Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Helium
aa r X i v : . [ m a t h - ph ] A ug Quantum Systems at The Brink. Existenceand Decay Rates of Bound States atThresholds; Helium.
Dirk Hundertmark, Michal Jex, Markus LangeAugust 15, 2019
Abstract
Existence and decay rates of eigenfunctions for Schrödinger opera-tors provide interesting and important questions in quantum mechan-ics. It is well known that for eigenvalues below the threshold of theessential spectrum eigenvectors exist and decay exponentially. How-ever, the situation at the threshold is much more subtle. In the presentpaper we propose a new method how to address both problems. Weshow how to calculate upper decay rate bounds at the threshold ex-plicitly. As an example of application we show that for helium atomthe decay rate of eigenvalues at the threshold of essential spectrumbehaves as exp (cid:16) − C p | x | ∞ (cid:17) where | x | ∞ = max {| x | , | x |} . Since the early days of quantum mechanics important questions about quan-tum system were in many cases related to the existence and behaviour of itsbound states. These states corresponds to the square integrable eigenstatesof the operator describing the quantum system. In this paper we considerSchrödinger operators of the form Hψ = − ∆ + V ( α ) (1)where − ∆ is a kinetic energy operator and V denotes a potential depend-ing on a parameter α . We are interested in the case when the eigenvalueapproaches the threshold of the essential spectrum. For such a case theeigenfunction can either exist or disappear for the critical coupling. The de-cision which case occurs is governed by the behaviour of the repulsive part1f the potential at infinity. For a fast decaying potential, i.e. decaying fasterthen | x | − the bound state disappears [5]. For long range slowly decayingpotentials, e.g. Coulomb potential, the bound state persists even for the crit-ical coupling [5]. We present the method how to calculate the eigenfunctiondecay rate at the critical coupling.It is well known that the eigenfunction corresponding to the discrete eigen-value λ of the operator defined in (1) decays at least as fast as exp( − ρ ( λ, inf σ ess ( H )) | x | ) where ρ ( λ, inf σ ess ( H )) denotes the distance of the eigenvalue to the thresh-old of the essential spectrum [1]. Unfortunately this type of estimate doesnot provide any information about eigenvectors at the threshold. There areseveral results in the literature dealing with the situation at the threshold.There is a result [5] describing the properties of the Green function for repul-sive potentials with slowly decaying tails which can be used to obtain certaininformation about the eigenstates near the threshold.Using our method proposed in this paper we are able to estimate the decayrate as exp( − F ( x )) where F ( x ) is related to the behaviour of the potential for | x | ≫ as |∇ F | < U . This estimate gives worse upper bound for subcritical cases. However it doesnot require a gap in the spectrum which allows it to be used also in thecritical case.As an example of the application we apply our method to the well-studiedHelium atom. The Hamiltonian for this system can be written as Hψ = − ∆ − ∆ − | x | − | x | + U | x − x | where the first four terms describe two noninteracting electrons around an in-finitely heavy nucleus and the last one describes electron-electron repulsion.The existence of the ground state for the critical coupling does depends onthe statistics imposed on the electrons. For the case of fermions without spinthere is no ground state even for the case U = 1 [6]. For any other statisticsthere is a critical U ≈ . and the ground state exists [3, 11]. Our method is2lso applicable for finite mass nucleus which is addressed in the appendix.Organization of our paper is as follows. We conclude Introduction with a stepby step overview of our method. In Section 2 we examine the decay behaviourof eigenfunctions of one-particle Hamiltonians at the threshold E = 0 . Inparticular we illustrate the power of our method by showing upper boundson the decay behaviour which depend solely on the repulsive term in theHamiltonian. A lower bound is constructed using the well-know comparisontheorem. In Section 3 we consider a Helium like atom and prove the mainresult of our paper. In the Appendix we summarize some technical detailsomitted in the paper and show how to avoid Born-Oppenheimer approxima-tion. Decay rate estimates for eigenfunctions of Hamiltionians usually require agap between the corresponding eigenvalue and the bottom of the essentialspectrum. In cases where the Hamiltonian has a repulsive part in its po-tential this gap is not required for our method to work. More precisely themain new idea of our method is to make use of this repulsive part and itspositive contributions to the overall energy of the system in order to removethe neccesity of a safety distance to the bottom of the essential spectrum.Omitting technical details we now describe our method.
The setup
Let H be a self-adjoint operator and ψ a normalized eigenvector such that Hψ = Eψ , for E ∈ R . We stress that E may be at the threshold of the essential spectrum of H . Our goal
We derive an upper bound for the decay rate of the eigenfunction ψ . (Projecting onto the region of interest)Since we are interested in the falloff behaviour of the eigenvector for large x we introduce a cutoff function χ R which is supported outside some compactregion. Moreover we define a sequence of bounded functions ζ ǫ that aredirectly related to the falloff behaviour. Obviously we have Re h ( χ R ζ ǫ ) ψ, Hψ i = E h ( χ R ζ ǫ ) ψ, ψ i = E k χ R ζ ǫ ψ k . nd step (Identifying the good and the bad part)Applying a variant of IMS formula we obtain h χ R ζ ǫ ψ, Hχ R ζ ǫ ψ i − h ψ, |∇ χ R ζ ǫ | ψ i = E k χ R ζ ǫ ψ k . Due to the cutoff function χ R most of the terms in |∇ χ R ζ ǫ | are compactlysupported. We denote these by G (the good part) and collect all other termsin B (the bad part). (Estimating and rearranging)Estimating G and rearranging the remaining terms we arrive at h χ R ζ ǫ ψ, ( H − E − B ) χ R ζ ǫ ψ i ≤ k Gψ k ≤ K .
