## Abstract

Within the Born–Oppenheimer approximation Lieb proved that the number of non-relativistic, spin- particles that can be bound to an atom of nuclear charge *Z* in the presence of an external magnetic field satisfies *N*_{max}<2*Z*+1, provided the magnetic field tends to zero at infinity and the coupling between the magnetic field and the spin is ignored. Assuming that the magnetic field is generic, we prove an upper bound which holds when the spin-field coupling is included; the set of generic magnetic fields contains an open, dense subset of .

## 1. Introduction

Experimental data tell us that the number *N*(*Z*) of electrons that can be bound to an atomic nucleus of charge *Ze* is at most *Z*+1. To give a mathematical proof of this Ionization Conjecture within the Schrödinger Theory has attracted many authors. Sigal (1982) proved that *N*(*Z*)≤const. *Z*, Ruskai (1982) proved that, and Sigal (1984) improved his result to *N*(*Z*)≤*α*(*Z*)*Z* with *α*(*Z*)≤12 and *α*(*Z*)→2 as *Z*→∞. Later, these results were improved in two directions: first, Lieb *et al*. (1988) proved the exact asymptotic formula *N*(*Z*)/*Z*→1 as *Z*→∞, which was further improved by Fefferman & Seco (1990), who showed that *N*(*Z*)=*Z*+*O*(*Z*^{β}) with *β*=47/56; second, thus far being the best result, Lieb (1984) proved the uniform bound *N*(*Z*)<2*Z*+1. Lieb's theorem even covers the presence of external magnetic fields which tend to zero at infinity. Within the Hartree–Fock Theory, Solovej (2003) has recently established the full conjecture.

In this paper, we address the question within the Schrödinger Theory, taking into account the spin-field coupling which Lieb ignored. Thus, we consider a non-relativistic atom (ion) with *N* identical spin- particles (electrons) in the Coulomb field of an infinite massive nucleus of charge *Z* and subject to an external magnetic field . Such an ion is usually modelled by the many-body Pauli operator(1.1)Here, are the positions of the particles, *j*=1, …,*N*, the magnetic vector potential fulfills , and is formally defined by(1.2)with * σ*=(

*σ*

_{1},

*σ*

_{2},

*σ*

_{3}) being the triple of Pauli spin matrices satisfying the anti-commutation relations

*σ*

_{j}

*σ*

_{k}+

*σ*

_{k}

*σ*

_{j}=2

*δ*

_{jk}

*I*. The Hamiltonian operates on the Fermionic (resp., Bosonic) subspace of the Hilbert space . The Fermionic subspace consists of all antisymmetric functions in , whereas the Bosonic subspace consists of all symmetric functions in (not ).

The ground state energy of this system is defined by(1.3)where denotes the spectrum of . Due to gauge invariance, the energy in equation (1.3) depends only on the magnetic field. Throughout we assume that is an eigenvalue of , i.e. the system is stable. Moreover, we assume that is a monotonically decreasing function of *N* for fixed *Z*.

One of the important features in the spectral properties of Pauli operators is the presence of zero modes, eigenfunctions corresponding to a zero eigenvalue; see, for example, the review in Melgaard & Rozenblum (2005). They seem to be rather common in the two-dimensional Euclidean space. It is found, however, that in dimension three zero modes are not a common feature. Balinsky & Evans (2001) showed that in magnetic fields without zero modes form a dense open set and, moreover, for any magnetic field , the set of *t*∈(0,∞) such that the Pauli operator with magnetic field has zero modes, is locally finite. We, henceforth, refer to these as *generic* magnetic fields.

For generic magnetic fields we shall prove an upper bound for the atomic Pauli operator (see theorem 5.1). Our bound improves the further away the generic magnetic field is from fields, which produce zero modes. We follow Lieb's strategy which is based on a clever use of the triangle inequality and the classical Hardy inequality, which appears in the proof of the Benguria–Lieb–Baumgartner (BLB) inequality (3.15) in Lieb (1984). Our work differs from Lieb's due to the coupling of the spin to the magnetic field expressed by the (Zeeman) term in equation (1.2). Ignoring the latter, Lieb's theorem holds for any bounded vector potential which tends to zero at infinity. This leaves out the case of a constant magnetic field which is explained by the fact that the ground state energy is not monotonically decreasing in *N* for such a field; adding a particle at infinity costs at least an energy . By taking into account the Zeeman term, the required monotonicity is ensured.

