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Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm
To cite this version:
Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm. The Feshbach-Schur map and perturbation theory. 2021, Partial Differential Equations, Spectral Theory, and Mathematical Physics, The Ari Laptev Anniversary Volume, �10.4171/ECR/18�. �hal-03216856�
theory
Geneviève Dusson Israel Michael Sigal Benjamin Stamm
To Ari with friendship and admiration
Abstract
This paper deals with perturbation theory for discrete spectra of linear opera- tors. To simplify exposition we consider here self-adjoint operators. This theory is based on the Feshbach-Schur map and it has advantages with respect to the stan- dard perturbation theory in three aspects: (a) it readily produces rigorous estimates on eigenvalues and eigenfunctions with explicit constants; (b) it is compact and elementary (it uses properties of norms and the fundamental theorem of algebra about solutions of polynomial equations); and (c) it is based on a self-contained formulation of a fixed point problem for the eigenvalues and eigenfunctions, al- lowing for easy iterations. We apply our abstract results to obtain rigorous bounds on the ground states of Helium-type ions.
Mathematics Subject Classification 2020.Primary 47A55, 35P15, 81Q15; Sec- ondary 47A75, 35J10
Keywords.Perturbation theory, Spectrum, Feshbach-Schur map, Schroedinger operator, Atomic systems, Helium-type ions, Ground state
1 Set-up and result
The eigenvalue perturbation theory is a major tool in mathematical physics and applied mathematics. In the present form, it goes back to Rayleigh and Schrödinger and became
G. Dusson: Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université Bourgogne Franche-Comté, 25030 Besançon, France; email:genevieve.dusson@math.cnrs.fr. Supported in part by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-0003).
I.M. Sigal: Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada;
email:im.sigal@utoronto.ca. Supported in part by NSERC Grant No. NA7901.
B. Stamm: Center for Computational Engineering Science, RWTH Aachen University, 52062 Aachen, Germany; email:best@acom.rwth-aachen.de.
1
a robust mathematical field in the works of Kato and Rellich. It was extended to quantum resonances by Simon, see [2,5,6,11,12,13,14,15] for books and a book-size review.
A different approach to the eigenvalue perturbation problem going back to works of Feshbach and based on the Feshbach-Schur map was introduced in [1] and extended in [4,3].
In this paper, we develop further this approach proposing a self-contained theory in a form of a fixed point problem for the eigenvalues and eigenfunctions. It is more compact and direct than the traditional one and, as we show elsewhere, extends to the nonlinear eigenvalue problem.
We show that this approach leads naturally to bounds on the eigenvalues and eigenfunctions with explicit constants, which we use in an estimation of the ground state energies of the Helium-type ions.
The approach can handle tougher perturbations, non-isolated eigenvalues (see [1, 4]) and continuous spectra as well as discrete ones. In this paper, we restrict ourselves to the latter. Namely, we address the eigenvalue perturbation problem for operators on a Hilbert space of the form
𝐻=𝐻0+𝑊 , (1.1)
where𝐻0is an operator with some isolated eigenvalues and𝑊 is an operator, small relative to𝐻0in an appropriate norm. The goal is to show that𝐻has eigenvalues near those of𝐻0and estimate these eigenvalues.
Specifically, withk · kstanding for the vector and operator norms in the underlying Hilbert space, we assume that
(A)𝐻0is a self-adjoint, non-negative operator (𝐻0≥ 𝛽 >0);
(B)𝑊is symmetric and form-bounded with respect to𝐻0, in the sense that k𝐻
−12 0 𝑊 𝐻
−12 0 k <∞.
(C)𝐻0has an isolated eigenvalue𝜆0>0 of a finite multiplicity,𝑚. Herek · kis the operator norm and𝐻−𝑠
0 ,0< 𝑠 <1,is defined either by the spectral theory or by the explicit formula
𝐻−𝑠
0 :=𝑐
∫ ∞
0
(𝐻0+𝜔)−1𝑑 𝜔 𝜔𝑠 ,
where𝑐:= h∫∞
0 (1+𝜔)−1𝑑 𝜔𝜔𝑠
i−1
.
It turns out to be useful in the proofs below to use the following form-norm k𝑊k𝐻0 := k𝐻
−12 0 𝑊 𝐻
−12 0 k,
Let 𝑃 be the orthogonal projection onto the span of the eigenfunctions of 𝐻0 corresponding to the eigenvalue𝜆0and let𝑃⊥ :=1−𝑃. Let𝛾0be the distance of𝜆0to
the rest of the spectrum of𝐻0, and𝜆∗ :=𝜆0+𝛾0. In what follows, we often deal with the expression
Φ(𝑊):= 𝜆0𝜆∗ 𝛾0
k𝑃⊥𝑊 𝑃k2𝐻
0. (1.2)
The following theorem proven in Section2is the main result of this paper.
Main Theorem 1.1. Let Assumptions (A)-(C) be satisfied and assume that, for some
0<𝑏 <1and0<𝑎 <1−𝑏,
k𝑃⊥𝑊 𝑃⊥k𝐻0 ≤ 𝑏 𝛾0 𝜆∗
, (1.3)
k𝑃𝑊 𝑃k +𝑘Φ(𝑊) < 𝑎 𝛾0, (1.4) 𝑘Φ(𝑊) < 1
2(𝑎 𝛾0− k𝑃𝑊 𝑃k), (1.5) where𝑘 := 1−𝑎1−𝑏. Then the spectrum of the operator𝐻 near𝜆0consists of isolated eigenvalues 𝜆𝑖 of the total multiplicity 𝑚 satisfying, together with their normalized eigenfunctions𝜑𝑖, the following estimates
|𝜆𝑖−𝜆0| ≤ k𝑃𝑊 𝑃k +𝑘Φ(𝑊), (1.6) k𝜑𝑖−𝜑0𝑖k ≤𝑘
√︄
Φ(𝑊) 𝛾0
, (1.7)
where𝜑0𝑖are appropriate eigenfunctions of𝐻0corresponding to the eigenvalue𝜆0. Remark 1(Comparison with [3]). A similar result was already proven in [3]. Here, the theory is made self-contained and formulated as the fixed point problem and the bounds are tightened.