Final step (Magic happens)The last step is to show that H − E − B is positive. This implies that ζ ǫ ψ has bounded norm independent on ǫ . In other words ψ decays at least asfast as ζ for x → ∞ where ζ ǫ → ζ for ǫ → pointwise. Note that in thisstep the repulsive part of the potential in H comes in handy if E sits at thethreshold of the essential spetrum of H . As an introductory example we consider one particle moving in an externalpotential. This external potential consists of an attractive and a repulsivepart. More precisely we consider the following Hamiltonian H = − ∆ − V + U (2)where, for all x ∈ R , V ( x ) ≥ and U ( x ) > .We assume that U is infinitesimally bounded with respect to − ∆ . For sim-plicity we also assume that supp V ( x ) = B R (0) := { x ∈ R | k x k ≤ R } forsome R > . However the proof works even for cases where the support isunbounded provides that the repulsion U dominates the attraction V outsidesome bounded region. For the considered case we have σ ess ( H ) = [0 , ∞ ) and H has only non-positive discrete eigenvalues. Using the Agmon method [1]one can easily show that eigenvectors corresponding to negative eigenvaluesdecay exponentially.We are interested in the decay behaviour of eigenfunctions of H correspond-ing to the critical eigenvalue E = 0 . We always assume that there exists4uch an eigenfunction ψ with Hψ = 0 . This purely technical assumptioncan be removed provided that − ∆ − V has discrete eigenvalues and U is nottoo big. The idea how to avoid this requirement is based on Tightness [8].We consider a weakly converging sequence of eigenstates ψ n correspondingto a given eigenvalue as a function of U n . We use that a weakly convergingsequence is in fact strongly converging provided that lim R →∞ lim sup n →∞ Z | x | >R | ψ n ( x ) | d x = 0 , lim L →∞ lim sup n →∞ Z | k | >L | ˆ ψ n ( k ) | d k = 0 , where ˆ ψ is the Fourier transform of ψ . The first condition follows directlyfrom our decay rate bounds. The second one is implied by finiteness of en-ergy and its relation to Sobolev norm. Details of this argument are for theconvenience of the reader given in Appendix A.In the following we show an upper bound on the decay behaviour of such aneigenfunction ψ and then a corresponding lower bound for the case that ψ isa ground state. We show that the decay rate of ψ is directly related to therepulsive potential U . In this subsection we provide an upper bound for the decay rate of ψ .Let Ω := R \ B R (0) and χ Rδ : R + → [0 , be a continuously differentiable,monotonically increasing function such that for < δ < , χ Rδ ( r ) = ( , if r ≤ R − δ , , if r ≥ R .
Remark . Note that supp ∇ χ Rδ ⊆ (cid:0) R − δ, R (cid:1) . Lemma 2.2.
Let H be given as in Eq. (2) and let ψ ∈ L ( R ) be normalizedeigenfunction such that Hψ = 0 . Moreover let ξ ( x ) = χ Rδ ( | x | ) e F ( x ) witharbitrary nonnegative F ∈ C ( R ) . If for all x ∈ Ω |∇ F | < U then D ξψ, (cid:16) U − |∇ F | (cid:17) ξψ E < ∞ . roof. In order to show the result we are going to apply a variant of IMSformula. But first we need to regularize the expression ξ and approximateit by L ∞ functions. We take ξ ǫ = χ Rδ exp (cid:16) F ( x )1+ ǫF ( x ) (cid:17) . It is easy to check that ξ ǫ converges pointwise to ξ for ǫ → . Hence we start by estimating ∇ ξ ǫ . Adirect calculation shows ∇ ξ ǫ = e F ( x )1+ ǫF ( x ) ∇ χ Rδ + χ Rδ e F ( x )1+ ǫF ( x ) (cid:18)
11 + ǫF ( x ) (cid:19) ∇ F ( x ) , and therefore |∇ ξ ǫ | ≤ e F |∇ χ Rδ | + 2 e F χ Rδ |∇ χ Rδ | |∇ F | + | ξ ǫ | |∇ F | . (3)This holds because < ǫF ( x ) ≤ . Note that due to Remark 2.1 e F |∇ χ Rδ | + 2 e F χ Rδ |∇ χ Rδ | |∇ F | ≤ C . (4)Since ψ ∈ L ( R ) satisfies Hψ = 0 we obtain using a version of IMS formula h ξ ǫ ψ, Hψ i = 0 ⇐⇒ h ξ ǫ ψ, Hξ ǫ ψ i − h ψ, |∇ ξ ǫ | ψ i = 0 . By plugging in Eq. (2) and (3) and rearranging we obtain h ξ ǫ ψ, (cid:0) − ∆ − V + U (cid:1) ξ ǫ ψ i≤ h ψ, (cid:0) e F |∇ χ Rδ | + 2 e F χ Rδ |∇ χ Rδ | |∇ F | (cid:1) ψ i + h ξ ǫ ψ, |∇ F | ξ ǫ ψ i Provided that R is big enough we have − V ξ ǫ ψ = 0 . Hence, using Eq. (4) wearrive at D ξ ǫ ψ, (cid:16) U − |∇ F | (cid:17) ξ ǫ ψ E ≤ C k ψ k . This holds for every ǫ and therefore D ξψ, (cid:16) U − |∇ F | (cid:17) ξψ E ≤ C k ψ k . Provided that
U > |∇ F | for some F ∈ C ( R ) we conclude using Lemma 2.2 Z | x |≥ R e F ( x )+ln (cid:0) U −|∇ F | (cid:1) | ψ ( x ) | d x < ∞ . Since ψ ∈ L ( R ) we deduce e F + ln (cid:0) U −|∇ F | (cid:1) ψ ∈ L ( R ) or in other words: Theorem 2.3.
