N . As a consequence, some additional care is required when studying the action of quadratic and cubic transformations that generate the correlation structure characterizing low-energy states. In particular, since cubic terms in the Hamilton operator carry large contributions to the energy (growing with N , as N → ∞) we are not able to prove a-priori bounds on moments of the number of excitations (nor on products of the energy with moments of the number of excitations operator), which were important in the three dimensional setting [8]. To overcome this problem, we are going to apply a localization on the number of particle argument (similarly to the one recently exploited in [22,33]), combined with a-priori bounds on the energy of the excitations. A second important difference, compared with the three-dimensional setting, is that even after quadratic and cubic conjugations, the quartic part V N of the (renormalized) excitation Hamiltonian is not negligible on uncorrelated states. While this is not a problem for the derivation of lower bounds (V N is the restriction of the potential energy on the orthogonal complement of ϕ 0 ; therefore, it is non-negative), it affects the proof of upper bounds for the eigenvalues of H N . To circumvent this problem, we need to implement an additional unitary transformation, defined by the exponential of a quartic expression in creation and annihilation operators. Through this quartic conjugation, we eliminate the low-momentum part of V N . This allows us to show upper bounds for the ground state energy and for low-energy excited eigenvalues of H N using uncorrelated states with low-momenta. This part is the main novelty of our work. We remark that unitary operators given by the exponential of quartic expressions in creation and annihilation operators have already been used in three dimensions in [1]. The action of the quartic operators used here, however, is quite different. In particular, they renormalize the interaction up to contributions which are only negligible on suitable low-momentum states (we will use such low-momentum states as trial states, to prove upper bounds on the eigenvalues of (1.1)).
The plan of the paper is as follows. In the next section, we introduce the formalism of second quantization and the map U N , factoring out the condensate. Moreover, we define the quadratic transformation e B and the cubic transformation e A that allow us to approximate the renormalized excitation Hamiltonian R N = e −A e −B L N e B e A by the sum of a quadratic Hamiltonian and of the quartic term V N . The action of the unitary operators e B , e A , the properties of R N and their implications for Bose-Einstein condensation in lowenergy states of (1.1) are discussed in Sect. 2. Up to this point, the analysis is similar to [16] (some adaptation is still required, because we need here slightly stronger bounds, compared with those established in [16]; for example we need an estimate for the energy of excitations, not only for their number). The real novelty of the present paper is in Sects. 3-5, where we show how to extract order one contributions to the ground state energy (to go from (1.8) to the much more precise estimate (1.3)) and to compute low-energy excitations. In Sect. 3, we introduce the quartic conjugation e D and we show how it can be used to get rid of the low-momentum part of V N . In Sect. 4, we diagonalize quadratic Hamiltonians that have been derived in Sect. 2 and in Sect. 3 (we will work with two different quadratic Hamiltonians, one for the upper bounds, one for the lower bounds). The results from Sects. 2-4 are combined in Sect. 5 Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2883 to complete the proof of Theorem 1.1; for the proof of the lower bounds, we apply here a localization argument.

