SEMICLASSICAL SZEGÖ LIMIT OF EIGENVALUE CLUSTERS FOR THE HYDROGEN ATOM ZEEMAN HAMILTONIAN

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1 SEMICLASSICAL SZEGÖ LIMIT OF EIGENVALUE CLUSTERS FOR THE HYDROGEN ATOM ZEEMAN HAMILTONIAN MISAEL AVENDAÑO-CAMACHO, PETER D. HISLOP, AND CARLOS VILLEGAS-BLAS Abstract. We prove a limiting eigenvalue distribution theorem LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian H V h, B) = /) ıh Ah)) x, defined on L R 3 ), in a constant magnetic field Bh) = Ah) = 0, 0, ɛh)b) in the weak field limit ɛh) 0 as h 0. We consider the Planck s parameter h taking values along the sequence h = /N + ), with N = 0,,,..., and N. We prove a semiclassical N LEDT of the Szegö-type for the scaled eigenvalue shifts and obtain both i) an expression involving the regularized classical Kepler orbits with energy E = / and ii) a weak limit measure that involves the component l 3 of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szegö-type for eigenvalue clusters for bounded perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this work is that the perturbation involves the unbounded, first-order, partial differential operator wh, B) = ɛh)b) x 8 + x ) ɛh)b hl 3, where the operator hl 3 is the third component of the usual angular momentum operator and is the quantization of l 3. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states Ψα,N, and their derivatives L 3h)Ψα,N, in the large quantum number regime N. Contents. Introduction: Limiting eigenvalue distribution theorems... Contents. 6. The basic operators and scaling Spectral analysis of the Zeeman hydrogen atom Hamiltonian and eigenvalue clusters The spectrum Norm resolvent estimates and the key lemma Eigenvalue stability A semiclassical trace identity for Zeeman eigenvalue clusters A trace estimate for the Zeeman perturbation of the hydrogen atom The angular momentum perturbation term Szegö-type theorem for the angular momentum operator Conclusion of the proof of Theorems. and Proof of Theorem.3. 30

2 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS 7. Alternate description of the limit measure in Theorem Eigenvalue approximation Alternate proof of Theorem Appendix : The Kepler problem and the Moser map Appendix : Coherent states for the hydrogen atom. 35 References 38. Introduction: Limiting eigenvalue distribution theorems. The behavior of eigenvalue clusters resulting from the perturbation of highly degenerate eigenvalues of elliptic operators on compact manifolds has been studied by many researchers, notably by V. Guillemin 9, 0], and by A. Weinstein ]. This is the second paper in which we study the behavior of resonance and eigenvalue clusters associated with a fixed eigenvalue of a family of hydrogen atom Hamiltonians, labeled by the Planck s parameter h, under perturbations by external electric and magnetic fields in the weak field limit. In ], we studied the resonance cluster associated with the hydrogen atom Stark Hamiltonian in the small electric field regime and proved a Szegö-type result for the resonance shifts. In this paper, we treat the eigenvalue clusters formed by a magnetic field the Zeeman effect). The behavior of eigenvalue clusters for smooth real-valued perturbations V of the Laplacian on rank one symmetric spaces was studied by V. Guillemin 9, 0]. A. Weinstein ] established a limiting eigenvalue distribution theorem LEDT) for the Laplacian M on a compact Riemannian manifold M all of whose geodesics are closed perturbed by a smooth real-valued potential V. The spectrum of the Laplacian M consists of eigenvalues E N with multiplicity d N that grows polynomially with N. In this case, the semiclassical parameter is the index N of the unperturbed eigenvalue E N. Since V is bounded, the spectrum of M + V consists of eigenvalues that form clusters around the unperturbed eigenvalues. To explain the LEDT introduced by Weinstein ], we denote by E N,j, j =,..., d N, the eigenvalues in the cluster around E N, and by ν N,j = E N,j E N the eigenvalue shifts. The LEDT states that for a continuous, real-valued function ρ : R R we have lim N d N d N j= ρ ν N,j ) = Γ ρ ˆV γ)) dµ Γ γ), ) where Γ denotes the space of oriented geodesics of M. These are the classical orbits of the unperturbed problem describing a particle moving on the manifold M with no potential. The function ˆV : Γ R is the Radon transform of the potential V. Namely, ˆV γ) denotes the average of V along the geodesic γ parameterized with respect to arc-length. The measure dµ Γ is the normalized measure on the space Γ obtained from the restriction to the unit cotangent bundle T M of the Liouville measure associated to the canonical simplectic

3 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 3 form on the symplectic manifold T M. Weinstein actually proves a LEDT for perturbations given by pseudo-differential operators of order zero. Here we only state the result for multiplicative potentials for simplicity. R. Brummelhuis and A. Uribe 4] extended these results to the study of the semiclassical Schrödinger operator H h V ) = h + V on L R n ). The potential V 0 is smooth with V lim inf x V x) > 0. They studied the semiclassical behavior of the eigenvalue cluster near an energy 0 < E < V. They proved an asymptotic expansion of T rρh h V ) / E)h ] as h 0 and related the coefficients to the classical flow for p + V on the energy surface E. For other results on the clustering of eigenvalues for h-pseudodifferential operators with periodic flows see, for example,, 6]. A. Uribe and C. Villegas-Blas 9] extended these results by considering perturbations of the family of hydrogen atom Hamiltonians H V h) = /)h x defined on L R n ), n, by operators of the form ɛh)q h where Q h is a zero-order pseudo-differential operator uniformly bounded in h and ɛh) = h +δ, for δ > 0. Here and in the sequel V denotes the Coulomb potential V = x. The spectrum of H V h) consists of discrete eigenvalues E k h) =, k N, with multiplicity d h k+ n )) k = Ok n ) together with the continuous spectrum 0, ). Here, we denote by N = N {0} the set of non-negative integers and by N the set of natural numbers {,,...}. They considered the Planck s parameter taking values along the following sequence converging to zero: h = /N + n ) with N N. Thus for N given and taking k = N, we have that the number E = E k=n h = /N + n )) = / is an eigenvalue of H V h = /N + n )) with multiplicity d N = ON n ). In this setting, Uribe and Villegas-Blas 9] established a LEDT similar to formula ) but with i) the clusters of eigenvalues around the number E = /, ii) the eigenvalue shifts scaled by ɛh),iii) the right-hand side involving averages of the principal symbol of Q h along the classical orbits of the regularized Kepler problem on the energy surface Σ /) = {x, p) T R 3 {0}) p } x = /, ) and iv) the integration being with respect to the normalized Liouville measure on the energy surface Σ /). The semiclassical limit is achieved by taking N or equivalently h 0. A novelty comes in the work of Uribe and Villegas-Blas 9] from the fact that for a fixed negative energy, there are two types of phase space) classical orbits for the classical Hamiltonian flow with Hamiltonian Gx, p) = p x the Kepler problem). Namely, i) bounded periodic orbits corresponding to nonzero angular momentum, and ii) unbounded collision orbits with zero angular momentum. Uribe and Villegas-Blas used Moser s regularization of collision orbits see Appendix, section 8) so that all the collision orbits on Σ /) can be considered periodic orbits after a time re-parametrization. In this regularization, all orbits on Σ /) correspond to geodesics on the sphere S n through the Moser map M : T R 3 ) T S 3 o) with S 3 o denoting the 3-sphere

