Bound States at Threshold Resulting from Coulomb Repulsion

# Bound States at Threshold Resulting from Coulomb Repulsion

Dmitry K. Gridnev FIAS, Ruth-Moufang Strasse 1, D–60438 Frankfurt am Main, Germany [
###### Abstract

The eigenvalue absorption for a many–particle Hamiltonian depending on a parameter is analyzed in the framework of non–relativistic quantum mechanics. The long–range part of pair potentials is assumed to be pure Coulomb and no restriction on the particle statistics is imposed. It is proved that if the lowest dissociation threshold corresponds to the decay into two likewise non–zero charged clusters then the bound state, which approaches the threshold, does not spread and eventually becomes the bound state at threshold. The obtained results have applications in atomic and nuclear physics. In particular, we prove that an atomic ion with the critical charge and electrons has a bound state at threshold given that , whereby the electrons are treated as fermions and the mass of the nucleus is finite.

On leave from: ] Institute of Physics, St. Petersburg State University, Ulyanovskaya 1, 198504 Russia

## I Introduction

In Refs. 1, ; 2, it was proved that a critically bound N–body system, where none of the subsystems has bound states with and particle pairs have no zero energy resonances, has a square integrable state at zero energy. The condition on the absence of 2–body zero energy resonances was shown to be essential in the three–body case 1 (). Here we consider the –particle system, where particles can be charged and apart from short–range pair–interactions may also interact via Coulomb attraction/repulsion. The formation of bound states at threshold in the two–particle case when the particles Coulomb repel each other is well–studied 3 (); gest (). In the three–particle case there is a well–known proof ostenhof () that a two–electron ion with an infinitely heavy nucleus has a bound state at threshold, when the nuclear charge becomes critical.

Our aim here is to investigate the general many–particle case. Here we generalize the result in Ref. ostenhof, to the case of many electron ions with Fermi statistics and finite nuclear mass. In the proofs we shall use the bounds on Green’s functions from Ref. 3, as well as the technique of spreading sequences from Ref. 1, , that is we prove the eigenvalue absorption by demonstrating that the wave functions corresponding to bound states do not spread, c.f. Theorem 1 in Ref. 1, . A different approach based on the calculus of variations was recently developed in Ref. frank, , where the authors give an alternative proof to the result in Ref. ostenhof, . The authors in Ref. frank, indicate that their approach could be generalized to the many–particle case. In the present paper as well as in Refs. ostenhof, ; frank, one uses essentially the same idea, namely, one uses the fact that the weak limit of ground state wave functions is a solution to the Schrödinger equation at the threshold. The hardest part is to prove that the weak limit is not identically zero. Our approach differs from the ones in Refs. ostenhof, ; frank, in that we use the upper bounds on the two–particle Green’s functions3 ().

The paper is organized as follows. In Sec. II we introduce notations, formulate the main theorem and prove a number of technical lemmas. In Sec. III we derive an upper bound on the Green’s function, which is used in Sec. IV for the proof of Theorem 1. In Sec. V we discuss two main applications of Theorem 1 concerning the stability diagram of three Coulomb charges (Theorem 2 in Sec. V.1) and negative atomic ions (Theorem  3 in Sec. V.2). In Appendix A we derive various criteria for non–spreading sequences.

Let us mention physical applications. The effect when a size of a bound system increases near the threshold and by far exceeds the scales set by attractive parts of potentials was discovered in neutron halos, helium dimer, Efimov states, for discussion see Refs.  fedorov, ; zhukov, ; efimov, ; hansen0, . Here we demonstrate that in a many–particle system similarly to the two–body case 3 (); gest () a Coulomb repulsion between possible decay products blocks the spreading of bound states and forces an bound state at threshold. In nuclear physics, this, in particular, explains why contrary to neutron halos no proton halos are found hansen ().

## Ii Formulation of the Main Theorem

We consider the –particle Hamiltonian ()

 H(λ)=H0+V(λ), (1) V(λ):=∑1≤i

where is a parameter, is the kinetic energy operator with the center of mass removed, are particles’ position vectors and denote the particles’ charges depending on . We shall assume that for each given . Here denotes the space of bounded Borel functions vanishing at infinity. We shall also take particle spins into account, though we shall consider only spin–independent Hamiltonians. The Hamiltonian acts in , where the direct sum has summands and denotes the spin of particle . Similar notation for the Hilbert space can be found in Refs. quotesimon, ; quotethaller, . By Kato’s theorem reed (); teschl () is self–adjoint on , where and denotes the corresponding Sobolev space teschl (); liebloss (). A function depends explicitly on the arguments as , where and are the spin variables.