Remark 2(Conditions on𝑊) The fine tuning the conditions (1.3)-(1.5) on𝑊is used in application of this theorem to atomic systems in Section3.
Note that because of the elementary estimate Φ(𝑊) ≤ 1
𝛾0
k𝑃𝑊 𝑃⊥𝑊 𝑃k,
see (3.11) below, the computation of k𝑃𝑊 𝑃k and Φ(𝑊) reduces to computing the largest eigenvalues of the simple𝑚×𝑚-matrices𝑃𝑊 𝑃,−𝑃𝑊 𝑃and𝑃𝑊 𝑃⊥𝑊 𝑃. Remark 3(Higher-order estimates and non-degenerate𝜆0). In fact, one can estimate 𝜆𝑖 (as well as the eigenfunctions 𝜑𝑖) to an arbitrary order in k𝑃⊥𝑊 𝑃k𝐻0/𝛾0. As a demonstration, we derive (after the proof of Theorem1.1) the second-order estimate of the eigenvalue in the rank-one (𝑚=1) case
|𝜆1−𝜆0− h𝑊i| ≤𝑘Φ(𝑊), (1.8)
where𝜑0is the eigenfunction of𝐻0corresponding to𝜆0,h𝑊i:= h𝜑0, 𝑊 𝜑0i.
For degenerate𝜆0, we would like to prove a similar bound on|𝜆𝑖−𝜆0−𝜇𝑖|, where 𝜇𝑖is the corresponding eigenvalue of the𝑚×𝑚-matrix𝑃𝑊 𝑃. Here, we have a partial result (proven at the end of the next section) for the lowest eigenvalue𝜆1of𝐻:
𝜆0+𝜇min−𝑘Φ(𝑊) ≤𝜆1≤𝜆0+𝜇min, (1.9) where𝜇mindenotes the smallest eigenvalue of the matrix𝑃𝑊 𝑃.
Remark 4(Non-self-adjoint 𝐻). With the sacrifice of the explicit constants in (1.6) and (1.7) (mostly coming from (2.5)), the self-adjointness assumption on 𝐻 can be removed. However, for the problem of quantum resonances, one can still obtain explicit estimates.
In the rest of this section𝐻is an abstract operator not necessarily self-adjoint or of the form (1.1). Our approach is grounded in the following theorem (see [4], Theorem 11.1).
Theorem 1.2. Let𝐻 be an operator on a Hilbert space and 𝑃 and 𝑃⊥, a pair of projections such that𝑃+𝑃⊥=1. Assume𝐻⊥:=𝑃⊥𝐻 𝑃⊥is invertible onRan𝑃⊥and the expression
𝐹𝑃(𝐻) := 𝑃(𝐻−𝐻 𝑅⊥𝐻)𝑃, (1.10) where𝑅⊥ := 𝑃⊥(𝐻⊥)−1𝑃⊥, defines a bounded operator. Then 𝐹𝑃, considered as a map on the space of operators, is isospectral in the following sense:
(a) 𝜆∈𝜎(𝐻) if and only if 0∈𝜎(𝐹𝑃(𝐻−𝜆)); (b) 𝐻 𝜓=𝜆𝜓 if and only if 𝐹𝑃(𝐻−𝜆)𝜑=0;
(c) dim𝑁 𝑢𝑙 𝑙(𝐻−𝜆)=dim𝑁 𝑢𝑙 𝑙 𝐹𝑃(𝐻−𝜆).
(d) 𝜓and𝜑in (b) are related as𝜑=𝑃𝜓and𝜓=𝑄𝑃(𝜆)𝜑, where
𝑄𝑃(𝜆) :=𝑃−𝑃⊥(𝐻⊥−𝜆)−1𝑃⊥𝐻 𝑃. (1.11) Finally, 𝐹𝑃(𝐻)∗ = 𝐹𝑃∗(𝐻∗) and therefore, if 𝐻 and 𝑃 are self-adjoint, then so is 𝐹𝑃(𝐻).
A proof of this theorem is elementary and short; it can be found in [1], Section IV.1, pp 346-348, and [4], Appendix 11.6, pp 123-125.
The map 𝐹𝑃 on the space of operators, is called the Feshbach-Schur map. The relation𝜓=𝑄𝑃(𝜆)𝜑allows us to reconstruct the full eigenfunction from the projected one. By statement (a), we have
Corollary 1.3. Assume there is an open setΛ ⊂ Csuch that𝐻−𝜆,𝜆 ∈Λ, is in the domain of the map𝐹𝑃, i.e.𝐹𝑃(𝐻−𝜆)is well defined. Define the operator-family
𝐻(𝜆) :=𝐹𝑃(𝐻−𝜆) +𝜆 𝑃,
and let𝜈𝑖(𝜆), for𝜆 inΛ, denote its eigenvalues counted with multiplicities. Then the eigenvalues of 𝐻 in Λ are in one-to-one correspondence with the solutions of the equations
𝜈𝑖(𝜆)=𝜆. (1.12)
Concentrating on the eigenvalue problem, Corollary1.3 shows that the original problem
𝐻 𝜓=𝜆𝜓 , (1.13)
is mapped into the equivalent eigenvalue problem,
𝐻(𝜆)𝜑=𝜆 𝜑, (1.14)
nonlinear in the spectral parameter𝜆, but on the smaller space Ran𝑃.
Since the projection𝑃 is of a finite rank, the original eigenvalue problem (1.13) is mapped into an equivalent lower-dimensional/finite-dimensional one, (1.14). Of course, we have to pay a price for this: at one step we have to solve a one-dimensional fixed point problem that can be equivalently seen as a non-linear eigenvalue problem and invert an operator in Ran𝑃⊥.
We call this approach theFeshbach-Schur map method, orFSM method, for short.
It is rather compact, as one easily see skimming through this paper and entirely elementary.
We call𝐻(𝜆)theeffective Hamiltonian(matrix) and write as
𝐻(𝜆)=𝑃 𝐻 𝑃+𝑈(𝜆), (1.15) with the self-adjointeffective interaction, or a Schur complement,𝑈(𝜆), defined as
𝑈(𝜆):=−𝑃 𝐻 𝑃⊥(𝐻⊥−𝜆)−1𝑃⊥𝐻 𝑃. (1.16) It is shown in Lemma2.1below that (1.16) defines a bounded operator family.