4 4 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS with the north pole removed see Appendix, section 8 for a review of M). Those passing through the north pole are the collision orbits. The geodesics on S n are parameterized by the quotient of the subset A = {α C n+ Rα = Iα =, Rα Iα = 0} of the null quadric in C n+ with respect to the circle action see Appendix, section 9). The set A corresponds to the unit cotangent bundle T Sn of S n under the map σ : A T Sn with σα) = Rα, Iα). In this paper, we extend these results to eigenvalue clusters of the hydrogen atom Zeeman Hamiltonian defined on L R 3 ). We prove a LEDT on the semiclassical behavior of the distribution of the eigenvalue shifts. To explain this in more detail, let us consider the same setting as Uribe and Villegas-Blas. We regard E = / as an eigenvalue of the family of hydrogen atom Hamiltonians H V h = /N + )) with multiplicity d N = N + ). We consider the atom in an external, constant magnetic field Bh) = 0, 0, ɛh)b), with the constant B 0 and ɛh) = h K+δ, δ > 0, for some suitably chosen K > 0, see Theorem.. We consider Bh) 0 as h 0. We refer to this as the weak field limit. The resulting Hamiltonian H V h, B) = /) ıh Ah)) x = H V h) + ɛh)b) 8 x + x ) ɛh)b hl 3 3) is called the hydrogen atom Zeeman Hamiltonian. Here, x = x, x, x 3 ) denotes Cartesian coordinates for R 3, = x, x, x 3 ) and the operator hl 3 = ) ıh x x x x is the component of the angular momentum operator hl = x ıh) along the direction of the magnetic field Bh). We are working in the symmetric gauge for the vector potential Ah) = ɛh)b x, x, 0). Although the perturbation ɛh)b) 8 x + x ) ɛh)b hl 3 is not bounded, we still have an eigenvalue stability theorem that follows from the work of J. Avron, I. Herbst, and B. Simon ]. Following ], we show that under the perturbation by the effective magnetic field ɛh)b, the eigenvalue E = / gives rise to a cluster of nearby eigenvalues E N,j h, B) = E N,j /N + ), B), j =,..., d N, with total geometric multiplicity equal to d N see Theorem 3. on eigenvalue stability). We obtain explicit relative bounds on the perturbation that allow an estimate on the size of the cluster. Our main result is the following LEDT for this eigenvalue cluster in the large N limit corresponding to a weak magnetic field: Theorem.. Let B > 0 be fixed, and let ρ be a continuous function on R. Let ɛh) = h 33/+δ, for some δ > 0, and take h = /N + ), with N N. For the eigenvalue cluster {E N,j /N + ), B)}, with j =,,..., d N, near E N /N + )) = /, we have lim N d N d N j= ) EN,j /N + ), B) E N /N + )) ρ ɛ/n + )) = ρ B ) l 3x, p) dµ L x, p), 4) Σ /)

5 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 5 where l 3 x, p) = x p x p is the component of the classical angular momentum vector l = x p along the direction of the magnetic field Bh) on the energy surface Σ /) with collision orbits treated as in 9]. The measure dµ L is the normalized restriction of the Liouville measure to the energy surface Σ /). Here T R 3 {0}) is endowed with its canonical symplectic form. We recall the proof in Appendix, section 8, that l 3 is conserved and bounded on the Kepler orbits on the energy surface Σ /), so there is no problem working with arbitrary continuous functions. Theorem. parallels and extends the result of Uribe and Villegas-Blas 9] on eigenvalue clusters formed by bounded perturbations Q h of the hydrogen atom Hamiltonian. These ideas were applied by two of the authors ] to the hydrogen atom Stark Hamiltonian. The main result of ] is a limiting resonance distribution theorem for resonance clusters associated with a hydrogen atom eigenvalue under the Stark perturbation by an external electric field. Theorem. considers the case of eigenvalue clusters formed when a hydrogen atom is placed in a constant magnetic field. Both of these works have in common an unbounded perturbation. As in ], control of the unbounded Zeeman perturbation is obtained through localization properties of coherent states of the hydrogen atom Hamiltonian. However, since the Zeeman perturbation is a first order differential operator, we have to extend these localization results to the derivatives of the coherent states. This requires an additional analysis see section 5.). We remark that the size of the exponent K in Theorem. is far from optimal. Roughly speaking, if we suppose that the perturbation proportional to x +x ) is bounded, the size of the eigenvalue cluster around the eigenvalue / is N K. For the eigenvalue clusters to be well separated, we need N K N, so K >. But, the perturbation is unbounded, and this forces us to take K much larger in order to control the error in the estimate of the difference of resolvents in Lemma 3. and Theorem 3.. Let dµ be defined as the normalized SO4)-invariant measure on A C n+ defined above. In Proposition 5., section 5, we show that the Liouville measure dµ L on the energy surface Σ /) is the push-forward measure of d µ by the map M σ, where d µ is the restriction of dµ to a subset à of A with µa Ã) = 0, see equation 08). Thus the right hand side of 4) can be written in terms of an integral over A. This allows the following reformulation of Theorem.. Theorem.. Under the same hypothesis as in Theorem., we have lim N d N d N j= ) EN,j /N + ), B) E N /N + )) ρ ɛ/n + )) = ρ B ) l 3α) dµα), 5) where, for all α A, we have l 3 α) = Rα) Iα) Rα) Iα). The function l 3 α) can be thought of as a continuous extension of the assignment α A