The two key ingredients in our approach are an equality for the one-particle magnetic Schrödinger operator (see proposition 3.3), which is of interest in itself, and an inequality which estimates the from above by the corresponding one-particle Pauli operator; see theorem 4.2(iii). The latter inequality, obtained by Balinsky *et al*. (2001), holds as long as the magnetic field is generic.

Seiringer (2001) has studied the same problem for the atomic Pauli operator. Since, he is mainly interested in a (strong) constant magnetic field, he cannot argue as we do. On the other hand, for a constant magnetic field his bound enables him to give upper bounds (separately for the Fermionic and Bosonic cases) with higher order terms expressed explicitly in terms of and *Z*.

Lieb (1984) also treated the molecular case. His approach relies on the afore-mentioned BLB inequality. For the Pauli operator one cannot establish such an inequality (or rather *positivity*) and thus, the molecular case is open.

## 2. One-particle Pauli operator

The Pauli operator in equation (1.2) can be expressed as on , where is the coupling between the magnetic field and the spin, and denotes the Schrödinger operator with an external magnetic field, i.e.(2.1)here, is the 2×2 identity matrix. Henceforth, we impose the condition , *k*=1,2,3. We let be the Friedrichs extension of the operator (2.1) on . It is a non-negative, self-adjoint operator in and its form domain is the completion of with respect to the norm given by(2.2)Since, we shall only consider magnetic fields belonging to , we may define as the form sum of and because the latter turns out to be a ‘small’ perturbation of in the following sense.

*Assume* *, k*=1*,*2*,*3*,* *and* . *Then,* *is infinitesimally form-bounded with respect to* .

Let , with and . For some we define if and 0 otherwise, and also . Evidently . If , then the Hölder inequality gives that(2.3)Furthermore, the Sobolev embedding theorem and the diamagnetic inequality (see, for example, Simon (1976)) yield(2.4)where *ν* is the embedding constant. By combining equations (2.3) and (2.4) we infer that provided . Select . Then,Since, , is infinite almost everywhere (a.e.) and, as *R*→∞, Lebesgue's dominated convergence theorem implies that the latter integral tends to zero. Hence, for any given , we can choose such that(2.5)and, since is a form core for both the form of and , we obtain the desired bound equation (2.5) for any . ▪

It follows from the KLMN theorem in Reed & Simon (1975) that the sesquilinear form is symmetric, closable and non-negative in . The corresponding non-negative, self-adjoint operator has form domain . The operators and the form sum(2.6)have no zero modes and thus, have dense domains and range in . Moreover, . Let , respectively , be the completion of with respect to the norm , respectively . Henceforth, we restrict ourselves to . In that case the vector potential can be chosen such that and ; see theorem A.1 in Fröhlich *et al*. (1986). Then, the following continuous embeddings hold:(2.7)Here, is the space with *A*_{k}=0, *k*=1,2,3.

## 3. An equality involving the magnetic Schrödinger operator

The first key ingredient is an equality involving the magnetic Schödinger operator . Let d*μ* be a (positive) measure which obeys . Define the function(3.1)Let be non-negative, with *r*(*x*)=0 if |*x*|≥1, and . Moreover, define *r*_{k}(*x*)=*k*^{3}*r*(*kx*). The finite Borel measure *μ* is, in particular, a distribution of order 0 and, consequently, the mollification of *μ*, i.e. *μ*_{k}=*r*_{k}**μ* is well defined, and *μ*_{k}→*μ* as *k*→∞. The Schwarz inequality and standard properties of the mollification give the following result.

*The function ρ defined in equation (**3.1**) belongs to*.*Define ρ*_{k}*as in equation (**3.1**) with μ replaced by the mollification μ*_{k}*of μ*.*Then*,*ρ*_{k}→*ρ in*as*k*→∞.

We, henceforth, impose the following conditions on the vector potential .

*Let the components of* *satisfy* *and let* .