The Renormalized Excitation Hamiltonian
We are going to describe excitations of the Bose-Einstein condensate on the truncated Fock space F ≤N + = N n=0 L 2 ⊥ϕ0 (Λ) ⊗sn constructed on the orthogonal complement of the zero-momentum orbital ϕ 0 (x) = 1 for all x ∈ Λ. As first observed in [24], we can define a unitary map U N : By definition, U N ψ N ∈ F ≤N + describes the orthogonal excitations of the condensate, in the many-body state ψ N .
For any p, q ∈ Λ * + = 2πZ 2 \{0}, we find (see [24,Prop. 4 for all p, q ∈ Λ * . With U N , we define the excitation Hamiltonian L N := U N H N U * N , acting on a dense subspace of F ≤N + . Expressing (1.1) in second quantized form and using (2.1), we find Here, we defined the Fourier transform of V by for all k ∈ R 2 , and we introduced the notation for the kinetic and potential energy operators, restricted to the orthogonal complement of the condensate wave function. In the rest of the paper, we are going to use the notation H N = K + V N . The Hamilton operator L N is the starting point for our analysis. As discussed in the introduction, we are going to conjugate L N by suitable unitary operators to extract large contributions to the energy that are still hidden in L N . To construct these unitary operators, we consider the ground state solution f of the eigenvalue problem on the ball |x| ≤ e N , satisfying Neumann boundary conditions and normalized so that f (x) = 1 for |x| = e N (for simplicity we omit here the N -dependence in the notation for f and for λ ). We will later choose = N −α with α > 0 so that e −N 1. The next Lemma (proven in Appendix B) collects properties of f , λ that will be important for our analysis. Lemma 2.1. Let V ∈ L 3 (R 2 ) be non-negative, compactly supported (with range R 0 ) and spherically symmetric, and denote its scattering length by a. For any 0 < < 1/2, N sufficiently large, let f denote the solution of (2.6). Then . (2.9) Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2885 (iv) Let w = 1 − f . Then there exists a constant C > 0 such that We rescale the solution of (2.6), setting f N, (x) := f (e N x) for |x| ≤ , and f N, (x) = 1 for x ∈ Λ, with |x| > . Then with χ denoting the characteristic function of the ball |x| ≤ . Setting with Fourier coefficients Notice that η p ∈ R (from the radial symmetry of f ). To express the scattering equation (2.10) in terms of the coefficients η p , it is useful to introduce the function ω N ∈ L ∞ (Λ), defined through the Fourier coefficients for all p ∈ Λ * + (here χ (p) and χ(p) denote the Fourier coefficients of the characteristic functions of the ball of radius and one respectively, and we used that χ (p) = 2 χ( p)). Again, we find ω N (p) ∈ R (by radial symmetry of χ ). In the next lemma, we list some properties ofη and of ω N . Lemma 2.2. Let V ∈ L 3 (R 2 ) be non-negative, compactly supported and spherically symmetric, and denote its scattering length by a. For any 0 < < 1/2, N sufficiently large, letη and ω N be defined as in (2.11) and (2.13), respectively. Then, we have |η 0 | ≤ C 2 and ω N (0) = πg N with |g N | ≤ C, uniformly in N . More precisely, we find (2.14) Moreover, we have ω N (p) ≥ 0 for all p ∈ Λ * + with |p| ≤ 1 and the pointwise bounds We also have the estimates Finally, for every p ∈ Λ * + , we can write (2.10) as Proof. The bounds for |η 0 |, |η p |, η 2 , η H 1 have been established in [16,Sect. 3]. The bounds for ω N (0) are a direct consequence of Lemma 2.1 (in particular, of parts (ii) and (iii)). To prove that ω N (p) ≥ 0 for p ∈ Λ * + with |p| ≤ 1 and to show the estimate for | ω N (p)|, we observe that, denoting by J 1 the Bessel function of the first kind of order 1, As mentioned above, we choose = N −α so that η 2 , |η 0 | ≤ CN −2α will be small factors. With the coefficients η p , introduced in (2.12) we define, following [16], the antisymmetric operators We will consider the unitary operators e B and e A . For our analysis, it will be important to control the growth of number of particles and energy with respect to the action of e B , e A ; the following lemma is proven in [16,. (2.17) and (2.18). Then, for any k ∈ N there exists a constant C > 0 (depending on k) such that

Lemma 2.3. Suppose that B, A are defined as in
Moreover, we also have the following bound for the growth of the energy w.r.t. e A (a similar estimate also holds for the action of e B , but we will not need it in the sequel): holds true on F ≤N + , for any α > 0 (recall the choice = N −α in the definition (2.12) of the coefficients η p ), for all α ≥ 1, s ∈ [0; 1] and N ∈ N large enough.
With A, B, we define the renormalized excitation Hamiltonian