6 6 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS x, p) l 3 x, p) thorough the map M σα) which is well defined as long as Rα is not the north pole of S 3. We can think of the right-hand side of 5) as a linear positive functional on C0 R). By a Riesz Representation Theorem, there exists a measure dκ on the real line such that the right-hand side of 5) can be written as the integral of ρ with respect to dκ. The measure dκ can be seen as the push-forward measure of dµα) under the map B l 3 : A R. By using an explicit expression for dµ in terms of coordinates for both the classical angular momentum vector l and the Runge-Lenz vector a, and the relative angle between the position vector x and a/ a see 0]), we can actually provide an explicit expression for dκ. This leads to another formulation of Theorem. : Theorem.3. Under the same hypothesis as in Theorem., we have: d N ) EN,j /N + ), B) E N /N + )) lim ρ N d N ɛ/n + )) j= = ρ B ) u u ) du, 6),] where du denotes the Lebesgue measure on the interval, ]. The variable u can be thought of as the component l 3 of the classical angular momentum vector l. The measure dκ = B B x ) dx is supported on the interval B, B ] for B > 0), where dx is the Lebesgue measure on R. This measure gives us a precise picture of how, for N large, the scaled eigenvalue shifts E N,j /N + ), B) E N /N + )), j =,..., d N, ɛ/n + )) are distributed in the interval B, B ]. The distribution around the origin in the interval B, B ] is determined according to the probability density function P x) = B B x ). Theorems.,. and.3 give a rather complete analytical and geometric description of the limiting eigenvalue distribution for the eigenvalue clusters formed by the Zeeman perturbation of the hydrogen atom Hamiltonian. We remark that Theorem.3 can actually be shown in a different way than using Theorem. and the expression for dµ mentioned above. One can use a suitable eigenvalue approximation for the cluster around E N /N + )) and then evaluate the left hand side of 6) by means of Riemann sums. This is shown in section 7. This procedure, however, completely masks the beautiful geometric foundations of the problem appearing in Theorems. and.... Contents. In section, we scale the hydrogen atom Zeeman Hamiltonian using the dilation group. This establishes a countable family of scaled hydrogen atom Zeeman Hamiltonians S V λ) = S V + W λ). The operator S V is a fixed, h-independent, hydrogen atom Hamiltonian S V = x. The magnetic perturbation is W λ) = λ 8 x + x ) λ L 3, where the effective magnetic field

7 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 7 strength is λh, B) = h 3 ɛh)b, and L 3 = ı x x x x ). In this new framework, we want to establish conditions on the size of ɛh) in order to show the existence of clusters of eigenvalues around E N =, for N sufficiently N+) large, with h = /N + ). In section 3, we provide the description of known results on the spectrum of the operator S V λ), with λ fixed, based on references ] and 8]. Then we mention and prove a key result of Avron, Herbst, and Simon, ] on the norm convergence V S 0 λ) z) V S 0 z) as λ 0 for z 0, ), with S 0 = and S 0λ) = S 0 + W λ), 7) by presenting several important resolvent estimates necessary for our work. Moreover, we study such a rate of convergence with λh, B) = h 3 ɛh)b and h = /N + ) when N and z is in a circle of radius ON 3 ) with center E N. Then we are able to show an eigenvalue stability theorem, Theorem 3., by estimating the difference between corresponding spectral projectors associated to the perturbed and unperturbed Hamiltonians on a small disk around E N. In section 4, we use the stability theorem in order to show that the averages appearing on the left hand side of equation 4) with a factor h included in the denominator due to scaling) can be approximated by the normalized trace of d N ρ Π N B hl ) 3) ΠN with ΠN the projector onto the eigenspace of the unperturbed operator S V with eigenvalue E N. Next, in section 5, we take the semiclassical limit N of this last trace by using the Stone-Weierstrass Theorem, the coherent states for the hydrogen atom introduced in 8], and the stationary phase method in order to estimate the expected value of B m, hl 3) m N, between coherent states. We use decay properties of coherent states shown in 8] but, in addition, we need to estimate decay of their derivatives. Finally, an alternate proof of Theorem.3 is presented in section 7. We include two appendices. The Kepler problem and the Moser map are briefly described in the first appendix in section 8. In the second appendix, section 9, details of the coherent states for the hydrogen atom are presented. Acknowledgment. PDH was partially supported by NSF grants and 0304 during the time this work was done. CV-B was partially supported by the projects PAPIIT-UNAM IN068, PAPIIT-UNAM IN0405 and thanks the members of the Department of Mathematics of the University of Kentucky for their hospitality during a visit. MA-C was supported by a fellowship of DGAPA-UNAM, by project PAPIIT-UNAM IN068, and by CONACYT under the Grants 963, CB-03-0 and 5830, CB The authors want to thank the referees for their suggestions to improve the presentation of the paper.. The basic operators and scaling. The hydrogen atom Hamiltonian H V h) with the semiclassical parameter h acts on the dense domain H R 3 ) in the Hilbert space L R 3 ). The operator is

8 8 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS self-adjoint on this domain and given by H V h) = h x. 8) We denote the Coulomb potential by V x) = / x. The discrete spectrum consists of an infinite family of eigenvalues E k h) E k h) = h, k = 0,,,... 9) k + ) each eigenvalue having multiplicity d k := k + ). The essential spectrum is 0, ). With the choice of h = /N + ) and k = N, we see that E k=n h = /N + )) = / is in the spectra of the countable family of Hamiltonians H V /N + )), N N. The multiplicity of the eigenvalue / is d N = N + ). We next consider a hydrogen atom in a constant magnetic field. We assume, without loss of generality, that the magnetic field has the form Bh) = 0, 0, ɛh)b). We keep B 0 fixed and use it only to control whether the magnetic field is on or not. We control the strength of the magnetic field by taking ɛh) = h K, with a constant K > 0 chosen below. We choose the gauge such that the vector potential Ah) is given by Ah) = ɛh)b x, x, 0). The unscaled Zeeman hydrogen Hamiltonian is H V h, B) = ıh Ah)) x = H V h) + wh, B). 0) where the unscaled Zeeman perturbation wh, B) is given by wh, B) = ɛh)b) 8 x + x ) ɛh)b hl 3, ) with L 3 = ı x x x x ). To implement scaling of these Hamiltonians, we use the dilation group D α, α > 0. The dilation group is a representation of the multiplicative group R + and has a unitary implementation on L R 3 ) given by D α f)x) = α 3/ fαx), f L R 3 ). ) Using the relation D α L 3 D α = L 3, we scale the Hamiltonian in 0) by α = h : D h H V h, B)D h = h h 3 x + ɛh)b ) ] x + x 8 ) h3 ɛh)b L 3 =: h S V λh, B)). 3) The scaled Zeeman hydrogen Hamiltonian S V λh, B)) is defined via the effective magnetic field λh, B) = h 3 ɛh)b and the operator S V λ) is given