It is well known that the magnetic Schrödinger operator is essentially self-adjoint on provided assumption 3.2 holds (Leinfelder & Simader (1981)).

Since, the following equality might be useful in other related contexts, we shall prove it in much more generality than we actually need it for our present purpose.

*Let* *be as in equation (**2.1**) with* *obeying* assumption 3.2. *Moreover, let ρ be defined in equation (**3.1**). If* *, ψ*≠0*,* *and, in addition,* *,* *,* *,* *then*(3.2)

We divide the proof into three steps. In the first step we establish the equality for smooth *μ* and bounded *ρ*. In the second step the equality is proved for general, finite *μ* and bounded *ρ*. This is done by an approximation argument using mollifications of *μ*. Finally, in the third step, we extend the equality to general *ρ* by a limiting procedure.

*Step 1*. Suppose that *μ* is a smooth measure and set *ρ*_{C}=*ρ*+*C* for some *C*>0. Then, *ρ*_{C}, . In this step we establish(3.3)for any . Note that and, likewise, all other entries are well defined. Straightforward calculations show thatNow, for the latter we have thatFurther,Hence,which, rewritten, verifies equation (3.3).

*Step 2*. Next *μ* is a general measure satisfying 0<∫d*μ*<∞. Moreover, assume that . Set *ρ*_{C}=*ρ*+*C* for some *C*>0. In this step we establish(3.4)for any . We make an approximation argument because *μ* is no longer smooth. According to lemma 3.1(ii) there exists a sequence {*ρ*_{k}}, , such that *ρ*_{k}→*ρ* in . We may pass to a subsequence, denoted by , such that pointwise a.e. as *l*→∞. In particular, pointwise a.e. as *l*→∞. Since is essentially self-adjoint on and , there exists a sequence {*ψ*_{k}}, with , such that *ψ*_{k}→*ψ*, and also in *L*^{2}. Again we may pass to subsequences, which converges pointwise a.e. The computation in Step 1 shows that(3.5)Since, , the dominated convergence theorem yields that in as *l*→∞. In particular, as *l*→∞. Likewise, let us consider the term on the right-hand side of equation (3.5) separately. The first term on the right-hand side converges to due to Lebesgue's dominated convergence theorem because and pointwise a.e. imply that in . The second term on the right-hand side of equation (3.5) can be rewritten as(3.6)which converges to 〈|(∇*ρ*_{C})/2*ρ*_{C}|^{2}*ψ*, *ψ*〉, because and its derivatives converge pointwise almost everywhere and, by hypothesis, .

*Step 3*. Finally, we prove the identity in its full version. The idea is to let *C*→0^{+} in equation (3.4). Recall that, by hypothesis, . The left-hand side converges because and . Similarly, for the first term on the right-hand side in because and, therefore, . To prove convergence of the second term on the right-hand side, observe that ∇*ρ*_{C}=∇*ρ*,|(∇*ρ*)/2*ρ*_{C}|^{2}≤|(∇*ρ*)/2*ρ*|^{2} and as above . Consequently, convergence follows by Lebesgue's convergence theorem because, by assumption, . ▪

## 4. Generic magnetic fields

Bear in mind the definition (2.6) of acting in . We summarize the following facts from Balinsky & Evans (2001) and Balinsky *et al*. (2001).

The operator can be extended to a unitary operator such that(4.1)By means of *U* one can define the operator . Then *RR** is a compact (Birman–Schwinger type) operator in and, by setting *T*≔1−*RR**, it turns out that if and only if *T*(*U*^{−1}*ψ*)=0; it is understood that . As a consequence, if and only if *Tφ*=0 implies that .

*The set* *of generic magnetic fields is defined by*(4.2)

*If*, then*except for a finite number of values of t in any compact subset of*[0, ∞).*The set**contains an open and dense subset of*.*If**and*

(4.3)*then* *and the inequality* *holds, in the sense of forms, for all* .

For the proofs of theorem 4.2(i) and (ii) we refer to Balinsky & Evans (2001). Since, the inequality in part (iii) plays an decisive role in the proofs of our main theorems, we give its proof here (cf. Balinsky *et al*. (2001)). The quantity can be interpreted as the length from to the nearest magnetic field which produces zero modes.