We mention here some additional properties discussed in [3].
Proposition 1.4. Let𝐻be self-adjoint, letΛbe the same as in Corollary1.3, let𝑚be the rank of𝑃and let the eigenvalues of𝐻(𝜆),𝜆 ∈ Λ∩R, be labeled in the order of their increase counting their multiplicities so that
𝜈1(𝜆) ≤. . .≤𝜈𝑚(𝜆). (1.17) Then we have that (a) a solution of the equation 𝜈𝑖(𝜆) = 𝜆, for 𝜆 ∈ Λ∩R, is the 𝑖-th eigenvalue,𝜆𝑖, of𝐻 and vice versa; (b)𝜈𝑖 is differentiable in𝜆and𝜈0
𝑖(𝜆)≤0, for 𝜆 ∈Λ∩R.
Proof. (a) By (1.17),𝜆𝑖 = 𝜈𝑖(𝜆𝑖) < 𝜈𝑖+1(𝜆) = 𝜆𝑖+1 (except for the level crossings), which proves the result. For (b), by the explicit formula
𝑈0(𝜆):=−𝑃 𝐻 𝑃⊥(𝐻⊥−𝜆)−2𝑃⊥𝐻 𝑃 ≤0,
we have𝑈0(𝜆) ≤ 0, which by the Hellmann-Feynman theorem (cf. (2.19)) implies 𝜈0
𝑖(𝜆) ≤0.
Remark 5 (Perturbation expansion). In the context of Hamiltonians of form (1.1) satisfying Assumptions (A)-(C), the FSM leads to a perturbation expansion to an arbitrary order. Indeed, in this case,𝑃is the orthogonal projection onto Null(𝐻0−𝜆0), 𝑃⊥:=1−𝑃and𝑈(𝜆)can be written as
𝑈(𝜆):=−𝑃𝑊 𝑃⊥(𝐻⊥−𝜆)−1𝑃⊥𝑊 𝑃.
Now, using the notation𝐴⊥:=𝑃⊥𝐴𝑃⊥|Ran𝑃⊥and expanding (𝐻⊥−𝜆)−1=(𝐻⊥
0 +𝑊⊥−𝜆)−1
in𝑊⊥and at the same time iterating fixed point equation (1.12), we generate a pertur- bation expansion for eigenvalues𝐻to an arbitrary order (see also Remark 2 above).
The paper is organized as follows. In Section2, we present the proof of the main result, Theorem1.1. Section 3uses this result to obtain bounds of the ground-state energy of Helium-type ions.
2 Perturbation estimates
We want to use Theorem1.2(b, c) to reduce the original eigenvalue problem to a simpler one. In this section, we assume that Conditions (A)-(C) of Section1are satisfied. Recall that𝛾0is the distance of𝜆0to the rest of the spectrum of𝐻0and the expressionΦ(𝑊)is defined in (1.2). First, we prove that the operator𝐹𝑃(𝐻−𝜆)is well-defined for𝜆∈Ω, where, with𝑎the same as in Theorem1.1,
Ω:={𝑧 ∈C: |𝑧−𝜆0| ≤ 𝑎 𝛾0}. (2.1) Recall that𝑃denotes the orthogonal projection onto Null(𝐻0−𝜆0)and𝑃⊥:=1−𝑃. Denote
𝐻⊥:=𝑃⊥𝐻 𝑃⊥|Ran𝑃⊥ and 𝑅⊥(𝜆):=𝑃⊥(𝐻⊥−𝜆)−1𝑃⊥ and recall𝑘= 1−𝑎1−𝑏. We have
Lemma 2.1. Recall𝜆∗ := 𝜆0+𝛾0and assume(1.3). Then, for𝜆 ∈ Ω, the following statements hold
(a) The operator𝐻⊥−𝜆is invertible onRan𝑃⊥;
(b) The inverse 𝑅⊥(𝜆) := 𝑃⊥(𝐻⊥−𝜆)−1𝑃⊥ defines a bounded, analytic operator- family;
(c) The expression
𝑈(𝜆) :=−𝑃 𝐻 𝑅⊥(𝜆)𝐻 𝑃 (2.2)
defines a finite-rank, analytic operator-family, bounded as
k𝑈(𝜆) k ≤ 𝑘Φ(𝑊). (2.3)
(d) 𝑈(𝜆)is symmetric for any𝜆∈Ω∩Rand therefore𝐻(𝜆)is self-adjoint as well.
Proof. With the notation𝐴⊥:=𝑃⊥𝐴𝑃⊥|Ran𝑃⊥, we write 𝐻⊥=𝐻⊥
0 +𝑊⊥. To prove (a), we let 𝐻𝜆 := 𝐻⊥
0 −𝜆 and use the factorization 𝐻𝜆 = |𝐻𝜆|12𝑉𝜆|𝐻𝜆|12, where𝑉𝜆is a unitary operator, and use that, for𝜆 ∈ Ω, the operator𝐻𝜆is invertible and therefore we have the identity
𝐻⊥−𝜆=|𝐻𝜆|12[𝑉𝜆+𝐾𝜆] |𝐻𝜆|12, (2.4) where𝐾𝜆 := |𝐻𝜆|−12𝑊⊥|𝐻𝜆|−12. Next, for𝜆 ∈ Ω, we have k𝐻⊥
0|𝐻𝜆|−1𝑃⊥k = 𝑓(𝜆), where
𝑓(𝜆):=sup
𝑠 𝑠−𝜆
:𝑠≥0,|𝑠−𝜆0| ≥𝛾0
. Assuming|𝜆−𝜆0| ≤𝑎 𝛾0and using
𝑠 𝑠−𝜆
≤1+
𝜆 𝑠−𝜆
≤1+ 𝜆0+𝑎 𝛾0 (1−𝑎)𝛾0
= 𝜆∗ (1−𝑎)𝛾0
, we obtain
𝑓(𝜆) ≤ 𝜆∗ (1−𝑎)𝛾0
.