9 by: EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 9 where we write S V the magnetic perturbation is S V λ) = x + λ 8 x + x ) λ L 3 = S V + W λ). 4) x for the scaled hydrogen atom Hamiltonian and W λ) = λ 8 x + x ) λ L 3. 5) The scaled Zeeman perturbation is then given by W λh, B)). Note that we can make the effective magnetic field λh, B) small by taking h 0. Equivalently, we may set h = /N + ) and take N. For B = 0, the eigenvalues of S V 0) are given by E k E k ) = /k+) ), with k N and multiplicity d k = k + ). Since the discrete spectra of the operators H V h, B) and S V λh, B)) are the same up to the factor h, Theorem. will be proved by establishing a LEDT theorem for the family of operators S V λh = /N + ), B)), N N, by studying the eigenvalue distribution in the cluster around E N ) = /N + ) ) and then taking the corresponding limit when N. Since the perturbation W λh, B)) is unbounded, the existence of these clusters of eigenvalues is by no means immediate. A suitable version of a stability theorem due to Avron, Herbst and Simon ], together with an adequate choice of the exponent K > 0 in the definition of ɛh), guarantee that, for N sufficiently large, the eigenvalue cluster around E N is well defined and the total multiplicity of the eigenvalues in the cluster is d N. This is the content of Theorem 3. proved in the next section. 3. Spectral analysis of the Zeeman hydrogen atom Hamiltonian and eigenvalue clusters. The main goal of this section is to show the existence of eigenvalue clusters C N for the operator S V λh, B)) around the unperturbed eigenvalues E N = /N +), taking h = /N +) with N sufficiently large. We will show that there exist circles Γ N with centers E N and radii r N N 3 such that the total number of eigenvalues of S V λ/n + ), B)) inside Γ N, including multiplicity, is equal to the multiplicity d N = N +) of the eigenvalue E N = /N +) of S V. This fact is a consequence of the main technical result of this section showing that the norm of the difference of the spectral projectors P N and Π N associated to the spectrum of the operators S V λh, B)) and S V, respectively, inside Γ N is ON σ ), σ > 0, and therefore smaller than one for N sufficiently large. This will give us the eigenvalue stability that we need in order to have well-defined clusters of eigenvalues. In subsection 3. we describe spectral properties of S V λh, B)) by summarizing some of the results of Avron, Herbst, and Simon in their papers, ]. As we are only concerned with the Coulomb potential, we state their results for this case.

10 0 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS The eigenvalue stability P N Π N 0 as N would be immediate if we had norm resolvent convergence of S V λ/+n), B)) to S V when N. However, this is not the case as it was shown in ]. Avron, Herbst, and Simon ] showed that we still can have eigenvalue stability due to the fact that for z 0, ) we have the norm convergence V S 0 λ) z) V S 0 z) as λ 0 see Lemma 3.) with S 0 and S 0 λ) given in Eq. 7). In subsection 3., we describe the work of Avron, Herbst and Simon about this point by refining some of their estimates in order to make the dependance on λh, B) explicit. We prove eigenvalue stability in subsection 3.3 by following reference ] and prove both suitable and finer estimates required for our purposes. 3.. The spectrum. The Hamiltonian obtained from the scaled Zeeman hydrogen Hamiltonian 4) by setting the Coulomb potential equal to zero is denoted by S 0 λh, B)) with S 0 λ) given by Eq.7) and λh, B) = h 3 ɛh)b. For λ > 0, the spectrum of S 0 λ) is purely absolutely continuous and equal to the half line λ/, ). We note that the operator S 0 λ) may be represented as a tensor product on the space L R 3 ) = L R x,x ) L R x3 ). For this purpose, we recall the two-dimensional Landau Hamiltonian S L λ) = x,x + λ 8 x +x ) λ L 3. This operator has pure point spectrum E n λ) = λ n+) with n N. Each Landau level E n λ) is an eigenvalue of infinite multiplicity. Then, the Hamiltonian S 0 λ) may be written as S 0 λ) = S L λ) I +I d /dx 3 ), where I j is the identity operator on L R j ), j =,, respectively. The Landau levels appear as thresholds of the operator S 0 λ). The spectrum of S 0 λ) can then be computed using a well-known result on the spectra of tensor products 7, section XIII.9, Theorem XIII.35]. It follows directly that σs 0 λ) = {E E n λ)+0, ) n N } = λ/, ), since inf σs L λ)) = λ/ and the spectrum of d /dx ) 3 is the closed half-line 0, ). We now consider S V λh, B)) defined via the operator S V λ) given in Eq. 4). The operator S V λ) is best understood by studying its restriction to the eigenspaces of L 3. These subspaces are S V λ)-invariant since S V λ) commutes with L 3. The eigenfunctions of the azimuthal angular momentum operator L 3, as an operator on the circle, are ϕ m φ) = e imφ, m Z. We write H m, m Z, for the subspace of L R 3 ) consisting of functions whose angular momentum decomposition contain only ϕ m φ). We then have the direct sum decomposition L R 3 ) = m Z H m. The restriction S m) λ) S V λ) H m of S V λ) to infinitedimensional subspaces H m, m Z, of constant azimuthal angular momentum m Z, has the form S m) λ λ) = / x + 8 x + x ) λ m). 6) H m We let E m) λ) inf σs m) λ)). This number is a simple isolated eigenvalue of S m) λ). We refer to E m) λ) as the ground state of S m) λ). For negative indices m < 0, these eigenvalues satisfy the relation E m) λ) = E m) λ) + mλ, for m 0. 7) Each operator S m) λ) has discrete spectra consisting of simple eigenvalues accumulating at the bottom of the essential spectrum see 8], page 5, Main