19)
Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2887 In the next proposition, we collect important properties of R N . Part a) isolates the important contributions to R N ; its proof follows closely the proof of Prop. 4 in [16] and is deferred to Appendix A. Part b) and c), on the other hand, are consequences of part a) and will be used to show upper and, respectively, lower bounds on the eigenvalues.
(a) There exists a constant C > 0 such that Let P L be the low-momenta set for an error term E R satisfying In contrast with the three-dimensional setting, V N is here of order one (on uncorrelated trial states); this is the reason why, to show upper bounds on the eigenvalues of R N , we will need an additional conjugation, with a quartic phase.
Proof of Proposition 2.4. As explained above, the proof of part (a) is sketched in Appendix A.
Part (b) follows from part (a). In fact, the cubic term appearing on the r.h.s. of (2.20) can be estimated by The Excitation Spectrum of Two-Dimensional 2889 where we used that and As for the off-diagonal quadratic contribution associated with momenta p ∈ P c L , we find, with Lemma 2.2, Finally, we show part c). Again, we start from (2.20) and we use (2.28) to bound the cubic term and (2.33) to control the off-diagonal quadratic contribution associated with p ∈ P c L . Instead of (2.31), we notice that, since This bound, combined with the observation that, by (2.13), Ann. Henri Poincaré and with V N ≥ 0 implies that, for any γ > 0, Later on, we will need to fix γ < 1/4 to control the error proportional to N 2 + . With this restriction and for ν ∈ (1/6; 1/2) there exists C such that Here, we divided the kinetic energy into the sum of two operators; in the one associated with p ∈ Λ * As shown in [16, Theorem 1.1], an important consequence of part a) of Prop. 2.4 is the emergence of Bose-Einstein condensation for low-energy states, with an optimal control on the number of orthogonal excitations. This also implies an upper bound for the expectation of the operator H N , on the excitation vectors associated with low-energy states; this is the content of the next proposition. Proposition 2.5. Let V ∈ L 3 (R 2 ) be non-negative, compactly supported and spherically symmetric. Let ψ N ∈ L 2 s (Λ N ) with ψ N = 1 belong to the spectral subspace of H N with energies below 2πN + ζ, i.e., for any α ≥ 5/2 and N large enough. The assumption (2.34), and the definition of ξ N imply therefore that Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2891 From the condensation estimate [16, Eq. (61)] and from Lemma 2.3, we conclude that

Quartic Conjugation
From (2.25), it is clear that to prove upper bounds on the eigenvalues of R N , we cannot ignore the contributions of V N on the r.h.s. of (2.25). Instead, we conjugate R N with a quartic phase, which (up to errors that can be neglected) removes the low-momentum part of V N , leaving us with an operator whose expectation vanishes on states generated by the action of creation operators a * p , with p ∈ P L , the low-momentum set defined in (2.23). At the end, this will allow us to show upper bounds for the eigenvalues of R N , making use of trial states involving only particles with low momentum.
We consider the quartic operator Here, η p is defined as in (2.12) and Since D commutes with the number of particles operator N + , we trivially obtain that We state now two lemmas that will be shown in the next subsections. In the first lemma, we control the action of the quartic transformation on the kinetic energy operator. Lemma 3.1. Let K and D be defined in (2.5) and in (3.1), respectively, with α ≥ 5/2 and ν ∈ (0, 1/2). Let κ ∈ N the smallest integer s.t. κ > 4(α+ν −1/2). Then there exists C > 0 such that Remark. Since N + commutes with D, (3.2) also implies that for all j ∈ N. In the second lemma, we bound the growth of the potential energy operator.
Then, we have Using the last two lemmas, we can describe the action of the quartic transformation on the renormalized excitation Hamiltonian R N . Our goal consists in proving that, on low-momentum states, the operator e −D R N e D is given, up to negligible errors, by a quadratic Hamiltonian which will be later diagonalized in Prop. 4.3.

Proposition 3.3.
Let R N be defined as in (2.25) and D defined as in (3.1) with α ≥ 5/2 and ν ∈ (0, 1/2). Let C R and Q (L) R be defined in (2.22) and (2.24), respectively. Suppose that ξ L ∈ F ≤N + is such that a p ξ L = 0, for all p ∈ P c L , with the low-momentum set P L defined as in (2.23). Then, we have where κ ∈ N is the smallest integer s.t. κ > 4(α + ν − 1/2) and N ∈ N is large enough.
Proof. From (2.25), we can write The Excitation Spectrum of Two-Dimensional 2893 and With Lemma 3.1 and Lemma 3.2, we find immediately that and, with (3.6), we find v+r a w δ w,r+v and using (2.2), we find We can estimate for all i = 1, . . . , 6. Indeed Finally, using that |η r | ≤ |r| −2 , together with (2.29), we end up with The Excitation Spectrum of Two-Dimensional 2895 The terms Z 5 and Z 6 can be bounded similarly. With Lemma 3.1, we conclude that where Combining (3.8), (3.9), (3.10) and (3.11), we obtain (3.7).