We treat the particles with integer spins as bosons and particles with half–integer spin as fermions. denotes the orthogonal projection operator on the subspace of functions, which are symmetric with respect to the interchange of bosons and antisymmetric with respect to the interchange of fermions. We denote the bottom of the continuous spectrum by

 Ethr(λ):=infσessH(λ)P. (3)

We shall use the function , which determines the asymptotic behavior at infinity

 ηα(r):=χ{r||r|≤1}+χ{r||r|>1}|r|α, (4)

where and always denotes the characteristic function of the set . Note that is continuous and . We make the following assumptions

• is defined for an infinite sequence of parameter values and , where . For all there is such that , where , and . Besides, .

• and , where is such that and , are fixed constants. Additionally, for all .

Let label all the distinct ways ims () of partitioning particles into two non–empty clusters and . We define the Jacobi intercluster coordinates for the clusters as and respectively, where and (the symbol denotes the number of particles in the corresponding cluster). By we denote the full set of intercluster coordinates and we set

 |xa|=#Ca1−1∑i=1|xa,1i|+#Ca2−1∑j=1|xa,2j|. (5)

points from the center of mass of to the center of mass of . The full set of Jacobi coordinates is .

We denote the sum of interaction cross terms between the clusters by

 Ia(λ):=∑i∈Ca1j∈Ca2Vij(λ). (6)

The product of net charges of the clusters is defined as

 Qa(λ):=∑i∈Ca1∑j∈Ca2qi(λ)qj(λ). (7)

The projection operators on the proper symmetry subspace for the particles within clusters and are and respectively. Namely, projects on a subspace of functions, which are antisymmetric with respect to the interchange of fermions in and symmetric with respect to the interchange of bosons in (). Naturally, and . We also define . The Hamiltonian (1) can be decomposed in the following way

 H(λ)=H(a)thr(λ)−ℏ22μaΔRa+Ia(λ), (8)

where is the Hamiltonian of the clusters’ intrinsic motion and denotes the reduced mass derived from clusters’ total masses. From now on without loss of generality we set .

It is convenient to treat the Hilbert space as the tensor product , where the first term in the product corresponds to the space associated with coordinates and spin variables, while the second one refers to the space associated with the coordinate. In such case the operator has the form , where is the restriction of to . The coordinate is unaffected by permutations of particles within the clusters or . Therefore, , where denotes the restriction of to the space associated with coordinates and spin variables.

The set of assumptions is continued as follows.

• For and one has . There is such that the following inequalities hold for

 (9) [Hathr(λ)−Ethr(λ)]Pa≥|Δϵ|Pa(a=N+1,…,2N−1−1). (10)

The requirement R3 says that the bottom of the continuous spectrum of is set by the decomposition into those two clusters that correspond to any of the decompositions . Inequality (9) introduces a gap between the ground state energy of the two clusters and other excited states. For and we define the projection operator acting on

 Pathr(λ)=Pa[Ethr(λ),Ethr(λ)+|Δϵ|], (11)

where are spectral projections of . Note that by R3 the projection operators , have a finite dimensional range.

The last assumption introduces the uniform control over the fall off of clusters’ wave functions

• There are constants such that

 ∥∥eβ|xa|Pathr(λ)∥∥≤A (12)

for and .

Due to R3 there must exist orthonormal for such that

 Pathr(λ)=na(λ)∑i=1Eai(λ)φai(λ)(⋅,φai(λ))for a=1,…,N and λ=λn,λcr, (13)

where . Note that , therefore, is uniformly bounded, c. f. Lemma 1 in Ref. 1, . Applying Lemma 1 below and using R4 we conclude that there exists an integer such that .

###### Lemma 1.

Suppose that the orthonormal set of function is such that and for , where are constants. If then

 N≤Cd(2T)d/2|ln2A|d(2β)dns, (14)

where is the Lieb’s constant in the Cwikel–Lieb–Rosenbljum bound.

###### Proof.

From it follows that

 (ϕi,χ{x||x|≤R}ϕi)≥12, (15)

where we set . Hence,

 (16)

By the min–max principle does not exceed the number of negative energy bound states of the operator in square brackets in (16). This number, in turn, is equal to the number of negative energy bound states of the operator in square brackets considered in times due to the spin degeneracy. Now (14) follows from the Cwikel–Lieb–Rosebljum bound reed (); cwikel (). ∎

Now we can formulate the main theorem.

###### Theorem 1.

Suppose that satisfies and

 Q0:=infa=1,…,Ninfλ=λn,λcrQa(λ)>0. (17)

Then the sequence does not spread and there exists such that , where and .

Let us remark that the term spreading was defined in Ref. 1, for sequences in . We shall say that a sequence spreads if spreads for all possible fixed values of the spin variables. We postpone the proof of Theorem 1 to Sec. IV.

Together with the upper bound on the Green’s function derived in the next section the following lemma is the key ingredient in the proof of Theorem 1.