11 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN Results, part b). The essential spectrum of S m) λ) consists of half-lines m < 0 σ ess S m) λ)) = m + ) λ, ), m 0 σ ess S m) λ)) = λ, ). 8) The spectrum of S V λ) is the union of the spectra of S m) λ), for m Z. It follows from 8) that λ = inf σ esss V λ)). The ground state eigenvalues of the operators S m) λ), for m 0, are strictly ordered: E 0) λ) < E ) λ) < E ) λ) <... λ = inf σ esss V λ)). 9) Because of this ordering 9) for m 0 and the relation 7), the ground state of S V λ) is E 0) λ) = inf σs V λ)). It is an isolated eigenvalue satisfying E 0) λ) = / + Oλ). From 9), the discrete spectrum of S V λ) consists of infinitely-many discrete eigenvalues {E m) λ) m 0} {E m) λ) E m) λ) < m) λ, m > 0}, less than λ, accumulating at λ = inf σ esss V λ)). There are infinitely-many embedded eigenvalues of finite multiplicity in the essential spectrum λ, ) since, by 7), for m > 0 large enough, E m) λ) >> λ. 3.. Norm resolvent estimates and the key lemma. We present a refined version of Lemma 6.6 of Avron, Herbst, and Simon ] on the norm convergence V S 0 λ) z) V S 0 z) as λ 0 for z 0, ) that gives the rate of the convergence. We will specialize to the case of the Coulomb potential V x) = / x and obtain finer estimates when z is close to an eigenvalue E N of the hydrogen atom Hamiltonian S V. We denote the resolvent of S 0 by R 0 z) = S 0 z), of S V λ) by R V,λ z) S V λ) z), so that for V = 0, we have R 0,λ z) = S 0 λ) z). The spectra of S 0 and S 0 λ) lie in the positive half-line, so both resolvents R 0 z) and R 0,λ z) exist as bounded operators for z 0, ). We have the basic bounds of their norms: R 0 z) distz, 0, ))], R 0,λ z) distz, λ/, ))]. 0) Avron, Herbst, and Simon, Lemma 6.4] proved that for z 0, ), R 0,λ z) converges strongly to R 0 z) ar λ 0. Moreover, in, Theorem 6.3], they showed that S V λ) = S 0 λ) + V does not converge in the norm resolvent sense to S V = S 0 + V as λ 0, which includes the fact that R 0,λ z) does not converge to R 0 z) in norm as λ 0. However, they show, Lemma 6.6] the norm convergence V S 0 λ) z) V S 0 z) as λ 0 for z 0, ), which plays the key role in the proof of eigenvalue stability. We prove this last result in Lemma 3. and obtain an estimate on the rate of convergence necessary in the proof of the eigenvalue stability theorem, Theorem 3.. In order to prove this last result, we introduce the cut-off function χ R as the characteristic function of the unit B R 0) of radius R > 0 centered at the origin. In the sequel, the symbol C will denote a constant whose value may differ from line-to-line but is independent of N. The first part of the following lemma is effectively, Lemma 6.6] and the second part gives the rate of convergence. The following notation will be used in the sequel: a bounded operator whose norm is ON α ), for some α R, will be denoted by ON α ) as well.

12 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS Lemma 3.. Key Lemma) Consider z 0, ). ) We have the following convergence in norm: V S 0 λ) z) V S 0 z), ) as λ 0. ) Consider λ = λh) with h = /N + ) and ɛh) = h q, q > 3/. For z E N = ON 3 ) we have V S 0 λh)) z) V S 0 z) q 3 = O N 5 ) ), ) as N. Proof.. For any fixed R > 0, we decompose V = / x as V = V + V, with V = V χ R and V = V χ R ), so that V has compact support and V is bounded. We choose R > 0 below. The contribution of V to ) is easy to treat. For z 0, ), both S 0 λ) z) and S 0 z) are bounded by /dz) with dz) distz, 0, )). Thus the contribution to ) from V is bounded by V S0 λ) z) S 0 z) ] V dz) = Rdz). 3) We note that the contribution 3) vanishes as R for dz) fixed.. As for the contribution of V to ), we write the difference of the resolvents using 7) as V S0 λ) z) S 0 z) ] = λv S 0 z) Λ ˆp λλ) S 0 λ) z) + λ V S 0 z) Λ S 0 λ) z) 4) where Λ = x, x, 0) and ˆp = ˆp, ˆp, ˆp 3 ) = ı. Since the spectrum of S 0 λ) = ˆp λλ) lies in the interval 0, ), the operators ˆp λλ) S 0 λ) z) and S 0 λ) z) are uniformly bounded for λ 0 and z 0, ): ˆp λλ) S 0 λ) z) max{s/ s / z s 0, )} = / Gz), 5) S 0 λ) z) /dz) 6) with Gz) z Rz). 3. We show that V S 0 z) Λ and V S 0 z) Λ are bounded operators in order to prove that the norm of the left hand side of 4) goes to zero as λ 0 with R fixed. We also estimate the rate of convergence. We use the following estimate 5] that is a consequence of the bound Ψ C Ψ H, for all Ψ H R 3 ). For β > 0, there exist a constant C such that for Ψ H R 3 ) we have ] Ψ C β / Ψ + β 3/ Ψ. 7) We use this inequality together with the simple bound fψ f Ψ, 8) for f L R 3 ) and Ψ H R 3 ). Since V Λ and V Λ are in L R 3 ), estimate 7) with Ψ = S 0 z) φ, φ L R 3 ), suggests to write both operators

13 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 3 V S 0 z) Λ and V S 0 z) Λ with the resolvent S 0 z) shifted to the right side by using commutator properties. Thus we consider the following expressions: V S 0 z) Λ = V ΛS 0 z) + ı V S 0 z) ˆp, ˆp, 0)S 0 z) 9) V S 0 z) Λ = V Λ S 0 z) + ı V x S 0 z) ˆp S 0 z) + x S 0 z) ˆp S 0 z) ] V S 0 z) ˆp + ˆp ) S0 z) + V S 0 z) 30) 4. We next estimate the right sides of 9) and 30). We first note that for q = 0,,, there exist a constant C such that, for j =,, 3, V Λ q j CR q+)/, with Λ j denoting the j th -component of the operator Λ. Thus, from estimates 7)-8), we have for β > 0 V Λ q j S 0 z) CR q+) ] β / S 0 z) + β 3/ S 0 z), j =,, 3 and q = 0,,. 3) Since S 0 z) max{s/ s z s 0, )} ηz), with ηz) =, if Rz) 0 and ηz) = z / Iz) if Rz) > 0, we obtain from 5)-6) with λ = 0) together with 3): V S 0 z) Λ C ] R 3/ + R/ Gz) ηz) β3/ + β/ dz) ], 3) ] V S 0 z) Λ C R 5/ + R3/ + R/ Gz) Gz) + R/ dz) ] ηz) β3/ +. 33) β/ dz) Therefore, we get from 4)-6), and 3)-33) that for β > 0 : ) V S0 λ) z) S 0 z) ] C λ R 3/ + R/ Gz) +λ ) R 5/ + R3/ + R/ Gz) Gz) + R/ dz) ] dz) Gz) ] ηz) β3/ +. 34) β/ dz) We conclude the proof of part ) of Lemma 3. from 34) and 3) by first taking the limit λ 0 with R fixed and then letting R. 5. For the proof of part ), let us consider λ = λh = /N + )), with N N, in 34) and 3). Let us take R = N γ /dz), γ > 0, in equation 3) in order to have V S0 λh)) z) S 0 z) ] = ON γ ). 35)