Growth of the Kinetic Energy
In this section, we show Lemma 3.1, establishing a-priori bounds on the growth of the kinetic energy under the action of the unitary operator e D . We will use the following preliminary estimate. With (3.14) To estimate the first term on the r.h.s of (3.14), we use (2.9) to estimate Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2897 Hence, As for K 3 , we use Eq. (2.9) in Lemma 2.1 to conclude: (3.16) Finally, to bound K 4 we write r·v = (r+v)·v−|v| 2 and we split correspondingly K 4 in two terms, denoted by K 41 and K 42 below. Recalling from Lemma 2.2 that On the other hand, a v+r a w−r (N + + 1) 1/2 e sD ξ 2 1/2 ≤ CN α+2ν−1 K 1/2 e sD ξ (N + + 1) 3/2 e sD ξ . Hence, applying Gronwall's lemma to the differential inequality we end up with (3.13).
With the help of Lemma 3.4, we can now show Lemma 3.1.

Proof of Lemma 3.1. We first show that the commutator [K, D] satisfies the bound
Indeed, the bounds for the terms K 1 , K 2 and K 4 defined in (3.14) can be all improved by using the kinetic energy operator. We have (recall the definition of P L in (2.23)) and, similarly, Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2899 To show that K 4 is also bounded from the r.h.s. of (3.17), we split as before K 4 into K 41 and K 42 . We get On the other hand, distinguishing the cases r + v ∈ P L and r + v ∈ P c L we find Eq. (3.17) then follows from the previous bounds, together with (3.14) and  Iterating κ − 1 times we obtain Estimating the error term with Lemma 3.4, we find Choosing κ > 4α + 4ν − 2 and N large enough, we obtain (3.2). Applying (3.17) to the identity in (3.19), we find With (3.2), we arrive at (3.3). with

Growth of the Interaction Potential
Moreover, Proof. First, we prove Eq. (3.20), (3.21). A straightforward computation leads us to [a * p+u a * q a p a q+u , a * v+r a * w−r a v a w ] = δ q+u,v+r a * p+u a * q a p a * w−r a v a w + δ q+u,w−r a * p+u a * q a p a * v+r a v a w + δ p,v+r a * p+u a * q a * w−r a q+u a v a w + δ p,w−r a * p+u a * q a * v+r a q+u a v a w − δ q,v a * v+r a * w−r a * p+u a w a p a q+u − δ q,w a * v+r a * w−r a * p+u a v a p a q+u − δ p+u,v a * v+r a * w−r a w a * q a p a q+u − δ p+u,w a * v+r a * w−r a v a * q a p a q+u .

(3.23)
Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2901 Normal ordering the terms in the first and in the last lines, we obtain: Using the definition η r = −N w N, (r), and w N, (r) = δ r,0 − f N, (r), we further split the first term on the r.h.s. of (3.24), thus getting To conclude the proof of (3.20), we are going to bound the terms V i , i = 1, . . . , 5. We notice that V 5 = −K 1 (see (3.14)), hence it satisfies the bound in (3.18). On the other hand, with |η 0 | ≤ N −2α and (which can be proved similarly as in (3.15)), with the difference that Next we bound V 1 . We split V 1 in two terms V 11 and V 12 , defined by restricting to the cases v + r ∈ P L and v + r ∈ P c L , respectively; we have Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2903 As for V 12 , we have Next, we focus on V 2 . We get: Finally to estimate V 3 , we consider the contributions coming from q ∈ P L and q ∈ P c L separately, which we denote with V 31 and V 32 , respectively. We get This concludes the proof of (3.20), (3.21). In order to show (3.22), we observe that We already proved that the operators V 1 , V 2 , V 3 on the r.h.s. of (3.24) can be bounded by the r.h.s. of (3.21); this will not change with the additional constraints. To conclude the proof of (3.22), we only have to show that also the first sum on the r.h.s. of (3.24), when restricted to momenta determined by (3.26), can be bounded by the r.h.s. of (3.21). This follows from Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2905 With Lemma 3.5, we can now show the validity of Lemma 3.2.