###### Lemma 2.

There is independent of such that

 ∣∣e−β|xa|ηδ(Ra)[Ia(λ)−Qa(λ)η−1(Ra)]∣∣≤Θa(x) (18)

for defined in R1, defined in R2 and defined in R4.

###### Proof.

The statement of the lemma is based on the following inequality, which can be checked directly. For all

 ∣∣∣χ{s,s′||s−s′|≥1}|s−s′|−η−1(s)∣∣∣≤2η2(s′)η−2(s). (19)

For fixed the term on the lhs of (19) falls off like . We write

 ∣∣Ia(λ)−Qa(λ)η−1(Ra)∣∣≤∑i∈Ca1j∈Ca2~Uij+∑i∈Ca1j∈Ca2q0|ri−rj|χ{x||ri−rj|≤1} +∑i∈Ca1j∈Ca2∣∣qi(λ)qj(λ)∣∣∣∣∣χ{x||ri−rj|≥1}|ri−rj|−η−1(Ra)∣∣∣. (20)

For any cluster decomposition and

 rj−ri=Ra+#Ca1−1∑i=1ca,1ixa,1i+#Ca2−1∑i=1ca,2ixa,2i, (21)

where , are numerical coefficients depending on masses. It is easy to see that the coefficient in front of is always by fixing and taking . Therefore, by (19) we have

 ∣∣∣χ{x||ri−rj|≥1}|ri−rj|−η−1(Ra)∣∣∣≤c0η2(|xa|)η−2(Ra), (22)

where is some constant. Substituting (22) into (20) we conclude that the inequality (18) would be true if we set , where

 Θa1(x):=e−β|xa|ηδ(Ra)∑i∈Ca1j∈Ca2[~Uij+q0|ri−rj|χ{x||ri−rj|≤1}], (23) Θa2(x):=c0N(N−1)q0e−β|xa|η2(|xa|)ηδ−2(Ra). (24)

Using R2 it is easy to see that . Because we have . ∎

## Iii Upper Bound on the Two Particle Green’s Function

Consider the following integral operator on

 Gck(A)=[−Δ+Aη−1(r)+k2]−1, (25)

for , whose integral kernel we denote as (the superscript “c” refers to “Coulomb”). Note that away from if either or , c. f. Lemma 1 in Ref. 3, . The following Lemma uses the upper bound on a two particle Green’s function from Ref. 3, .

###### Lemma 3.

There is such that for all ,

 supk>0∥∥Gck(A)χ{r||r|≤n}∥∥≤b(A)n, (26)

where the norm on the lhs is the operator norm.

###### Proof.

The operator is an integral operator with a positive kernel lpestim () and, hence, it suffices to consider (26) for . For a shorter notation we denote . Obviously

 ∥Gck(A)χn∥≤∥χ4nGck(A)χn∥+∥(1−χ4n)Gck(A)χn∥ ≤∥χ4nGck(A)χ4n∥+∥(1−χ4n)Gck(A)χn∥, (27)

where the last inequality follows from being an integral operator with a positive kernel 3 (); lpestim (). We shall derive the following estimates and for , from which the the statement of the Lemma follows. The first term on the rhs of (27) is the norm of the self–adjoint operator, which can be estimated as follows

 ∥χ4nGck(A)χ4n∥=sup∥f∥=1(χ4nf,Gck(A)χ4nf)≤ (28)

where we have used the inequality for non–negative self–adjoint operators and (see, for example Ref. glimmjaffe, , Proposition A.2.5 on page 131). Thus as claimed.

Let us now consider the second term on the rhs of (27). We shall need the bound on the Green’s function from Ref. 3, . Let denote the integral kernel of the following operator on

 ~Gk(a)=[−Δ+(a24|r|−1+a4|r|−3/2)χ{r||r|≥1}+k2]−1. (29)

Lets us set equal to the positive root of the equation . Then we get

 Aη−1(r)≥(a24|r|−1+a4|r|−3/2)χ{r||r|≥1}, (30)

which means that pointwise for all , see Ref.  3, . The upper bound on from Ref. 3, (Eqs.(42)–(43) and Eqs. (39)–(40) in Ref. 3, ) reads

 (31)

where have to be chosen to satisfy the following inequalities

 ~R0≥1+|r′|, (32) ~a≤a~R3/20(~R0+|r′|)−3/2. (33)

From the inequality (31) we obtain the bound

 ~Gk(a;r,r′)χ{|r′|≤n}≤14π|r−r′|×⎧⎨⎩1for |r−r′|≤2nexp{a2(√2n−√|r−r′|)}for |r−r′|>2n, (34)

where we have set and . It is straightforward to check that this choice of indeed satisfies (32)–(33). Taking into account that we finally get from (34) the required bound