Let H be given as in Eq. (2) and assume that there existsa normalized ψ ∈ L ( R ) such that Hψ = 0 . Then e F ( x )+ ln (cid:0) U −|∇ F | (cid:1) ψ ∈ L ( R ) for any F ∈ C ( R ) that satisfies |∇ F | < U , for all x ∈ Ω . .2 The lower bound To show the lower bound, we assume, in addition to the existence of a nor-malized eigenfunction ψ with Hψ = 0 , that ψ > a.e. in Ω := R \ B R (0) .This especially holds if ψ is a ground state for the critical eigenvalue E = 0 .To obtain a lower bound we apply a version of the comparison lemma [4, 12]. Theorem 2.4 (Comparison Lemma, [7]) . Let Ω be an open subset of R n , let ψ > a.e. in Ω and let ψ , ϕ satisfy ψ, ϕ ∈ C (Ω) ; ψ, ϕ ≥ in Ω and ψ, ϕ → for | x | → ∞ if Ω isunbounded. ϕ ≤ ψ for all x ∈ ∂ Ω . ( − ∆ + W ) ϕ ≤ and ( − ∆ + W ) ψ ≥ in the weak sense in Ω . W > W a.e. in Ω . ∆ ψ , ∆ ϕ ∈ L (Ω) .Then ψ ≥ ϕ in all of Ω . Due to our assumptions we directly get that ψ ∈ C (Ω) and that ψ → for | x | → ∞ . Next we choose ϕ := N e − F such that ϕ ∈ H ( R ) . This imposecertain conditions on the function F especially that lim | x |→∞ F ( x ) = ∞ and lim | x |→∞ F ( x ) / log | x | = ∞ . It is obvious that ϕ → for | x | → ∞ and ϕ ∈ C (Ω) . In addition we can use the parameter N to obtain ϕ ≤ ψ forall x ∈ ∂ Ω = { x ∈ R | | x | = R } since ψ is bounded from below on ∂ Ω .Moreover ∆ ψ , ∆ ϕ ∈ L (Ω) due to our assumptions and the above choice for ϕ . Hence, in order to use the comparison theorem it remains to show that Hϕ ≤ in Ω . Lemma 2.5.
Let H be given as in Eq. (2) and let ϕ := N e − F such that ϕ ∈ H ( R ) . Then if U ≤ |∇ F | − ∆ F , for all x ∈ Ω , then Hϕ ≤ for x ∈ Ω .Proof. By assumption V ( x ) = 0 for all x ∈ Ω , hence in Ω we have Hϕ = ( − ∆ + U ) ϕ . Using that ∇ ϕ = − N e − F ∇ F and ∆ ϕ = N e − F |∇ F | − N e − F ∆ F ,
7e obtain ( − ∆ + U ) ϕ = (cid:0) U + ∆ F − |∇ F | (cid:1) ϕ . Thus we have Hϕ ≤ if − ∆ F + |∇ F | ≥ U .
Therefore by the comparison theorem we have that ψ ≥ ϕ in all of Ω . Henceas a direct consequence we obtain Theorem 2.6.
Let H be given as in Eq. (2) and let ψ ∈ L ( R ) be such that Hψ = 0 and ψ > . Moreover let F be such that e − F ∈ H ( R ) and U ≤ |∇ F | − ∆ F , for all x ∈ Ω . Then there exists an
N > such that N e − F ≤ ψ for all x ∈ Ω . For x ∈ R we consider the Hamiltonian H = − ∆ − χ B R (0) + C | x | where χ B R (0) denotes the characteristic function of an open ball with radius R in R and < C < . As a function for Theorem 2.3 we can use F = K p | x | . Adirect calculation shows that |∇ F | = K √ | x | . This implies that eigenfunctionsat the threshold converge to faster then exp (cid:16) − p C | x | (cid:17) . In a similarfashion we can show using Theorem . that the suitable lower bound forthe ground state eigenfunction is exp (cid:16) − p (4 C + ǫ ) | x | (cid:17) . We remark thatexplicit calculation of the true eigenfunction at the threshold has asymptoticbehaviour in the form ψ ( x ) = N K (2 p c | x | ) p | x | ∼ N r π e − √ c | x | √ c | x | / O p c | x | !! for | x | → ∞ where K is the modified Bessel function of the second kind. In the following we consider a helium-like atom consisting of an infinitelyheavy nucleus at the origin and two distinguishable electrons. We provideupper and lower bound estimates for decay rates of eigenstates at the thresh-old of the essential spectrum. We denote by x i the operator of position forthe two electrons, i ∈ { , } . 8he Hamiltonian of this system is given by H U = p + p − | x | − | x | + U | x − x | (5)where p i = − i ∇ x i is the momentum operator of the i -th electron. It iswell-defined and self-adjoint on D ( H U ) ⊂ L ( R ) .We denote the ground state energy of H U by E U . It is well-known that E U ismonotonically increasing with respect to U . Moreover using classical resultsby Bethe [3], HVZ Theorem [13] and Lieb [ ? ] there exists a critical < U c ≤ such that for U < U c E U < − , and for U ≥ U c inf σ ( H U ) = − . we note that − = inf σ ( p − | x | − ) is the infimium of the energy of thehydrogen atom. Goal:
We are interested in the fall-off properties of the normalized groundstate ψ U of H U for the critical case U = U c .The existence of such a ground state was proved in [10]. Nevertheless we wantto mention that similar to the one particle case we can obtain the existenceof a ground state using tightness arguments [8]. For more details we referthe reader to Appendix A. Before we formulate the main theorem of this section we define the region A δ : | x | ≥ δ | x | ∞ ,where | x | ∞ := max {| x | , | x |} , | x | := min {| x | , | x |} and < δ < . Theorem 3.1 (Fall-off properties of the eigenstate at the threshold) . Let H U be given by Eq. (5) and let ψ U ∈ L ( R ) be such that H U ψ U = − ψ U .Then e F ψ U ∈ L ( R ) , where for > ⋔ > and > K > F := √ π ⋔ δ | x | ∞ , in the interior of the region A δ ,K q U − − δ δ p | x | ∞ , otherwise . (6)9 emark . A direct consequence of this theorem is that any eigenstate sat-isfying the assumptions has to decay at least as fast as e − F . In the appendixwe give a simple way how to show this behaviour pointwise. Remark . The upper bound obtained in this way works for all eigenfunc-tions which are at the threshold. Hence also for the subcritical case, i.e.
U < U c . In the proof of our theorem we apply our method introduced in Subsec-tion 1.1. In order to apply it we prepare several useful estimates regardingthe action of our Hamiltonian. We summarize these estimates in followingtwo Lemmata.