Proof of Lemma 3.2.
We write Expanding once more the integral and using ( where Here κ ∈ N is the smallest integer such that κ > 4(α + ν − 1/2) and N is large enough. which we are going to diagonalize in Sect. 4.1; this will be used later to prove upper bounds on the eigenvalues of (1.1).

Diagonalization of Quadratic Hamiltonians
On the other hand, by Prop. 2.4, the excitation Hamiltonian R N can be bounded below by the quadratic operator appearing on the r.h.s. of (2.27), which will be diagonalized in Sect. 4.2; this will allow us later to establish lower bounds on the eigenvalues of (1.1).

Diagonalization of (4.1)
For p ∈ Λ * + , we introduce the notation be non-negative, compactly supported and spherically symmetric. Let F p and G p be defined as in (4.2). Then there exists a constant C > 0 such that . On the one hand, we have F p − G p = p 2 > 0; on the other hand, it is easy to show that F p + G p = p 2 + 2 ω N (p) ≥ p 2 /2 > 0, arguing as we did for the lower bound in part (i). Thus, |G p | < F p .
By Lemma 4.1, part (iii), we can introduce, for an arbitrary p ∈ Λ * , the coefficient τ p , requiring that We define the antisymmetric operator with the low-momentum set P L defined in (2.23). The generalized Bogoliubov transformation e Bτ has the following properties.

(4.4)
Proof. We proceed similarly as in [8,Lemma 5.2]. From Lemma 4.1 and from |τ p | ≤ C|G p |/F p , we easily obtain To show the first bound in (4.4), for k = 1, we consider, for a fixed ξ ∈ F ≤N and using τ 2 ≤ C, we obtain |f ξ (s)| ≤ Cf ξ (s). With Gronwall, we obtain the first bound in (4.4), for k = 1. The case k > 1 can be handled similarly.
As for the second estimate in (4.4), let us consider the case k = 0. For ξ ∈ F ≤N + , we set g ξ (s) = ξ, e −sBτ Ke sBτ ξ Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2907 and we compute Using where we used the estimate for the growth of N + , shown above. By Gronwall, we obtain the second bound in (4.4), for k = 0. The case k > 0 can be treated analogously (in this case, g ξ (s) contains an additional contribution, arising from the commutator of B τ with (N + + 1) k , which can also be treated similarly; for more details, see [8,Lemma 5.2]). Finally, let us show the last estimate in (4.4), focussing again on the case k = 0. For fixed ξ ∈ F ≤N + , we define We have Switching to position space, we find Using τ 1 ≤ C log N , τ ≤ C and the first estimate in (4.4) (for k = 2), we find By Gronwall, we obtain the last bound in (4.4), for k = 0. The case k > 0 can be treated similarly.
In the next proposition, we show that conjugation with the generalized Bogoliubov transformation e Bτ diagonalizes the quadratic Hamiltonian (4.1), up to negligible errors. defined in (2.22) and, respectively, (2.24) with parameters α ≥ 5/2 and ν ∈ (0; 1/2)). Let Then where and the error term δ Bog is bounded by for N large enough.
Let us now consider the constant term on the r.h.s. of (4.9). From (2.22) and (4.2), we obtain (adding and subtracting the factor p∈PL ω 2 N (p)/(4p 2 )) (4.11) Expanding the square root, we find (4.12) uniformly in N . Up to an error vanishing as N −1/2 , we can therefore restrict the sum on the last line of (4.11) to |p| < N 1/4 . After this restriction, we can use | ω N (p)−4π| ≤ C|p|N −α +C(log N )/N , to replace ω N (p) by 4π. Comparing with (4.6) (and noticing that (4.12) remains true, if we replace ω N (p) with 4π), we conclude that (4.13) Let us now consider the terms on the second line of (4.11). First of all, we observe that, by (2.9), As for the second term on the r.h.s. of (4.11), we use the scattering equation (2.15) and the definition (2.13) to write (4.14) Since ω N * η ∞ ≤ ω N η ≤ C, the last term on the r.h.s. of (4.14) is negligible, of order (log N )/N . The second term on the r.h.s. of (4.14), on the other hand, cancels with the third term on the r.h.s. of (4.11), up to a small error of order N −3ν (because p∈P c L ω 2 N (p)/p 2 ≤ N −3ν , from Lemma 2.2 and by the definition (2.23) of the set P L ). Finally, to estimate the first term on where we introduced the notation E(p) = |p| 4 + 8π|p| 2 , for p ∈ P γ = {p ∈ Λ * + : |p| < N γ/2 }, and E(p) = N γ , for p ∈ P L \P γ . Next, we use (see [8,Lemma 5.3] for a proof) that where the remainder operator D p satisfies (4.26) This implies that, after some algebraic manipulations, that With (4.26) and (4.20) we easily bound where we used (4.20). The term δ (2) N can be bounded similarly. Using again (4.20), we also obtain Summarizing, from (4.23)-(4.27) we obtain that Inserting on the r.h.s. of (2.27), we find With Eq. (4.16) and (4.17) (choosing = a) and using the a-priori bound (4.20) we have Observing that, by (2.2) (in particular, the last two commutators), [B v , a * p a p ] = 0 for all p ∈ Λ * + \P L (because, from (4.19), B v only contains the operators b p , b * p with p ∈ P L ), we arrive at (4.21).