By hypothesis, dim Ker *T*=0 and, therefore, 1∉*σ*_{pp}(*R***R*), which verifies that is positive. In particular, for any with we have that(4.4)Setting , we infer that *Uφ*=*ψ* and, in view of equation (4.1), . Invoking equation (4.4) and using equation (4.1) we deduce that(4.5)Furthermore, the simple estimate implies thatfor any . The latter, in conjunction with , implies that ‖*R*‖≤1. Hence, ‖*R**‖=‖*R*‖≤1 and, therefore, equation (4.5) reduces towhich verifies the assertion because . ▪

## 5. Maximal ionization for atoms

The Coulomb potential *V*_{C}(*y*)=1/|*y*|, , belonging to , is infinitesimally small with respect to the self-adjoint operator −*Δ* in , which has domain and, moreover, −*Δ* is essentially self-adjoint on . This is Kato's theorem (see, e.g. theorem X.16 in Reed & Simon 1975), which implies that the atomic Schrödinger operator is a self-adjoint operator in (respectively, ) if one takes into account spin) with domain (respectively, ) and it is essentially self-adjoint on (respectively, ). Under the assumption we have shown that is infinitesimally form-bounded with respect to (provided ). Together with the properties of *V*_{C} above, the latter enables us to introduce the atomic Pauli operator (1.1) by means of the KLMN theorem, which generates a self-adjoint operator in with form domain so that is a form core. In particular, this guarantees that the ground state energy of is finite.

*Let* *assumption 3.2* *be satisfied for* *and let* *be a generic magnetic field*. *Suppose,* *moreover,* *that* *is an eigenvalue of the Pauli operator* *and that* *is monotonically decreasing in N for fixed Z*. *Then,* *the number of identical particles,* *N*_{max}*,* *that can be bound to the nucleus with charge Z satisfies*(5.1)

*independent*of

*N*because the ground state energy is monotonically decreasing and superadditive in

*N*for fixed

*N*. Furthermore, the concavity of with respect to

*Z*(as an infimum over linear functions in

*Z*) implies that the limit

*l*↓1 exists and this yields the best estimate in equation (5.1).

By hypothesis, the *N*-particle system has a bound state, i.e. there exists such that for some (normalized) . Moreover, , where denotes the ground state energy of the (*N*−1)-particle system with the *k*th particle removed; the corresponding Pauli operator is denoted by . Similarly, the one-particle Pauli operator associated with the *k*th particle will be denoted by in the sequel. Then,(5.2)where we used the variational principle for and .

For the first term on the right-hand side of equation (5.2), we apply the equality in proposition 3.3 with the particular choice *ρ*(*x*_{k})=|*x*_{k}|^{−1}. This yields(5.3)From theorem 4.2(iii) and the diamagnetic inequality we obtain the (Hardy-type) inequalityfor any . Invoking the latter on the right-hand side of equation (5.3) yields1Returning to equation (5.2) we infer thatDue to the symmetry (of the identical particles, namely electrons) lemma 5.2 (see below) applies and thus we obtain, replacing *x*_{k} by *x*_{i} and summing over *i*,Using the triangle inequality, |*x*_{i}−*x*_{j}|≤|*x*_{i}|+|*x*_{j}|, we get thatThe latter, in conjunction withproves equation (5.1). ▪

*Let* *be a non-negative function such that u*>0 *on a set of positive measure*. *Then,*

Let *v*(*x*, *y*)=(|*x*|+|*y*|)|*x*−*y*|^{−1}−1, then *v*(*x*, *y*)≥0, since |*x*−*y*|≤|*x*|+|*y*| and *v* is zero if and only if |*x*−*y*|=|*x*|+|*y*|, which holds if and only if *y*=−*bx* for a non-negative . Hence, *v*>0 a.e. because has zero measure. This gives , because *u* is positive on a set of non-zero measure. ▪

## Footnotes

↵Noting that the integration with respect to the variables

*x*_{j},*j*≠*k*, can be done after the*x*_{k}integration.- Received December 19, 2004.
- Accepted June 23, 2005.

- © 2005 The Royal Society