Lemma 3.4. If | x | ≥ δ | x | ∞ we have − | x | − | x | + U | x − x | ≥ (cid:18) U − δδ (cid:19) | x | ∞ , (7) and if | x | ≤ δ | x | ∞ we have − | x | − | x | + U | x − x | ≥ − | x | + 1 | x | ∞ U − − δ δ . (8) Proof.
We begin with the estimate in region A . Assume that | x | = | x | ∞ ,then | x | ≥ δ | x | and 2 | x | ≥ | x | + | x | ≥ | x − x | Hence we obtain − | x | ≥ − δ | x | and 1 | x − x | ≥ | x | and therefore − | x | − | x | + U | x − x | ≥ (cid:16) − δ − U (cid:17) | x | = − δ | x | ∞ . For the case | x | = | x | ∞ we obtain the inequality analogously. Hence we haveshown Inequality (7). Outside of the region A we write (1 + δ ) | x | ∞ ≥ | x − x | i . e . | x − x | ≥ δ ) | x | ∞ and hence we obtain − | x | − | x | + U | x − x | ≥ − | x | − | x | + (1 − δ ) U | x | ∞ , (9)which is Inequality (8). 10ext we specify χ R , ξ ǫ as described in 1.1[Step 1]. We define a continuouslydifferentiable, monotonically decreasing function ϕ ⋔ : R + → [0 , such thatfor < ⋔ < , ϕ ⋔ ( r ) = ( , if r ≤ − ⋔ , , if r ≥ . Note that supp ∇ ϕ ⋔ ⊆ (cid:0) − ⋔ , (cid:1) .Next we define for R > and < ⋔ < the function χ R, ⋔ : R × R → [0 ,
1] ; ( x , x ) (cid:18) − ϕ ⋔ (cid:18) | x | ∞ R (cid:19)(cid:19) , (10)where one directly sees that χ R, ⋔ acts as the identity if R < | x | ∞ . Moreover, supp ∇ χ R, ⋔ ⊆ { ( x , x ) ∈ R | | x | ∞ ∈ ( R − R ⋔ , R ) } . We also need a partitionof unity which will map to the neighborhood of A δ in the following form ς A δ := ( , if x ∈ A δ , , if x / ∈ A δ . and its complement as ς ⊥ A δ := q − ς A δ The construction of these functions is summarized in Appendix B. Further-more, we define for ≤ η ≤ F η : R × R → R + ; ( x , x ) C p | x | ∞ η p | x | ∞ + Dγ A δ | x | ∞ η | x | ∞ , (11)where γ A δ := ( ϕ ⋔ (cid:16) δ | x | ∞ | x | (cid:17) , if x ∈ A δ , , if x / ∈ A δ . Last but not least we specify two multiplication operators. Let δ, ⋔ , η ∈ (0 , and R > , ξ : L ( R ) → L ( R ) ; ψ ς A δ χ R, ⋔ e F η ψ ,ξ ⊥ : L ( R ) → L ( R ) ; ψ ς ⊥ A δ χ R, ⋔ e F η ψ , (12)Using Lemma 3.4 we conclude the following11 emma 3.5. Let H U be given as in Eq. (5) and let ψ ∈ H ( R ) . Then h ξψ, H U ξψ i ≥ D ξψ, " −
14 +
14 + U − δ δ ! | x | ∞ ! ξψ E , h ξ ⊥ ψ, H U ξ ⊥ ψ i ≥ D ξ ⊥ ψ, (cid:20) −
14 + U − − δ (1 + δ ) | x | ∞ (cid:21) ξ ⊥ ψ E . In the proof of our main theorem we use a variant of IMS error for ξ and ξ ⊥ i.e. we need to calculate |∇ ξ | and |∇ ξ ⊥ | respectively. In order to make theexpressions more readable we set I := δ | x | ∞ | x | and I c := | x | δ | x | ∞ . A straightforward calculation yields |∇ ξ | = (cid:12)(cid:12)(cid:12) ς A δ χ R, ⋔ e F η (cid:2) ∇ F η (cid:3) + (cid:2) ∇ ς A δ (cid:3) χ R, ⋔ e F η + ς A δ (cid:2) ∇ χ R, ⋔ (cid:3) e F η (cid:12)(cid:12)(cid:12) = G δ ( ς A δ , F η , χ R, ⋔ ) + (cid:12)(cid:12)(cid:12) ξ (cid:2) ∇ F η (cid:3) + (cid:2) ∇ ς A δ (cid:3) χ R, ⋔ e F η (cid:12)(cid:12)(cid:12) . (13)where ∈ { , ⊥} . By consulting the definitions of the appearing terms oneeasily sees, that all terms in which ∇ χ R, ⋔ appear, are bounded and can beestimated by some constant not depending on | x | ∞ . In Eq. (13) we collectedall these term into G δ ( ς A δ , F η , χ R, ⋔ ) which we call the good part of |∇ ξ | .The remainder is called the bad part. Lemma 3.6.