 Gck(A;r,r′)χ{r,r′||r|≥4n,|r′|≤n}≤ea2(√2n−√|r|−n)4π(3n)χ{r,r′||r|≥4n,|r′|≤n}. (35)

Note that the rhs of (35) does not depend on . Using the upper bound (35) and estimating the operator norm through the Hilbert–Schmidt norm we get

 ∥(1−χ4n)Gck(A)χn∥2≤∫|r|≥4ndr∫|r′|≤ndr′|Gck(A;r,r′)|2 ≤n27ea√2n∫∞4ne−a√t−nt2dt. (36)

The integral in (36) can be calculated explicitly and we obtain as claimed. ∎

We shall need the following corollary of Lemma 3

###### Lemma 4.

For fixed the following inequality holds

 supk>0∥∥Gck(A)η−α∥∥<∞. (37)
###### Proof.

For an arbitrary we have

 ∥∥Gck(A)η−αf∥∥=limN→∞∥∥N∑n=1Gck(A)η−α(χn−χn−1)f∥∥ ≤limN→∞N∑n=1∥∥Gck(A)χnη−α(χn−χn−1)2f∥∥, (38)

where we have used and . For the operator norms we have and for . Substituting these into (38) and using Lemma 3 we rewrite (38) as

 ∥Gck(A)η−αf∥≤b(A)limN→∞(∥∥χ1f∥∥+N∑n=2n(n−1)−α∥∥(χn−χn−1)f∥∥). (39)

Now using that and applying the Cauchy-Schwartz inequality we get from Eq. (39)

 ∥Gck(A)η−α∥≤b(A)(1+∞∑n=2n2(n−1)−2α)1/2. (40)

For the series on the rhs of Eq. (40) obviously converge. ∎

## Iv Proof of the Main Theorem

We shall need an analogue of the IMS localization formula, see Ref. ims, . The functions form the partition of unity and are homogeneous of degree zero in the exterior of the unit sphere, i.e. for , (this makes fall off at infinity). Additionally, there exists a constant such that

 suppJa∩{x||x|>1}⊂{x||xi−xj|≥C|x|%for$i∈Ca1,j∈Ca2$}. (41)

The functions of the IMS decomposition can be chosen sigalsays1 (); sigalsays2 () invariant under permutations of particle coordinates both in and in , hence . Note also that for all one has , c. f. Lemma 7.4 in Ref. liebloss, (the proof in Ref. liebloss, easily extends to the case of Sobolev spaces of higher order). The following version of the IMS localization formula can be verified by the direct substitution

 Δ=Δ∑a,bJ2aJ2b=∑a,bJaJbΔJbJa+2∑a|∇Ja|2, (42)

where is the Laplace on . Rescaling (42) we get

 H0=∑a,bJaJbH0JbJa+2∑a3N−3∑s=1ms|∂sJa|2, (43)

where are real coefficients depending on masses and the second term is relatively compact ims (). We introduce

 Ha(λ)=H(λ)−Ia(λ)=H(a)thr(λ)−ΔRa. (44)

From (43) it follows that

 H(λ)=∑aJ2aHa(λ)J2a+∑a≠bJaJbHab(λ)JbJa+K(λ), (45)

where we define

 K(λ):=∑a≠bJ2aJ2bIab(λ)+∑a[J4aIa(λ)+23N−3∑s=1ms|∂sJa|2], (46) Hab(λ):=H0+2∑s=12∑p=1∑i,j∈Cas∩CbpVij(λ), (47) Iab(λ):=H(λ)−Hab(λ). (48)

The Hamiltonian defined for contains intercluster interactions of the following four clusters , , , , while all interaction cross–terms between these four clusters are contained in . (For some partitions it might happen that one of the four clusters is empty). If we define by the projection operator on the proper symmetry subspace for particles within the cluster then by the HVZ theorem reed (); teschl (); beattie ()

 infσ(Hab(λ)P(ab))≥Ethr(λ), (49)

where we define

 P(ab):=P(ab)11P(ab)12P(ab)21P(ab)22. (50)

Note that .

###### Lemma 5.

Suppose that satisfies . If then

 limn→∞∥∥[1−P(a)thr(λn)]J2a(ψn−ϕ0)∥∥=0(a=1,…,N), (51) limn→∞∥∥J2a(ψn−ϕ0)∥∥=0(a=N+1,…,2N−1−1). (52)
###### Proof.

Note that by Lemmas 1, 2(a) in Ref. 1, . For every we have

 (g,[1−P]ϕ0)=([1−P]g,ϕ0) =limn→∞([1−P]g,ψn)=limn→∞(g,[1−P]ψn)=0. (53)

Because in (53) is arbitrary we conclude that . Consequently, .

Following the arguments of the proof of Lemma 10 in Ref. 1, we get

 limn→∞((ψn−ϕ0),[H(λn