Since𝐻
1 2
0|𝐻𝜆|−12𝑃⊥=(𝐻0|𝐻𝜆|−1|Ran𝑃⊥)12𝑃⊥, we have for𝜆∈Ω, k𝐻
1 2
0|𝐻𝜆|−12𝑃⊥k ≤ 𝜆∗
(1−𝑎)𝛾0 12
, (2.5)
which implies in particular thatk𝐾𝜆k ≤ 𝜆∗
(1−𝑎)𝛾0k𝑊⊥k𝐻0. By the assumption (1.3), i.e., k𝑊⊥k𝐻0 ≤ 𝑏 𝛾𝜆0
∗ , we have
k𝐾𝜆k ≤ 𝑏 1−𝑎
. (2.6)
Since 1−𝑎 > 𝑏, by (2.4), the operator𝐻⊥−𝜆is invertible and its inverse is analytic in𝜆 ∈Ω, which proves (a) and (b).
We show that statement (c) is also satisfied. Since𝑃 𝐻0 = 𝐻0𝑃and 𝑃 𝑃⊥ =0, we have
𝑃 𝐻 𝑃⊥=𝑃𝑊 𝑃⊥, 𝑃⊥𝐻 𝑃=𝑃⊥𝑊 𝑃.
These relations and definition (2.2) yield
𝑈(𝜆)=−𝑃𝑊 𝑅⊥(𝜆)𝑊 𝑃. (2.7)
Inverting (2.4) on Ran𝑃⊥and recalling the notation𝑅⊥(𝜆):=𝑃⊥(𝐻⊥−𝜆)−1𝑃⊥gives 𝑅⊥(𝜆)=|𝐻𝜆|−12𝑃⊥[𝑉𝜆+𝐾𝜆]−1𝑃⊥|𝐻𝜆|−12. (2.8) Now, using identity (2.8), estimate (2.5) and (2.6), we find, for𝜆 ∈Ω,
k𝐻
1 2
0𝑅⊥(𝜆)𝐻
1 2
0k ≤ 𝑘 𝜆∗ 𝛾0
. (2.9)
Furthermore, by the eigen-equation𝐻0𝑃=𝜆0𝑃, we have k𝐻
1 2
0𝑃k2=𝜆0. (2.10)
Using expression (2.7) and estimates (2.9) and (2.10), we arrive at inequality (2.3).
The analyticity follows from (2.7) and the analyticity of𝑅⊥(𝜆).
For (d), since𝐻0, 𝑊 and𝑃 are self-adjoint, then so are𝑈(𝜆), for any𝜆 ∈ Ω∩R, and, since𝑈(𝜆)is bounded,𝐻(𝜆)is self-adjoint as well.
Proof of Theorem1.1. LetΩbe given by equation (2.1). Recall that, by Lemma2.1 and Theorem 1.2, the 𝑚 × 𝑚 matrix-family 𝐻(𝜆) := 𝐹𝑃(𝐻 −𝜆) +𝜆 𝑃, with 𝐹𝑃 given in (1.10), is well defined, for each𝜆 ∈ Ω, and can be written as (1.15). Since 𝑃 𝐻 𝑃=𝜆0𝑃+𝑃𝑊 𝑃, Eq. (1.15) can be rewritten as
𝐻(𝜆)=𝜆0𝑃+𝑃𝑊 𝑃+𝑈(𝜆). (2.11) Eqs (2.3) and (2.11) imply the inequality
k𝐻(𝜆) −𝜆0𝑃−𝑃𝑊 𝑃k ≤𝑘Φ(𝑊). (2.12) By a fact from Linear Algebra, for each𝜆 ∈ Ω∩R, the total multiplicity of the eigenvalues of the𝑚×𝑚self-adjoint matrix𝐻(𝜆)is𝑚.
Denote by𝜈𝑖(𝜆), 𝑖 =1,2, . . . , 𝑚,the eigenvalues of 𝐻(𝜆),repeated according to their multiplicities. Eq (2.12) yields
|𝜈𝑖(𝜆) −𝜆0| ≤ k𝑃𝑊 𝑃k +𝑘Φ(𝑊). (2.13)
Indeed, let𝑃𝑖(𝜆)be the orthogonal projection onto Null(𝐻(𝜆) −𝜈𝑖(𝜆)). Then 𝐻(𝜆)𝑃𝑖(𝜆)=𝜈𝑖(𝜆)𝑃𝑖(𝜆),
which, due to (2.11) and𝑃𝑖(𝜆)𝑃=𝑃𝑖(𝜆), can be rewritten as (𝜈𝑖(𝜆) −𝜆0)𝑃𝑖(𝜆)=(𝑃𝑊 𝑃+𝑈(𝜆))𝑃𝑖(𝜆).
Equating the operator norms of both sides of this equation and using (2.12) and k𝑃𝑖(𝜆) k=1 gives (2.13).
By Corollary1.3, the eigenvalues of 𝐻 in the interval Ω∩R are in one-to-one correspondence with the solutions of the equations
𝜈𝑖(𝜆)=𝜆 (2.14)
inΩ∩R. If this equation has a solution, then, due to (2.13), this solution would satisfy (1.6). Thus, we address (2.14). Let
Ω0:={𝑧∈R:|𝑧−𝜆0| ≤𝑟 𝛾0}, (2.15) with
𝑟:= 1 𝛾0
k𝑃𝑊 𝑃k +𝑘Φ(𝑊) . By our assumption (1.4),𝑟 < 𝑎 <1−𝑏.
Recall that by the definition, a branch point is a point at which the multiplicity of one of the eigenvalue families (branches) changes. One could think on a branch point as a point where two or more distinct eigenvalue branches intersect. Our next result shows that the eigenvalue branches of𝐻(𝜆) could be chosen in a differentiable way and estimates their derivatives.
Proposition 2.2. The following statements hold.
i) The eigenvalues 𝜈𝑖(𝜆) of 𝐻(𝜆) and the corresponding eigenfunctions can be chosen to be differentiable for𝜆∈Ω0.
ii) The derivatives𝜈0
𝑖(𝜆),𝜆∈Ω0, are bounded as
|𝜈𝑖0(𝜆) | ≤ 𝑘 𝑎−𝑟
Φ(𝑊) 𝛾0
. (2.16)
iii) 𝜈𝑖(𝜆)maps the intervalΩ0into itself.