There exist constants c , c > such that |∇ F η | ≤ c I supp( γ A ) + c p | x | ∞ . (14) provided that | x | ∞ is large enough.Proof. From the definition of F η in Eq. (11) outside of the region A δ , i.e.outside of the support of γ A , we obtain ∇ F η = (cid:18) C (cid:0) p | x | ∞ (cid:1) − η p | x | ∞ − C p | x | ∞ η (cid:0) p | x | ∞ (cid:1) − (cid:0) η p | x | ∞ (cid:1) (cid:19) ∇| x | ∞ = C ∇| x | ∞ p | x | ∞ (1 + η p | x | ∞ ) . Hence, we get |∇ F η | ≤ C p | x | ∞ . c := C . Similarly we obtain from the definition of F η inEq. (11) that ∇ F η = (cid:18) C (cid:0) p | x | ∞ (cid:1) − η p | x | ∞ − C p | x | ∞ η (cid:0) p | x | ∞ (cid:1) − (cid:0) η p | x | ∞ (cid:1) (cid:19) ∇| x | ∞ + (cid:18) D η | x | ∞ − D η | x | ∞ (cid:0) η | x | ∞ (cid:1) (cid:19) γ A ∇| x | ∞ + D | x | ∞ η | x | ∞ ∇ γ A where ∇ γ A = ϕ ′ ⋔ ( I ) (cid:18) ∇| x ∞ || x | − δ | x | ∞ ∇| x | | x | (cid:19) . It is easy to check that in region A δ we have |∇ γ A | ≤ | ϕ ′ ⋔ ( I ) || x | . which means |∇ F η | ≤ C p | x | ∞ + Dγ A + 2 D | ϕ ′ ⋔ ( I ) || x | ∞ | x | . Therefore setting c := D + D k ϕ ′ ⋔ k ∞ δ and c := C completes the proof. For the convenience of the reader we indicate the steps of our method asdescribed in Subsection 1.1 in the course of the proof. Let H U be given byEq. (5) and suppose there exists ψ ∈ L ( R ) such that H U ψ = − ψ . We get H U ψ = − ψ Step 1 ==== ⇒ h ( ξ + ξ ⊥ ) ψ, H U ψ i = − h ( ξ + ξ ⊥ ) ψ, ψ i . Step 2 : Using IMS formula we obtain h ξψ, Hξψ i + h ξ ⊥ ψ, Hξ ⊥ ψ i − h ψ, ( |∇ ξ | + |∇ ξ ⊥ | ) ψ i = −
14 ( k ξψ k + k ξ ⊥ ψ k ) . Step 3 : Rearranging the terms and using Lemma 3.5 we obtain h ψ, |∇ ξ | ψ i + h ψ, |∇ ξ ⊥ | ψ i≥ * ξψ,
14 + U − δ δ ! | x | ∞ ! ξψ + + (cid:28) ξ ⊥ ψ, U − − δ (1 + δ ) | x | ∞ ξ ⊥ ψ (cid:29) . (15)13ow splitting the l.h.s of Eq. (15) into good and bad parts, bringing the badparts to the r.h.s. we arrive at h ψ, ( G δ ( ς A δ , F η , χ R, ⋔ ) + G δ ( ς ⊥ A δ , F η , χ R, ⋔ )) ψ i≥ (cid:28) ψ, (cid:20)(cid:18)
14 + δU − − δ δ | x | ∞ − |∇ F η | (cid:19) ξ + (cid:18) U − − δ (1 + δ ) | x | ∞ − |∇ F η | (cid:19) ξ ⊥ (cid:21) ψ (cid:29) − (cid:10) ψ, (cid:0) |∇ ς A δ | + |∇ ς ⊥ A δ | (cid:1) χ R, ⋔ e F η ψ (cid:11) . Now we evaluate the terms on the right hand side in 3 disjoint regions(1) ς A δ = 1 ,(2) ς ⊥ A δ = 1 and(3) ς A δ ∈ (0 , ∧ ς ⊥ A δ ∈ (0 , We denote A (1) := (cid:18)
14 + δU − − δ δ | x | ∞ − |∇ F η | (cid:19) χ R, ⋔ e F η , A (2) := (cid:18) U − − δ (1 + δ ) | x | ∞ − |∇ F η | (cid:19) χ R, ⋔ e F η , A (3) := (cid:2) A (1) ( ς A δ ) + A (3) ( ς ⊥ A δ ) − (cid:0) |∇ ς A δ | + |∇ ς ⊥ A δ | (cid:1)(cid:3) χ R, ⋔ e F η . and obtain h ψ, ( G δ ( ς A δ , F η , χ R, ⋔ ) + G δ ( ς ⊥ A δ , F η , χ R, ⋔ )) ψ i ≥ X (cid:10) ψ, ( A ( j ) ) ψ (cid:11) ( j ) where h· , ·i ( j ) is restriction of the scalar product to appropriate regions. UsingLemma 3.6 we show that A ( j ) are positive. We estimate A (1) in region (1) as A (1) =
14 + δU − − δ δ | x | ∞ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c I supp( γ A ) + c p | x | ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ ≥ (cid:18)
14 + δU − − δ δ | x | ∞ − c − c | x | ∞ (cid:19) ξ where we assumed that | x | ∞ is sufficiently large. Next we evaluate A (2) inregion (2) A (2) = (cid:18) U − − δ (1 + δ ) | x | ∞ − |∇ F η | (cid:19) ξ ⊥ ≥ (cid:18) U − − δ (1 + δ ) | x | ∞ − c | x | ∞ (cid:19) ξ ⊥ γ A is outside of region (2) . Lastwe estimate A (3) in region (3) . We use |∇ ς A δ | + |∇ ς ⊥ A δ | ≤ L | x | ∞ for given < L < ∞ which is shown in Appendix B. Therefore, for bigenough | x | ∞ we have A (3) = (cid:2) A (1) ( ς A δ ) + A (2) ( ς ⊥ A δ ) − (cid:0) |∇ ς A δ | + |∇ ς ⊥ A δ | (cid:1)(cid:3) χ R, ⋔ e F η ≥ (cid:20) U − − δ (1 + δ ) | x | ∞ − c | x | ∞ − L | x | ∞ (cid:21) χ R, ⋔ e F η where we used that ς A δ + ( ς ⊥ A δ ) = 1 and
14 + δU − − δ δ | x | ∞ − c − c | x | ∞ ≥ U − − δ (1 + δ ) | x | ∞ − c | x | ∞ for large enough | x | ∞ . Now choosing c and c appropriately small, whichcorresponds to C and D small enough in Eq. (11), we obtain that A ( j ) ispositive in region ( j ) . Last step : We now conclude that ξ ψ has a bounded L norm up to loga-rithmic correction in the exponent exp( c log( | x | ∞ )) for big enough R whichcompletes the proof. Finally we provide a lower bound for the ground state in the critical case.Before we state and prove our bound we introduce a following auxiliary func-tion M ( x ) := ( x + m ) m , x ≤ m ( t ( x ) + m ) m , m < x < m (3 m ) m , m ≤ x (16)where m ∈ N and t ( x ) denotes smooth increasing function such that t ( x ) := ( x , ≤ x ≤ m m , m ≤ x (17)for which ≤ t ′ ( x ) ≤ and | t ′′ ( x ) | < (cid:0) π m (cid:1) holds.15 heorem 3.7 (Fall-off properties of the critical ground state) . Let H U begiven by Eq. (5) and let ψ U ∈ L ( R ) be a positive ground state functionsuch that H U ψ U = − ψ U . Furthermore denote | x | = min {| x | , | x |} , | x | ∞ =max {| x | , | x |} . Then there exist suitable constants m, R, N, C > such thatfor every ( x , x ) ∈ R satisfying | x | ∞ > Rψ U ≥ N M m ( | x − x | ) exp (cid:18) − | x | − C | x | ∞ (cid:19) holds.Remark . In other words ground state eigenfunction can not decay fasterthen exponentially. We strongly believe that it is possible to show subex-ponential lower bound in appropriate regions. However, construction of thecomparison function is more elusive because it is required to smoothly andsufficiently slowly connect exponential and subexponential decay regions.Furthermore subexponential regions corresponds to tubular regions where | x | < C . Proof.