Proof of Theorem 1.1.
In this section we focus on the low energy spectrum of H N . We fix α = 5/2 and ν = 1/5 (recall the definitions of = N −α and P L = {p ∈ Λ * + : |p| ≤ N α+ν } entering in the definitions of the operators B, A and D defined in (2.17), (2.18) and (3.1) respectively).
First of all, we observe that, from Prop. 3.3 (choosing ξ L = e Bτ Ω, with B τ defined as in (4.3)) and Prop. 4.3, the ground state energy E N satisfies for any δ > 0, if N is large enough. Recall here the definition (4.8) of E Bog N . Next, we prove lower bounds for the ground state energy and for the excited eigenvalues of H N below the threshold E N + ζ. For k ∈ N, let λ k be the k-th eigenvalue of H N − E Bog N and μ k the k-th eigenvalue of the quadratic operator with P γ = {p ∈ Λ * + : |p| ≤ N γ/2 }, as defined in Prop. 4.4 (note that eigenvalues are counted with multiplicity). We claim that Vol. 24 (2023) The with n p ∈ N for all p ∈ Λ * + and n p = 0 for finitely many p ∈ Λ * + only (to stay below the threshold ζ > 0, we cannot excite modes with |p| > N γ/2 ).
To prove (5.3), we apply a localization argument similar to those recently used in [22,33] for all N ∈ N sufficiently large (here, we used that 2πN ≥ E Bog N ). We will make use of the following lemma, which is proven in App. A.
for all α > 1, M ∈ N and N ∈ N large enough.
Let now Y ⊂ F ≤N + denote the subspace spanned by the eigenvectors of G N − E Bog N associated with its first (k + 1) eigenvalues λ 0 ≤ λ 1 ≤ · · · ≤ λ k . Since, by assumption, λ k ≤ ζ + 1, we find Y ⊂ P ζ+1 (F ≤N + ). We have where we introduced the notation N + := e −Bυ e −A N + e A e Bυ and