14 4 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS We will choose γ below. Since z E N = ON 3 ) then /dz) = z = ON ), and Rz) < 0 for N sufficiently large, which implies ηz) =. Whence R = ON +γ ). Thus we can estimate the two terms appearing in the first factor in square brackets in 34) by using both estimates λ = h 3 ɛh)b = ON 3 q ) and /Gz) / z : ) λ R 3/ + R/ ) = O N q+ 3γ, 36) Gz) Gz) ) λ R 5/ + R3/ + R/ Gz) Gz) + R/ dz) dz) ) = O N q+ 5γ. 37) ] Now we replace the factor ηz) + β3/ β / dz) appearing in 34) by its minimum value Cdz)) /4 = ON / ). Thus we have V S 0 λ) z) S 0 z) ] = O N 3 q+ 3γ ) + ON 3 q+ 5γ ) + ON γ ). 38) 6. If we take q > 3/ given then there exist 0 < γ < 3q such that both exponents 3 q+ 5γ and 3 q+ 3γ are negative. Moreover, since γ < 3 q < q then 3 q + 5γ < 3 q + 3γ, which implies that in the regime q > 3/ and 0 < γ < 3q the contribution from the linear term in λ dominates the quadratic one in equation 34). From equations 38) and 35) V S 0 λh)) z) S 0 z) ] ) = O N 3/ q+ 3γ + O N γ) 39) Let us write the right hand side of equation 39) as ON Eqγ) ) with E q γ) = min{q 3/ 3γ, γ}. Working in the regime specified above, we actually have that the maximum value of E q γ) is E q 5 q 3 5 ) = 5 q 3 5. Hence we finally have V S 0 λh)) z) S 0 z) ] ) = O N q ) This completes the proof of part ). We prove some resolvent estimates that are needed in the proof of the stability theorem in the next section. For an eigenvalue E N of S V, let Γ N be a circle with center E N and radius r N = cn 3, with c a suitable constant independent of N in such a way that r N is smaller than half the distance to the nearest eigenvalue, which is ON 3 ). Lemma 3.. Uniformly for all z Γ N, we have ) VS 0 z) = ON) ) VS V z) = ON 3 ) Proof.. for R > 0, we decompose V = V χ R + V χ R ) = V + V, as in the proof of Lemma 3., with V L R 3 ), V L R 3 ), V = πr / and

15 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 5 V = /R. Using estimate 7), we have for all β > 0 V S 0 z) C V β / ] C V z + β / + C V β 3/ + V S 0 z) ] CR/ CR / z β / + β / + CR / β 3/ + S 0 z) R CN µ + CN µ z + CN µ β + ] N µ S 0 z), 4) β where we have written R = βn µ, with µ R. Then we optimize the function gβ) = CN µ β + N µ β by its minimium value, with N µ fixed, and use the estimates z = ON ) and S 0 z) = ON ), in order to get the estimate V S 0 z) = ON µ ) + ON µ ), which is optimal when µ =.. As for part ), from the estimate 7) with C > 0 as there, we obtain for all β > 0: V S V z) γ V, β) {C V ) ] } z β / + + β3/ S β/ V z) + V S V z), 4) as long as γv, β) := C V β / upper bound in equation 4), we take R = factor satisfies γv, β /. Thus we have: is strictly positive. In order to optimize the β 8 πc). With this choice, the { V S V z) E + F β + G } 43) β ) with the coefficients E = + z S V z) /, F = S V z) /4 and G = 8 πc) S V z). Since the minimum value of the function gβ) := E +F β + G β on the interval 0, ) is E + G F ) ) /3 + /3 then using the /3 estimates z = ON ) and S V z) = ON 3 ) we conclude the proof Eigenvalue stability. We next prove the main result on eigenvalue stability by following ] adapted to our setting. Let us first recall from Kato 4, chapter VIII, section, part 4] that an isolated eigenvalue E o of a closed operator T o with finite multiplicity N o is stable with respect to a family of closed perturbations {T n n N} if ) There exists an ɛ > 0, so that any z with 0 < z E o < ɛ is not in the spectrum of T n, for all n large depending on z), and for such a z, we have T n z) T o z), n, strongly; ) The total multiplicity of the eigenvalues of T n in a neighborhood of E o given by {z 0 z E o < µ}, with 0 < µ < ɛ, is precisely N o for all n large.

16 6 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS It is proven in 4, chapter VIII, section, part 4, Lemma.4] that if E o is stable with respect to the family T n, and all the operators are self-adjoint, then, in fact, the spectral projectors converge in norm. That is, by part of the definition and 0 < µ < ɛ, the contour Γ Eo,µ = {z z E o = µ} is in the resolvent sets of T n, for all n large. We can then define the projectors P n = πi Γ Eo,µ T n z) dz, 44) and, similarly, we define the projector P o for T o using the same contour Γ Eo,µ. The self-adjointness of T n and T o imply that these are orthogonal projectors. By part, we have P n P o strongly, and by part, dim RanP n = N o, for all n large. Under these conditions, Kato proves that P n P o 0, as n. We are interested in studying a situation that is not exactly the one described in the above definition by Kato but very much in the same spirit. Namely, given N N, let us consider the operator S V λh = /N + ), B)) = S V + W λh = /N + ), B)) see equation 4)). Here we have the fixed N-independent) operator S V = x plus a perturbation W λh = /N + ), B)) = λh=/n+),b)) 8 x + x ) λh=/n+),b) L 3 indexed by N. We want to look at the eigenvalues E N = /N + ) of the hydrogen atom Hamiltonian S V remember that each eigenvalue E N has multiplicity d N = N + ) ). We want to show that, for N sufficiently large, the spectrum of S V λh = /N + ), B)) inside some neighborhood around E N consists of a cluster C N of d N eigenvalues including multiplicity). The size r N of such a neighborhood should decrease with N. Notice that in this situation, both the eigenvalue E N and r N change with N, which is not the case in the definition of stability by Kato where the eigenvalue E o is fixed and the size ɛ of the neighborhood around E o can be kept fixed as well. Since the cluster C N can be thought of as splitting off of the unperturbed eigenvalue E N into several eigenvalues of total multiplicity d N, we will refer to the existence of C N as a stability property. In order to show the existence of the cluster C N, we first regard E N as an element of the discrete spectrum of S V and notice that the distance ρn) between E N and its nearest neighbors is ON 3 ). Thus we want to consider a circle Γ N with center E N and radius r N = cn 3, with c a suitable constant independent of N in such a way that r N is smaller than ρn)/. Then we have the following: Theorem 3. Stability theorem). Given B 0 and suppose that the constant q in part of Lemma 3. satisfies q > 9. The following spectral projectors are well-defined for N sufficiently large: P N = S V λh = /N + ), B)) z) dz, 45) πı Γ N Π N = S V z) dz. 46) πı Γ N Moreover, these projectors are orthogonal and satisfy P N Π N = ON q 33 5 ). 47)