We want to use Theorem 2.4. In order to satisfy the assumptions weset the following: • Ω = { ( x , x ) ∈ R (cid:12)(cid:12) | x | ∞ > R } , • W = − | x | − | x | + U | x − x | + , • W = W + ǫ | x − x | , • ψ ( x ) = ψ U ( x ) and • ϕ ( x ) = N M m ( | x − x | ) exp (cid:16) − | x | − C | x | ∞ (cid:17) for all x ∈ Ω .By this choice the assumptions 1), 2), 4) and 5) of Theorem 2.4 are satisfiedfor appropriate choice of the constant N . It remains to check assumption 3).Since Hψ = − ψ holds trivially we obtain ( − ∆ + W ) ψ = 0 for free. Itremains to show ( − ∆ + W ) ϕ ≤ . This is done by a direct calculation.With slight abuse of notation we obtain − ∆ ϕN = " −
14 + 1 | x | − C + 2 C | x | ∞ − m ( m − (cid:18) t ′ ( | x − x | ) | x − x | + m (cid:19) − m | x − x | + m (cid:18) t ′′ ( | x − x | ) + 2 t ′ ( | x − x | ) | x − x | (cid:19) +2 mt ′ ( | x − x | ) | x − x | + m (cid:18) h x , x − x ∞ i| x | | x ∞ − x | + C h x ∞ , x ∞ − x i| x | ∞ | x ∞ − x | (cid:19) ϕ
16e can estimate this from above after adding W ϕ as − ∆ + W N ϕ ≤ " − C + 2 C − | x | ∞ + U + ǫ | x − x | − m ( m − (cid:18) t ′ ( | x − x | ) | x − x | + m (cid:19) + m | x − x | + m (cid:18) (2 C + 1) t ′ ( | x − x | ) − t ′′ ( | x − x | ) − t ′ ( | x − x | ) | x − x | (cid:19) ϕ ≤ (cid:0) − C + C − R + U + ǫ m (cid:1) ϕ, for | x − x | > m (cid:16) − C + C − R + U − | x − x | − m − m + 2 C + 1 (cid:17) ϕ, for | x − x | < m (cid:0) − C + C − R + U + ǫm + C + + π m (cid:1) ϕ, otherwise ≤ where the last inequality holds for C big enough since the critical value of U is smaller than . Acknowledgement
We would like to thank our collegue Ioannis Anapolitanos for fruitful discus-sions and comments regarding the content of this manuscript.
A Tightness argument
In this section we show application of Tightness [8]. In our setting we areinterested in existence of the eigenfunction at the threshold. We consider anSchrödinger operator − ∆ + V ( U ) depending on parameter U with potential V infinitesimally bounded with respect to − ∆ . Our task is to show existenceof a ground state for the situation U → U c when the discrete spectrumdisappears. The tightness argument is based on the following equivalence. Theorem A.1. ([8])
Let ( ψ n ) n ∈ N be a sequence in L ( R d ) . Then the followingare equivalent:1. the sequence ( ψ n ) n ∈ N is converging strongly,2. the sequence ( ψ n ) n ∈ N is converging weakly and satisfies lim R →∞ lim sup n →∞ Z | x | >R | ψ n ( x ) | d x = 0 , (18) lim L →∞ lim sup n →∞ Z | k | >L | ˆ ψ n ( k ) | d k = 0 , (19) where ˆ ψ is the Fourier transform of ψ , . the sequence ( ψ n ) n ∈ N is converging weakly and there exist functions H, F ≥ with lim | x |→∞ H ( x ) = ∞ = lim | k |→∞ F ( k ) such that lim sup n →∞ Z R d H ( x ) | ψ n ( x ) | d x < ∞ , (20) lim sup n →∞ Z R d F ( k ) | ˆ ψ n ( k ) | d k < ∞ , (21) where ˆ ψ is the Fourier transform of ψ .Remark A.2 . In the previous theorem it is also possible to replace conditionEq. (18) by Eq. (20) or Eq. (19) by Eq. (21).
Lemma A.3.
Let H U = − ∆ + V ( U ) be an operator such that V ( U ) isinfinitesimally bounded with respect to − ∆ . Then each eigenstate satisfies Z R d F ( k ) | ˆ ψ ( k ) | d k < ∞ where F ≥ with lim | k |→∞ F ( k ) = ∞ .Proof. Using the assumption there exists a normalized function ψ such that H U ψ = Eψ thus h ψ, − ∆ ψ i = −h ψ, V ( U ) ψ i + E .
Using infinitesimal boundness of V ( U ) we obtain (1 − ǫ ) |h ψ, − ∆ ψ i| ≤ C + | E | for ǫ ∈ (0 , and C < ∞ . This implies |h ψ, − ∆ ψ i| ≤ C + | E | − ǫ . (22)Rewriting l.h.s. of Eq. (22) using Fourier transform we obtain |h ψ, − ∆ ψ i| = Z R d k | ˆ ψ n ( k ) | dk . Hence adding 1 to both sides of Eq. (22) we conclude Z R d ( k + 1) | ˆ ψ n ( k ) | dk ≤ C + E − ǫ + 1 < ∞ which completes the proof. 18ow we are prepared to prove existence of eigenvector at the threshold foroperators defined by Eq. (2). Lemma A.4.