Now, with Lemma 2.3 and Eq.(4.20) we have
Hence, for any normalized ξ ∈ Y ⊂ P ζ+1 (F ≤N + ), we find, with (5.5), We conclude that Next we observe that, for any normalized ξ ∈ Y ⊂ P ζ+1 (F ≤N + ), we have (again, with (5.5)) This immediately implies that the linear subspace X = f M ( N + )e −Bυ e −A Y ⊂ F ≤N + has dimension (k + 1) (like Y ) and that Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2917 Thus, by the min-max principle for the eigenvalues of D γ , (5.12) Choosing M = N 3/4+1/20 and γ = 1/4 − 3/20, we obtain (using that (5.12) in particular implies that μ k ≤ Cλ k ≤ C(ζ + 1)) for any δ > 0. Finally, we show upper bounds for all the excited eigenvalues with λ k ≤ ζ + 1 (we already proved an upper bound for the ground state energy, with k = 0, at the beginning of this section). We are going to use trial states given by eigenvectors of the operator D γ , defined in (5.2). Fix k ∈ N\{0}, with λ k < ζ. For j = 1, . . . , k, the j-th eigenvalue μ j of D has the form with ε p = |p| 4 + 8πp 2 and n (j) p ∈ N, for all p ∈ P L (since we consider eigenvalues below a fixed ζ > 0, there is no contribution from the second sum in (5.2), running over p ∈ Λ * + \P γ , and there are only finitely many p ∈ P L with n (j) p = 0). The eigenvector associated with μ j has the form for an appropriate normalization constant C j > 0 (if the eigenvalue has multiplicity larger than one, eigenvectors are not uniquely defined, but they can always be chosen in this form). We denote by span(ξ 1 , . . . , ξ k ) the linear space spanned by the eigenvectors defined in (5.14). From the min-max principle, we have Since a p e Bτ ξ = 0 for all p ∈ P c L and all ξ ∈ span(ξ 1 , . . . , ξ k ), we can apply Prop. 3.3 with κ = 10 (so that κ > 4(α + ν − 1/2)) to conclude that (recall for α > 1, and N ∈ N sufficiently large. with K and V N defined in (2.5), and where for any α > 1 and ξ ∈ F ≤N + . With the scattering equation (2.15), we rewrite On the other hand, using that p∈Λ * + | V (p/e N )|/|p| 2 ≤ CN and the bound |η 0 | ≤ CN 2 α (see Lemma 2.2), we have Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2921 Finally, by the definition (2.12) of η p and using again |η 0 | ≤ CN −2α (since we need to add the zero momentum mode), we rewrite where ±E 3 ≤ CN 1−2α . Using again (2.15), the fourth line of (A.4) reads: We focus on the last two terms on the r.h.s. of (A.6). With p∈Λ * The second term on the right hand side of (A.6) can be bounded in position space: The term in parenthesis can be bounded as (see [16,Eq. (80)] for details) for any q > 2 and 1 < q < 2 with 1/q + 1/q = 1. Choosing q = log N , we get Combining the previous bounds with (A.6) and using the definition (2.13) we obtain: Here, we take advantage of the analysis in [16,Sec. 6] where properties of e −A (O N + Z N + C N + K + V N )e A were established, with Vol. 24 (2023) The Excitation Spectrum of Two-Dimensional 2923 In fact, since the operators O N and O N only differs for some constant terms and for the fact that ω N (0) in O N is replaced by N V (·/e N ) * f N, (0) in O N , one can easily check that the analysis of [16,Sec. 6] also apply here. One conclude (see [16,Sec. 6.6]): Finally, we show Lemma 5.1, which is used in Sect. 5 to localize in the number of excitations and to prove lower bounds on the spectrum of the excitation Hamiltonian.
Proof of Lemma 5.1. For simplicity, we omit the argument of the functions f M (N + ) and g M (N + ). From a direct computation, we find N ξ (N + + 1) 1/2 ξ for any α > 1. The point here is that the proof of (A.3) is based on an expansion of E G in a sum of terms given by products of creation and annihilation operators, whose commutator with N + has exactly the same form, up to a constant (given by the difference between the number of creation and annihilation operators in the term). Thus, each contribution to the commutator can be estimated as the corresponding term in the expansion for E G (the only difference is that terms where the number of creation operators match the number of annihilation operators do not contribute to the commutator).
As for the quadratic off-diagonal term appearing in (A.9) we have This implies (5.7).

B. Properties of the Scattering Function
For a potential V with finite range R 0 > 0 and scattering length a, and for a fixed R > R 0 , we establish properties of the ground state f R of the Neumann problem on the ball |x| ≤ R, normalized so that f R (x) = 1 for |x| = R. Lemma 2.1, parts (i)-(iv) follows by setting R = e N N −α 0 in the following lemma.
Lemma B.1. Let V ∈ L 3 (R 2 ) be non-negative, compactly supported and spherically symmetric, and denote its scattering length by a. Fix R > 0 sufficiently Vol. 24 (2023) The Passing to polar coordinates, and using that Δf R (x) = |x| −1 ∂ r |x|∂ r f R (x), we find that the first term vanishes. Hence