17 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 7 For q > 33/, the difference of the orthogonal projectors P N Π N converges in norm to zero. Consequently, the spectrum of S V λh = /N + ), B)) inside the circle Γ N consist of a cluster C N of eigenvalues with total multiplicity d N provided N is sufficiently large. Proof. We follow Avron, Herbst and Simon ] in obtaining specific upper bounds on the difference P N Π N. For the purpose of the proof of Theorem 3., we will only write λ to actually specify λh = /N + ), B), assuming B > 0.. We first establish the existence of the resolvent operator S V λ) z) for z Γ N. Since both resolvents S V z) and S 0 z) exist for z Γ N, the equality S V z = I + V S 0 z) ] S 0 z) implies that the operator I +V S 0 z) is invertible. Thus for λ small, we expect from Lemma 3. that the operator I +V S 0 λ) z) is invertible as well. This, and the invertibility of S 0 λ) z for z Γ N, imply that S V λ) z is invertible for z Γ N since S V λ) z = S 0 λ) + V z = I + V S 0 λ) z) ] S 0 λ) z).. To establish the invertibility I + V S 0 λ) z), we write { I + V S 0 λ) z) = I + V S 0 λ) z) S 0 z) ] I + V S 0 z) ] } I + V S 0 z) ].48) Because of the estimate in part ) of Lemma 3., we need to estimate I + V S 0 z) ] 49) in order to have S V 0 λ) z) S 0 z) ] I + V S 0 z) ] <, 50) for z Γ N, which would imply the invertibility of I + V S 0 λ) z) for N large. We write I + V S 0 z) ] = S 0 z + V ) S 0 z) ] = S V z V )S V z) = I V S V z). 5) From part ) of Lemma 3., we have I + V S 0 z) ] = ON 3 ). 5) Using Lemma 3. and 5), we obtain V S 0 λ) z) S 0 z) ] I + V S 0 z) ] = ON q 8 5 ). 53) Thus, in order to satisfy condition 50), and then to have the existence of the resolvent S V λ) z) with z E N = r N, we need to take q > 9.

18 8 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS 3. We next apply these estimates to bound the difference of the projectors from above. As the resolvents have been shown to exist on the contour Γ N, we have P N Π N = S V λ) z) S V z) ] dz. 54) πı Γ N We know the operator S V λ) does not converge to S V in the norm resolvent sense as λ 0 ]. The key ideas necessary to obtain norm convergence of the left hand side of 54) consist in i) inserting in the right hand side of equation 54) the integrals πı Γ N S 0 λ) z)) dz and πı Γ N S 0 z)) dz which are zero due to the analyticity of the resolvents S 0 λ) z) and S 0 z) on C\0, ), and then ii) using the convergence of S V λ) z) S 0 λ) z) to S V z) S 0 z) as λ 0. In order to estimate this last convergence, we write S V λ) z) S 0 λ) z) ] S V z) S 0 z) ] = S 0 λ) z) V S V λ) z) S 0 z) V S V z) = S 0 λ) z) V S V λ) z) V S V z) ] S 0 λ) z) V S 0 z) V ] S V z) + S 0 λ) z) V SV λ) z) V S V z) + S 0 λ) z) V S 0 z) V S V z). 55) In light of resolvent estimates 0), and the estimate ), we can control the norm of the difference of the resolvents in 55) provided we can estimate V S V λ) z) V S V z). 56) 4. To obtain an upper bound on the norm in 56), we write V S V λ) z) = V S 0 λ) z) I + V S 0 λ) z) ], 57) V S V z) = V S 0 z) I + V S 0 z) ]. 58) The invertibility of I +V S 0 z), appearing in 58), was established in 5). We prove that I +V S 0 λ) z) in 57) is invertible as follows. We write this factor as I + V S 0 λ) z) ] = I + V S 0 z) ] I + I + V S 0 z) ) { V S 0 λ) z) V S 0 z) }]. 59) It follows from 5) and part ) of Lemma 3. that I + V S 0 z) ) {V S 0 λ) z) V S 0 z) } q 8 = ON 5 ) ), 60)

19 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN 9 so each factor in square brackets on the right side of 59) is invertible. Furthermore, it follows from 59) 60) that I + V S 0 λ) z) ] = ON 3 ). 6) Returning to 57) 58), we have V S V λ) z) V S V z) V S 0 λ) z) V S 0 z) + V S 0 λ) z) ] + V S 0 z) + V S 0 λ) z) ] + V S 0 z) ]. 6) The first term on the right in 6) is bounded from part ) of Lemma 3. and 6). As for the second term on the right, we note that from equations 5)-53) we obtain I + V S 0 λ) z) ] I + V S 0 z) ] ) = O N q ) and V S 0 z) = ON) from part ) of Lemma 3.. Consequently, we have the bound V S V λ) z) V S V z) = ON q 38 5 ). 64) 5. We conclude the convergence of the projectors as follows. From Lemma 3., equation 64), and the norm estimates S 0 λ) z) = ON ) and S V z) = ON 3 ), we finally get the estimate for the difference of the spectral projectors P N Π N S π V λ) z) S 0 λ) z) ] Γ N S V z) S 0 z) ] dz = ON q 33 5 ). 65) Thus taking q > 33/ we have, for N sufficiently large, P N Π N < and then that the dimension of the range of both projectors P N and Π N is the same see reference 4]) which in turn implies the existence of the cluster C N. 4. A semiclassical trace identity for Zeeman eigenvalue clusters. From Theorem 3., we know that for q > 33/ the size of the eigenvalue cluster C N around E N is no larger than r N = ON 3 ). We need to get a better estimate on the size of C N in order to scale the shifts of eigenvalues within C N. Let us first consider the case of the eigenvalue cluster formed by the perturbation S V λ) of S V, where λ 0 is the magnetic field strength independent of any other parameters. In this case, Theorem 5.6 in reference ], when applied to the Coulomb potential, shows that if we take a fixed eigenvalue E M of the scaled hydrogen atom Hamiltonian S V with multiplicity d M = M +), then we have for λ sufficiently small that the operator S V λ) has a cluster of eigenvalues E M,j λ), j =,..., d M around E M inside a small but fixed circle with center E M. The eigenvalues in the cluster can be written in the following way: Let