Let H U be defined by Eq. (2) . Then there exists a ground stateat the threshold of the essential spectrum for the case of critical value U = U c .Proof. We take sequence of normalized eigenfunctions ψ U corresponding tothe ground state eigenvalue E U of operator H U where U → U c . Existence ofsuch eigenfunctions is guaranteed by the existence of a gap in the spectrumfor every subcritical value of U . Due to reflexivity of L spaces we know that ψ U contains a weakly converging subsequence. The task is to show that thissubsequence converges strongly.With Theorem A.1 in mind we need to show conditions Eq. (18) and Eq. (21)in order to prove strong convergence of a given weakly convergent subse-quence. The condition (18) can be obtained by mimicking the proof of The-orem 2.3. This is possible since the proof does not rely on precise choice of U and works also for each subcritical case U < U c .The condition (21) is a direct consequence of Lemma A.3. Uniformity of theestimate follows from V ( U ) ≤ V ( U c ) for all U ≤ U c . B Partition of Unity
For our main proof we need the partition of unity. We start by introducingtwo auxiliary functions ˜ ς A δ := , if x ∈ A δ , − ϕ (cid:16) | x | δ | x | ∞ (cid:17) , if x ∈ A δ \ A δ , , if x / ∈ A δ and ˜ ς ⊥ A δ := 1 − ˜ ς A δ . We define ς A δ := ˜ ς A δ q | ˜ ς A δ | + | ˜ ς ⊥ A δ | and ς ⊥ A δ := ˜ ς ⊥ A δ q | ˜ ς A δ | + | ˜ ς ⊥ A δ | .
19e need to check that | ˜ ς A δ | + | ˜ ς ⊥ A δ | > which will prove that ς A δ and ς ⊥ A δ are well defined and | ς A δ | + | ς ⊥ A δ | = 1 . We have | ˜ ς A δ | + | ˜ ς ⊥ A δ | = , if x ∈ A δ , (cid:16) − ϕ (cid:16) | x | δ | x | ∞ (cid:17)(cid:17) + (cid:16) ϕ (cid:16) | x | δ | x | ∞ (cid:17)(cid:17) , if x ∈ A δ \ A δ , , if x / ∈ A δ . The second expression is positive because (1 − x ) + x = 1 − x + 2 x = (1 − √ x ) + 2( √ − x > where we used that x ∈ [0 , . We also check |∇ ς A δ | + |∇ ς ⊥ A δ | ≤ L | x | ∞ for given < L < ∞ . We write ∂ A A √ A + B = 1 √ A + B − A ( √ A + B ) = B ( √ A + B ) ∂ B A √ A + B = − AB ( √ A + B ) which implies |∇ ς A δ | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (˜ ς ⊥ A δ ) ((˜ ς A δ ) + (˜ ς ⊥ A δ ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |∇ ˜ ς A δ | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ ς A δ ˜ ς ⊥ A δ ((˜ ς A δ ) + (˜ ς ⊥ A δ ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |∇ ˜ ς ⊥ A δ | , |∇ ˜ ς ⊥ A δ | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ ς A δ ˜ ς ⊥ A δ ((˜ ς A δ ) + (˜ ς ⊥ A δ ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |∇ ˜ ς A δ | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (˜ ς A δ ) ((˜ ς A δ ) + (˜ ς ⊥ A δ ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |∇ ˜ ς ⊥ A δ | . Combining the above with |∇ ˜ ς A δ | ≤ (cid:13)(cid:13)(cid:13)(cid:13) ϕ ′ (cid:18) | x | δ | x | ∞ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ δ | x | ∞ , |∇ ˜ ς ⊥ A δ | ≤ (cid:13)(cid:13)(cid:13)(cid:13) ϕ ′ (cid:18) | x | δ | x | ∞ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) ∞ δ | x | ∞ we obtain |∇ ς A δ | + |∇ ς ⊥ A δ | ≤ L | x | ∞ . Avoiding Born-Oppenheimer approximation
Our method is also applicable for the case without Born-Oppenheimer ap-proximation with additional assumptions. The first one is that we considerelectrons to be bosons or fermions. Unfortunately our method is not appli-cable for distinguishable electrons in this setting. The second assumption isthat the nucleus has at least the same mass as an electron. We consider afinite mass nucleus and transform our system to the center of mass picture.Our system is described by H U = p + p + p N M − | x − x N | − | x − x N | + U | x − x | (23)where p N = − i∂ N is the momentum of the nucleus, M its mass in multiplesof the weight of the electron and x N its position. The domain of the operator(23) is D ( H U ) = H a/s ( R ) ⊗ H ( R ) . We transform the Hamiltonian into thenew coordinates using the following x a := x − x N ,x b := x − x N ,x c := x + x + M x N M .
One can easily check that this change of variables induces a unitary trans-form. Our Hamiltonian then becomes ˜ H U = − ∆ a − ∆ b − M + 2 ∆ c − M ( ∂ a + ∂ b ) − | x a | − | x b | + U | x a − x b | . In this new coordinates it is possible to rewrite the operator in a directintegral decomposition after Fourier transform in the x c coordinate as ˜ H U = Z R H ( P ) dP where H ( P ) = − ∆ a − ∆ b + P M + 2 − M ( ∂ a + ∂ b ) − | x a | − | x b | + U | x a − x b | . In order to show the fall-off behaviour for H ( P ) at the threshold we firstneed to identify the threshold. First we consider Hamiltonian describing oneelectron and the nucleus of the weight M , i.e. H ,U = p + p N M − | x − x N | (24)21ith the domain D ( H ,U ) = H ( R ) ⊗ H ( R ) . We transform this operatorusing x a := x − x N ,x c := x + M x N M .
Again we acquire an operator which can be written in a direct integral de-composition after Fourier transform in the x c coordinate as ˜ H = Z R H ( P ) dP = Z R (cid:18) − ∆ a + P M + 1 − M ∆ a − | x a | (cid:19) d P .