Consequently, since by (1.5) the r.h.s. of (2.16) is<1, the equations 𝜈𝑖(𝜆) = 𝜆have unique solutions inΩ0.
Proof. Proof of (i) for simple𝜆0. For a simple eigenvalue𝜆0,𝑃is a rank-one projection on the space spanned by the eigenvector𝜑0of𝐻0corresponding to the eigenvalue𝜆0 and therefore Eq. (2.11) implies that𝐻(𝜆)=𝜈(𝜆)𝑃, with
𝜈(𝜆) :=𝜆0+ h𝜑0,(𝑊+𝑈(𝜆))𝜑0i. This and Lemma2.1show that the eigenvalue𝜈(𝜆)is analytic.
Proof of (i) for degenerate𝜆0. We pick anarbitrarypoint𝜇inΩ0and let𝑃𝜇𝑖be the orthogonal projections onto the eigenspaces of𝐻(𝜇)corresponding to the eigenvalues 𝜈𝑖(𝜇), i.e. for a fixed𝑖, 𝑃𝜇𝑖 projects on the spans of all eigenvectors with the same eigenvalue 𝜈𝑖(𝜇). We now show that, in a neighbourhood of 𝜇, the eigenfunctions 𝜒𝑖(𝜆)can be chosen in a differentiable way. We introduce the system of equations for the eigenvalues𝜈𝑖(𝜆)and corresponding eigenfunctions𝜒𝑖(𝜆):
𝜈𝑖(𝜆) = h𝜒𝑖(𝜆), 𝐻(𝜆)𝜒𝑖(𝜆)i
k𝜒𝑖(𝜆) k2 , (2.17)
𝜒𝑖(𝜆)= 𝜒𝑖(𝜇) −𝑅⊥(𝜈𝑖(𝜆), 𝜆)𝑊𝜆𝜒𝑖(𝜇), (2.18) where
𝑅⊥(𝜈, 𝜆):=𝑃⊥
𝜇𝑖(𝑃⊥
𝜇𝑖𝐻(𝜆)𝑃⊥
𝜇𝑖−𝜈)−1𝑃⊥
𝜇𝑖, 𝑊𝜆:=𝑈(𝜆) −𝑈(𝜇).
The expression on the right side of (2.18) is the𝑄-operator for𝐻(𝜆)and𝑃𝜇𝑖defined according to (1.11) and applied to 𝜒𝑖(𝜇). Note that systems (2.17)-(2.18) for different indices 𝑖 are not coupled. Furthermore, since 𝜒𝑖(𝜇) and 𝑅⊥(𝜈𝑖(𝜆), 𝜆)𝑊𝜆𝜒𝑖(𝜇) are orthogonal, we havek𝜒𝑖(𝜆) k ≥ k𝜒𝑖(𝜇) k; and𝜒𝑖(𝜆)and 𝜒𝑗(𝜆)are almost orthogonal for𝑖 ≠ 𝑗. Assuming this system has a solution(𝜈𝑖(𝜆), 𝜒𝑖(𝜆)), we see that, by Theorem 1.2(d), with𝑃=𝑃𝜇𝑖,𝜒𝑖(𝜆)are eigenfunctions of𝐻(𝜆)with the eigenvalues𝜈𝑖(𝜆)and (2.17) follows from the eigen-equation𝐻(𝜆)𝜒𝑖(𝜆)=𝜈𝑖(𝜆)𝜒𝑖(𝜆).
For each𝑖, we can reduce system (2.17)-(2.18) to the single equation for𝜈𝑖(𝜆), by treating𝜒𝑖(𝜆)in (2.17) as given by (2.18). This leads to the fixed point problem for the functions𝜈𝑖(𝜆):
𝜈=𝐹𝑖(𝜈, 𝜆), where
𝐹𝑖(𝜈, 𝜆):= h𝜒𝑖(𝜈, 𝜆), 𝐻(𝜆)𝜒𝑖(𝜈, 𝜆)i k𝜒𝑖(𝜈, 𝜆) k2 , 𝜒𝑖(𝜈, 𝜆):= 𝜒𝑖(𝜇) −𝑅⊥(𝜈, 𝜆)𝑊𝜆𝜒𝑖(𝜇). Notice that since𝑃⊥
𝜇𝑖𝐻(𝜆)𝑃⊥
𝜇𝑖are self-adjoint, the resolvent𝑅⊥(𝜈, 𝜆)and its derivatives in𝜈are uniformly bounded in a neighbourhood of(𝜈𝑖(𝜇), 𝜇), which does not contain branch points, except, possibly, for𝜇.
To be more specific, the resolvent𝑅⊥(𝜈, 𝜆)is uniformly bounded in a neighbour- hoodΩ𝜇𝑖 of (𝜈𝑖(𝜇), 𝜇) whether 𝜇 is a branch point or not, as long asΩ𝜇𝑖 does not contain any other branch point than𝜇.
Hence𝐹𝑖(𝜈, 𝜆)is differentiable in𝜈and𝜆and |𝜕𝜈𝐹𝑖(𝜈, 𝜆) | →0, as𝜆→𝜇(since
𝜕𝜈𝜒𝑖(𝜈, 𝜆)=𝑅⊥(𝜈, 𝜆)2𝑊𝜆𝜒𝑖(𝜇)and therefore
k𝜕𝜈𝜒𝑖(𝜈, 𝜆) k ≤ k𝑅⊥(𝜈, 𝜆)2(𝐻0+𝛼)12k k𝑊𝜆k𝐻0, 𝛼k (𝐻0+𝛼)12𝜒𝑖(𝜇) k →0, as𝜆→𝜇). Moreover,𝜈𝑖(𝜇) −𝐹𝑖(𝜈𝑖(𝜇), 𝜇)=0. Let 𝑓𝑖(𝜈, 𝜆):=𝜈−𝐹𝑖(𝜈, 𝜆). Then, by the above, 𝑓𝑖(𝜈𝑖(𝜇), 𝜇) = 0 and|𝜕𝜈𝑓𝑖(𝜈, 𝜆) −1| →0, as𝜆 → 𝜇. Thus, the implicit function theorem is applicable to the equation 𝑓𝑖(𝜈, 𝜆) = 0 and shows that there is a unique solution 𝜈𝑖(𝜆) in a neighbourhood of (𝜈𝑖(𝜇), 𝜇) and that this solution is differentiable in𝜆.