20 0 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS m = M ),..., M be the eigenvalues of Π M L 3 Π M, where Π M projects onto the eigenspace of S V and eigenvalue E M. Then, for a given j, there exist an index m so that E M,j λ) = E M λm + Oλ ). 66) This indicates that the size of the cluster of eigenvalues around E M is Oλ), with M fixed. In our setting, the parameter N controls both the strength of the effective magnetic field λ = λh, B) = h 3 ɛh)b, h = /N + ), and the radius r N of the circles C N where our clusters of eigenvalues are well defined. Thus neither the center E N nor the radius r N stay fixed as it is the case of the mentioned result of reference ]. So we need to get estimates considering that fact. We begin with norm estimates of the operators Π N L 3 Π N and Π N x + x )Π N : Lemma 4.. Let Π N be the projector to the eigenspace associated to the eigenvalue E N of the scaled hydrogen atom Hamiltonian S V as above. Then for N and k N fixed we have Π N L 3 Π N = ON), 67) Π N x + x ) k Π N = ON 4k ). 68) Remark. The physical intuition for equation 68) comes from the Kepler problem. In that case, the maximum apogee distance r max E) for an orbit in configuration space of negative energy E is / E including collision and noncollision orbits). So, r max E = /) = and r max E = /N )) = N. Thus we expect, semiclassically speaking, that for a Kepler orbit in configuration space, x + x = ON 4 ). This property is implemented via coherent states Ψα,N with N large. Before presenting the proof of Lemma 4., we briefly recall some facts about the coherent states Ψα,N, complete details are presented in Appendix, section 9. The index α C 4 is an element of A = {α C 4 Rα) = Iα) =, Rα) Iα) = 0}. The set A can be thought of as the unit cotangent bundle T S3 of the 3-sphere S 3 with Rα S 3 and Iα an element of the cotangent space to S 3 at the point Rα. The inverse of the Moser map see Appendix, section 8) then relates T S3 o the unit cotangent bundle T S3 minus the north pole) with the energy surface Σ /) of the Kepler problem. The states Ψα,N belong to the range of Π N and provide a resolution of the identity giving an expression for the projector Π N in terms of them, see equation 46) in Appendix, section 9. Proof.. Equation 67) comes from the well known fact that the eigenvalues of L 3 restricted to the range of Π N are N,..., N.. We use the coherent states Ψα,N, for α A in the proof of 68). In 8], it is shown that the dilated coherent state D N+) Ψα,N has a fast decay outside of a ball of radius r 0 >. Specifically, we have the following result shown in 8, 4.9]:

21 EIGENVALUE CLUSTERS FOR THE HYDROGEN ZEEMAN HAMILTONIAN Lemma 4.. Let Ṽ : R3 R be a polynomially bounded continuous function. Then for r 0 > we have Π N D N+) Ṽ D N+) Π N = Π N D N+) Ṽ χ x r 0 D N+) Π N + ON ) 69) where χ x r0 is the characteristic function of the ball x r 0 and the dilation operator D α is defined in ). The proof of equation 68) is then a consequence of Lemma 4. and the following equalities: D N+) x + x ) k D N+) = N + ) 4k x + x ) k Π N x + x ) k Π N = N + ) 4k Π N D N+) x + x ) k D N+) Π N. 70) Thus we finally have Π N D N+) x + x )k D N+) Π N = O) and then the proof of 68). Remark. We can actually say more about the coherent states D N+) Ψα,N in terms of concentration. It can be shown that for α A given, the state D N+) Ψα,N is highly concentrated along the classical orbit in configuration space associated to α by the inverse of the Moser map and the classical flow of the Kepler problem on the energy surface Σ /). See references 8] and 0] for details. Let us denote by ẼN,j, j =,..., d N, the eigenvalues of S V λ) inside the circle Γ N this notion is well defined for N sufficiently large). Now we consider the eigenvalue shifts ν N,j = ẼN,j E N thinking of them as the eigenvalues of the operator P N S V λ) E N )P N. We write P N S V λ) E N )P N = P N S V λ) E N )P N Π N + P N S V λ) E N )P N P N Π N ) 7) which in turn implies P N S V λ) E N )P N I P N Π N )] = P N S V λ) E N )Π N = P N W λ)π N. 7) For q > 33/, we have from Theorem 3. that I P N Π N ) is invertible for N sufficiently large. Moreover I P N Π N )] I = ON σ ) with P N Π N = ON σ ) and σ = q 33 5 > 0. Thus we have P N S V λ) E N )P N = Π N W λ)π N + P N Π N ) W λ)π N +P N W λ)π N {I P N Π N )] I }, = Π N W λ)π N + ON σ )W λ)π N +P N W λ)π N ON σ ). 73)

22 M. AVENDAÑO-CAMACHO, P. D. HISLOP, AND C. VILLEGAS-BLAS From Lemma 4., we have L 3 Π N = Π N L 3 Π N = ON), x + x ) ΠN = x + x ) ) ΠN x + x ) ΠN / = Π N x + x ) ΠN / = ON 4 ) 74) which implies W λ)π N = O h ɛh) ). Hence, we have from equation 73) P N S V λ) E N )P N h = Π NW λ)π N ɛh) h + ON σ ) = Π N B ) ɛh) hl 3 Π N + Oɛh)) + ON σ ) = Π N B ) hl 3 Π N + ON σ ). 75) Since Π N B hl 3) ΠN = O) then P N S V λ) E N )P N h ɛh) = O). Due to the self-adjointness of P N S V λ) E N )P N, then the spectral radius of the operator P N S V λ) E N )P N h ɛh) h ɛh) is O) as well. Thus we see that the size of the eigenvalue shifts ν N,j is Oh ɛh)). Moreover, equation 75) is the basis to establish the following theorem: Theorem 4.. Let h = /N + ) and σ = q 33)/5. For any polynomial Q, we have d N d N j= ) νn,j Q h = Tr Q Π N B ) ) ɛh) d N hl 3 Π N + ON σ ). 76) So for q > 33/, the remainder term in 76) vanishes as N. Proof. For N sufficiently large, there exist a fixed interval A, A] containing both ν N,j h ɛh), j =,..., d N, and the eigenvalues of the operator Π N B hl 3) ΠN. Since a polynomial is a finite linear combination of monomials, then we only need to show equation 76) for a monomial. We consider a monomial of degree k N and write d N d N j= ) k νn,j h = Tr ɛh) d N ) PN S V λ) E N ) P k N. 77) h ɛh) To simplify notation, let A N := Π N B hl 3) ΠN. From equation 75), we have ) PN S V λ) E N )P k N = P N A N ) k + P N ON σ ), h ɛh) = A N ) k + P N Π N ) A N ) k + P N ON σ ). 78) To evaluate the trace of the second term on the right of the last line of 78), consider an orthonormal basis {φ j } d N j= for the range of Π N and extend it to an