Now we are almost ready to repeat the proof given in the main body of thepaper with two corrections. One thing which is missing is the estimate onIMS error corresponding to the term − M ( ∂ a + ∂ b ) , i.e. M h ψ, ( | ( ∂ a + ∂ b ) ξ | + | ( ∂ a + ∂ b ) ξ ⊥ | ) ψ i . This can be written as M h ψ, ( | ∂ a ξ | + | ∂ b ξ | + | ∂ a ξ ⊥ | + | ∂ b ξ ⊥ | ) ψ i ≤ M h ψ, ( |∇ ξ | + |∇ ξ ⊥ | ) ψ i where the last term on the right is well known from the previous case. Thesecond change in the proof is the lower bound for the kinetic energy of theoperator (23) by the kinetic energy terms in (24). This can be achieved inthe following way − ∆ a − ∆ b − M ( ∂ a + ∂ b ) = − ∆ a − M ∂ a − ∆ b − M ∂ b − M ∂ a ∂ b . We show − ∆ b − M ∂ b − M ∂ a ∂ b ≥ . This is equivalent to (cid:18) M (cid:19) k∇ b ψ k − M h ∂ a ψ, ∂ b ψ i ≥ . Using the symmetry or antisymmetry of the functions in the domain of ouroperator we have k∇ a ψ k = k∇ b ψ k . This means (cid:18) M (cid:19) k∇ b ψ k − M h ∂ a ψ, ∂ b ψ i ≥ (cid:18) M (cid:19) k∇ b ψ k − M k∇ a ψ kk∇ b ψ k (cid:18) M (cid:19) k∇ b ψ k − M k∇ a ψ kk∇ b ψ k = (cid:18) − M (cid:19) k∇ b ψ k ≥ . The last inequality holds provided that the nucleus has at least the mass ofthe electron. At this point we are able to repeat the proof in the main body ofthe paper step by step for a fixed fiber P = 0 in the integral decomposition. D Construction of Point Bounds
There is obvious discrepancy in the description of our upper and lower bound.Our upper bound is integral one and our lower bound is a point one. Usingthe method described in [2] we can transform integral bounds to point boundsprovided that our eigenfunction ψ is positive. We summarize the argumentwhich is based on Harnack inequality. Lemma D.1.
Let H U be defined by Eq. (5) . Furthermore assume that ψ isa ground state of H U and e F ψ ∈ L ( R ) . Then there exists a constant c > such that ψ ( x ) ≤ ce − F ( x ) . Proof.
Using standard arguments we can show that ψ is positive and con-tinuous. Then by Harnack inequality for each compact subset U of R thereexists C > s.t. inf x ∈ U ψ ( x ) ≥ C sup x ∈ U ψ ( x ) . This implies C vol( U ) Z U ψ ( y ) dy ≥ inf x ∈ U ψ ( x ) C vol( U ) Z U dy = u ( x ) . For each point x ∈ R there exists a unit ball U away from origin such that x ∈ ∂U and F ( x ) ≤ F ( y ) . for every y ∈ U . Combining above estimates we obtain u ( x ) e F ( x ) ≤ e F ( x ) C vol( U ) Z U ψ ( y ) dy ≤ C vol( U ) Z U e F ( y ) ψ ( y ) dy . This implies u ( x ) ≤ e − F ( x ) C vol( U ) Z U e F ( y ) ψ ( y ) dy ≤ C vol( U ) vol( U ) k e F ψ k e − F ( x ) ≤ ce − F ( x ) where we denoted c := k e F ψ k C . 23 eferences [1] Shmuel Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N -body Schrödingeroperators , Mathematical Notes, vol. 29, Princeton University Press,Princeton, NJ; University of Tokyo Press, Tokyo, 1982. MR 745286[2] M. Aizenman and B. Simon, Brownian motion and Harnack inequalityfor Schrödinger operators , Comm. Pure Appl. Math. (1982), no. 2,209–273. MR 644024[3] Hans Bethe, Berechnung der Elektronenaffinität des Wasserstoffs ,Zeitschrift für Physik (1929), no. 11-12, 815–821.[4] Percy Deift, Walter Hunziker, Barry Simon, and Egon Vock, Pointwisebounds on eigenfunctions and wave packets in N -body quantum systems.IV , Comm. Math. Phys. (1978/79), no. 1, 1–34. MR 516993[5] Dmitry K. Gridnev and Martin E. Garcia, Rigorous conditions for theexistence of bound states at the threshold in the two-particle case , J.Phys. A (2007), no. 30, 9003–9016. MR 2344533[6] Maria Hoffmann-Ostenhof and Thomas Hoffmann-Ostenhof, Absence ofan L -eigenfunction at the bottom of the spectrum of the Hamiltonian ofthe hydrogen negative ion in the triplet S -sector , J. Phys. A (1984),no. 17, 3321–3325. MR 771622[7] Thomas Hoffmann-Ostenhof, A comparison theorem for differential in-equalities with applications in quantum mechanics , J. Phys. A (1980),no. 2, 417–424. MR 558638[8] Dirk Hundertmark and Young-Ran Lee, On non-local variational prob-lems with lack of compactness related to non-linear optics , J. NonlinearSci. (2012), no. 1, 1–38. MR 2878650[9] Elliott H. Lieb, Bound on the maximum negative ionization of atomsand molecules , Phys. Rev. A (1984), 3018–3028.[10] Thomas Hoffmann-Ostenhof Maria Hoffmann-Ostenhof and Barry Si-mon, A multiparticle Coulomb system with bound state at threshold , J.Phys. A (1983), no. 6, 1125–1131. MR 706689[11] Frank H. Stillinger Jr, Ground-state energy of two-electron atoms , TheJournal of Chemical Physics (1966), no. 10, 3623–3631.2412] Maria Hoffmann-Ostenhof Thomas Hoffmann-Ostenhof and ReinhartAhlrichs, "schrödinger inequalities" and asymptotic behavior of many-electron densities , Phys. Rev. A (1978), 328–334.[13] Grigorii M. Zhislin, Discussion of the spectrum of schrödinger operatorsfor systems of many particles , Trudy Moskovskogo matematiceskogo ob-scestva9