Next, we have
Lemma 2.3. We have, for𝜆∈Ω0, 𝜈0
𝑖(𝜆)= h𝜒𝑖(𝜆), 𝑈0(𝜆)𝜒𝑖(𝜆)i
k𝜒𝑖(𝜆) k2 . (2.19)
(Eq.(2.19)is closely related to the widely used Hellmann-Feynman theorem.) Proof. Let ˆ𝜒𝑖(𝜆):= k𝜒𝜒𝑖(𝜆)
𝑖(𝜆) k. Writing equation (2.17) as 𝜈𝑖(𝜆)=h𝜒ˆ𝑖(𝜆), 𝐻(𝜆)𝜒ˆ𝑖(𝜆)i and differentiating this with respect to𝜆, we obtain
𝜈0
𝑖(𝜆)=h𝜒ˆ𝑖(𝜆), 𝐻0(𝜆)𝜒ˆ𝑖(𝜆)i +𝜂(𝜆), where𝜂(𝜆) := h𝜒ˆ0
𝑖(𝜆), 𝐻(𝜆)𝜒ˆ𝑖(𝜆)i + h𝜒ˆ𝑖(𝜆), 𝐻(𝜆)𝜒ˆ0
𝑖(𝜆)i. Now, moving 𝐻(𝜆)in the last term to the l.h.s. and using that𝐻(𝜆)𝜒ˆ𝑖(𝜆)=𝜈𝑖(𝜆)𝜒ˆ𝑖(𝜆), we find
𝜂(𝜆)=𝜈𝑖(𝜆) [h𝜒ˆ0
𝑖(𝜆),𝜒ˆ𝑖(𝜆)i + h𝜒ˆ𝑖(𝜆),𝜒ˆ0
𝑖(𝜆)i] =𝜈𝑖(𝜆)𝜕𝜆h𝜒ˆ𝑖(𝜆),𝜒ˆ𝑖(𝜆)i=0,
which implies (2.19).
To prove Eq. (2.16), we use formula (2.19) above and the normalization of ˆ𝜒𝑖(𝜆) to estimate𝜈0
𝑖(𝜆)as
|𝜈0
𝑖(𝜆) | ≤ k𝑈0(𝜆) k. (2.20) To bound the r.h.s. of (2.20), we use the analyticity of𝑈(𝜆)inΩand estimate (2.3).
Indeed, by the Cauchy integral formula, we have k𝑈0(𝜆) k ≤ 1
𝑅 𝛾0 sup
|𝜆0−𝜆|=𝑅 𝛾0
k𝑈(𝜆0) k,
with𝑅≤𝑎−𝑟, so that{𝜆0∈C: |𝜆0−𝜆|< 𝑅𝛾0} ⊂ Ω, for𝜆∈Ω0. This, together with (2.3) and under the conditions of Lemma2.1, gives the estimate
k𝑈0(𝜆) k ≤ 𝑘 𝑎−𝑟
Φ(𝑊) 𝛾0
, (2.21)
for𝜆∈Ω0. Combining Eqs. (2.20) and (2.21), we arrive at (2.16).
Due to the definition of𝑟after (2.15), we can rewrite estimate (2.13) as
|𝜈𝑖(𝜆) −𝜆0| ≤𝑟 𝛾0.
By (2.15), this shows that𝜈𝑖(𝜆)maps the intervalΩ0into itself which proves (iii).
Hence, under condition (1.4), the fixed point equations 𝜆 = 𝜈𝑖(𝜆) have unique solutions on the intervalΩ0, proving Proposition2.2.
By Corollary1.3, the eigenvalues of𝐻 inΩ0 are in one-to-one correspondence with the solutions of the equations𝜈𝑖(𝜆)=𝜆. By Proposition2.2, these equations have unique solutions, say,𝜆𝑖. Then, estimate (2.13) implies inequality (1.6).
To obtain estimate (1.7), we recall from Theorem1.2that𝑄(𝜆𝑗)𝜑0𝑗 = 𝜑𝑗, where the operator𝑄(𝜆)is given by
𝑄(𝜆):=1−𝑅⊥(𝜆)𝑃⊥𝑊 𝑃,
𝜆𝑗 are the eigenvalues of 𝐻 and 𝜑0𝑗 are eigenfunctions of𝐻0corresponding to𝜆0. This gives
𝜑0𝑗 −𝜑𝑗 =𝜑0𝑗 −𝑄(𝜆𝑗)𝜑0𝑗 =𝑅⊥(𝜆𝑗)𝑃⊥𝑊 𝑃. (2.22) Now, as in the derivation of (2.9), using identity (2.8), estimates (2.5) and
k |𝐻𝜆|−12k ≤ 1
√︁(1−𝑟)𝛾0 and the estimatek𝐾𝜆k ≤ 𝑏
1−𝑎, see (2.6), we find, for𝜆∈Ω0, k𝑅⊥(𝜆)𝐻
1 2
0k ≤ 𝑘 𝜆
1
∗2
𝛾0
, (2.23)
noting that𝑟 < 𝑎. sing inequalities (2.23) and (2.10), we estimate the r.h.s. of (2.22) as k𝑅⊥(𝜆𝑗)𝑃⊥𝑊 𝑃k ≤𝑘
√︄
Φ(𝑊) 𝛾0
.
This, together with (2.22), gives (1.7). This proves Theorem1.1.
Remark 6. Differentiating (2.18) with respect to𝜆, setting𝜆=𝜇, using𝑊𝜆=𝜇=0 and 𝑊0
𝜆=𝑈0(𝜆)and changing the notation𝜇to𝜆, we find the following formula for𝜒0
𝑖(𝜆):
𝜒0
𝑖(𝜆)=−𝑅⊥(𝜈𝑖(𝜆), 𝜆)𝑈0(𝜆)𝜒𝑖(𝜆).
Finally we prove the relations (1.8) and (1.9) stated in the introduction.
Proof of (1.8). Eq. (2.11) gives
𝐻(𝜆)=(𝜆0+ h𝑊i + h𝑈(𝜆)i)𝑃,
where we use the notationh𝐴i:=h𝜑0, 𝐴𝜑0i. Since𝑃is a rank one projection, it follows that𝐻(𝜆)has only one eigenvalue and this eigenvalue is
𝜈1(𝜆)=𝜆0+ h𝑊i + h𝑈(𝜆)i.
This formula and estimate (2.3) show that𝜈1(𝜆)obeys the estimate
|𝜈1(𝜆) −𝜆0− h𝑊i| ≤𝑘Φ(𝑊).
By Proposition2.2, the eigenvalue𝜆1of𝐻satisfies the equation𝜆1=𝜈1(𝜆1). Then
the estimate above implies inequality (1.8).
Proof of Eq(1.9). Let𝑊 be symmetric and denote by 𝜇min the smallest eigenvalue of the matrix𝑃𝑊 𝑃. Then𝑈(𝜆) ≤ 0 and (2.11) implies 𝐻(𝜆) ≤ 𝜆0𝑃+𝑃𝑊 𝑃. This, together with inf𝐴≤inf𝐵for𝐴 ≤𝐵, gives the upper bound𝜈1(𝜆) ≤𝜆0+𝜇minon the smallest eigenvalue-branch𝜈1(𝜆)of𝐻(𝜆).
For the lower bound, Eqs (2.3) and (2.11) imply the following estimate 𝜆0+𝜇min−𝑘Φ(𝑊) ≤𝜈1(𝜆).
The last two estimates and the equation𝜈𝑖(𝜆)=𝜆(see (1.12)) yield (1.9).
3 Application: The ground state energy of the Helium- type ions
In this section, we will use the inequalities obtained above to estimate the ground state energy of the Helium-type ions, which is the simplest not completely solvable atomic quantum system. For simplicity, we assume that the nucleus is infinitely heavy, but we allow for a general nuclear charge𝑧 𝑒, 𝑧 ≥ 1. Then the corresponding Schrödinger operator (describing 2 electrons of mass𝑚and charge−𝑒, and the nucleus of infinite mass and charge𝑧 𝑒) is given by
𝐻𝑧 ,2=
2
∑︁
𝑗=1
− ℏ2 2𝑚
Δ𝑥𝑗− 𝜅 𝑧 𝑒2
|𝑥𝑗|
+ 𝜅 𝑒2
|𝑥1−𝑥2|, (3.1) acting on the space𝐿2
𝑠 𝑦 𝑚(R6)of𝐿2-functions symmetric (or antisymmetric) w.r.t. the permutation of𝑥1and𝑥22. Here𝜅= 4𝜋 𝜀1
0 is Coulomb’s constant and𝜀0is the vacuum
2By the Pauli principle, the product of coordinate and spin wave functions should antisymmetric w.r.t. permutation of the particle coordinates and spins. Hence, in the two particle case, after separation of spin variables, coordinate wave functions could be either antisymmetric or symmetric.
permittivity. For𝑧=2,𝐻𝑧 ,2describes the Helium atom, for𝑧=1, the negative ion of the Hydrogen and for 2< 𝑧≤94 (or≤103, depending on what one counts as stable elements), Helium-type positive ion. (We can call (3.1) with𝑧 >103 a 𝑧-ion.)
It is well known that 𝐻𝑧 ,2has eigenvalues below its continuum. Variational tech- niques give excellent upper bounds on the eigenvalues of𝐻𝑧 ,2, but good lower bounds are hard to come by. Thus, we formulate
Problem 3.1. Estimate the ground state energy of𝐻𝑧 ,2.
The most difficult case is of𝑧=1, the negative ion of the hydrogen, and the problem simplifies as𝑧increases.
Here we present fairly precise bounds on the ground state energy of𝐻𝑧 ,2implied by our actual estimates. However, the conditions under which these estimates are valid impose rather sever restrictions in𝑧. We introduce the reference energy
𝐸ref:= 𝜅2𝑒4𝑚
ℏ2 =𝛼2𝑚 𝑐2
(twice the ground state energy of the hydrogen, or 2Ry), where𝑐is the speed of light in vacuum and
𝛼=𝜅 𝑒2 ℏ𝑐
is the fine structure constant, whose numerical value, approximately 1/137. Let 𝐸(
𝑧)
#
stand for either 𝐸sym
𝑧 ,2/𝐸ref or 𝐸as
𝑧 ,2/𝐸ref, the ground state energy of 𝐻𝑧 ,2 on either symmetric or anti-symmetric functions. We have
Proposition 3.2. Assume 𝑧 ≥ 31.25 for the symmetric space and 𝑧 ≥ 170 for the anti-symmetric one. Then the ground state energy of𝐻𝑧 ,2is bounded as
−𝑐#𝑧2+𝑤#
1𝑧−10𝑤#
2
𝛾#
0
≤𝐸(
𝑧)
# ≤ −𝑐#𝑧2+𝑤#
1𝑧, (3.2)
where, for symmetric functions,𝑐=1,𝛾0= 38and𝑤1≈0.6 and𝑤2≈0.27, defined in equations (3.13) and (3.15), and, for anti-symmetric functions,𝑐as = 58,𝛾as
0 = 725 and 𝑤as
1 ≈0.20 and𝑤as
2 ≈0.01, defined in (3.12).
The approximate values of 𝑤1, 𝑤2, 𝑤as
1 and 𝑤as
2 are computed numerically in AppendixB. Here we report the stable digits of our computations.
The inequalities𝑧 ≥31.25 and𝑧≥170 come from condition (1.3), while estimates (3.2), from (1.8), with𝑏=0.8 and𝑎=0.1 (which give𝑘=10).
Table1compares the result for symmetric functions with computations in quantum chemistry. (We did not find results for the antisymmetric space.)
We observe that, except for 𝑧 = 50, the results of [16,17,18] lie in the interval provided by the estimation (3.2).
𝑧 10 20 30 40 50
−𝐸exact(from [16,17,18]) 93.9 387.7 881.4 1575.2 2468.9 main part−𝐸sym
,lead
𝑧 ,2 in (3.2) 94 388 882 1576 2470
relative difference err𝑧 0.11% 0.077% 0.068% 0.051% 0.045%
relative error termΔ𝑧 in (3.2) 8.51% 2.06% 0.91% 0.51% 0.32%
Table 1: Comparison of non-rigorous computations (see [8,9,16,17,18]) with the main part−𝐸sym
,lead
𝑧 ,2 := 𝑧2−𝑤1𝑧 in equation (3.2), its relative contribution of the error estimation Δ𝑧 := 10𝑤2/(−𝐸sym
,lead
𝑧 ,2 𝛾0) and the relative difference defined by err𝑧 :=|𝐸exact−𝐸sym
,lead
𝑧 ,2 |/(−𝐸sym
,lead 𝑧 ,2 ).
For computations of the relativistic and QED contributions, see [10,18,19].
Now, we derive some consequences of estimates (3.2).
Let𝑧∗be the smallest𝑧for which the error bound in the symmetric case is less than or equal to the smallest explicit (subleading) term𝑤1. Inequality (3.2) shows that the latter bound is satisfied for𝑧such that 𝑤1𝑧 ≥ 10𝛾𝑤2
0 , which shows that𝑧∗ = 10𝑤𝑤2
1𝛾0 and consequently,
𝑧∗≈12.
According to equation (3.2), the symmetric ground state energy is lower than the anti-symmetric one, if −𝑧2+𝑤1𝑧 < −5
8𝑧2+𝑤as
1𝑧 −160𝑤as
2. Using the values 𝑤1 ≈ 0.6, 𝑤2 ≈ 0.3, 𝑤as
1 ≈ 0.20 and 𝑤as
2 ≈ 0.01, we find that, based on (3.2), the ground state is symmetric and therefore its spin is 0 for
𝑧 ≥ 4 3
2 5 +
√︂4 25+96
5
!
= 8 3 ≈2.6.
We conjecture that the symmetric ground state energy is lower than the anti-symmetric one for all𝑧’s.
Finally, note that on symmetric functions, the eigenvalue of the unperturbed oper- ator is simple and on anti-symmetric ones, has the degeneracy 4, see below.
Proof of Proposition3.2. First, we rescale the Hamiltonian (3.1) as𝑥→𝑥/𝜇, with 𝜇:= 𝑧 𝜅 𝑒2𝑚
ℏ2 = 𝑧
the Bohr radius, to obtain
𝑈𝜇−1𝐻𝑧 ,2𝑈𝜇= 𝑧2𝜅2𝑒4𝑚 ℏ2 𝐻(𝑧),
where𝑈𝜇:𝜓(𝑥) →𝜇3𝜓(𝜇𝑥)and𝐻(𝑧)is the rescaled Hamiltonian given by 𝐻(𝑧) =
2
∑︁
𝑗=1
−1
2Δ𝑥𝑗− 1
|𝑥𝑗|
+ 1/𝑧
|𝑥1−𝑥2|.
Thus, it suffices to estimate the ground state energy of𝐻(𝑧). We consider 𝑊= 1/𝑧
|𝑥1−𝑥2| as the perturbation andÍ2
𝑗=1(−1
2Δ𝑥𝑗 − 1
|𝑥𝑖|)as the unperturbed operator.
First, we consider the rescaled Hamiltonian𝐻(𝑧)on thesymmetricfunctions sub- space. On symmetric functions, the ground state energy ofÍ2
𝑗=1(−12Δ𝑥𝑗− |𝑥1
𝑗|)is−1 (see below), so we shift the operator𝐻(𝑧)by 1+𝛽, for some𝛽 >0, so that
𝐻(𝑧)+1+𝛽=𝐻0+𝑊 , with
𝐻0:=
2
∑︁
𝑗=1
−1
2Δ𝑥𝑗− 1
|𝑥𝑗|
+1+𝛽 and 𝑊= 1/𝑧
|𝑥1−𝑥2|.
Now, the ground state energy of𝐻0is𝛽and we can use inequality (1.8) and Proposition 1.4(c) to estimate the ground state energy of𝐻(𝑧)+1+𝛽.
By the HVZ theorem, the spectrum of𝐻0on symmetric functions consists of the continuum[𝑒1+1+𝛽,∞)and the eigenvalues𝑒𝑚+𝑒𝑛+1+𝛽, 𝑚, 𝑛 ≥1, where𝑒𝑛, 𝑛= 1,2, . . . ,denote the discrete eigenvalues of the Hydrogen Hamiltonian−1
2Δ𝑥− 1
|𝑥|. The eigenvalues𝑒𝑛, 𝑛=1,2, . . . ,are known explicitly:
𝑒𝑛 =− 1 2𝑛2
, (3.3)
with the multiplicities of𝑛2 and the corresponding eigenfunctions are given by Eq (A.1) below.
Then the ground state energy of 𝐻0is𝜆0 = 2𝑒1+1+𝛽 = 𝛽, as claimed, and the gap,𝛾0=𝑒1+𝑒2−2𝑒1=𝑒2−𝑒1=3/8.
Next, we show that the condition (1.3) is satisfied for𝑧 ≥31.25. We begin with the really rough estimate:
1
|𝑥1−𝑥2| ≤ ℎ𝑥
1+ℎ𝑥
2+10, (3.4)
whereℎ𝑥 :=−1
2Δ𝑥− 1
|𝑥|. First, we use Hardy’s inequality,−Δ≥ 1
4|𝑥|2, and the estimate
1 4𝑚|𝑥|2 − 1
|𝑥| ≥ −𝑚to obtain
− 1 𝑚
Δ− 1
|𝑥| ≥ −𝑚 . (3.5)
Next, passing to the relative and centre-of-mass coordinates,𝑥−𝑦and 12(𝑥+𝑦), we find
−Δ𝑥−Δ𝑦=−2Δ𝑥−𝑦−1 2Δ1
2(𝑥+𝑦) ≥ −2Δ𝑥−𝑦.
which, together with (3.5), yields 1
|𝑥1−𝑥2| ≤ −1
2Δ𝑥1−𝑥2 +2≤ −1
4(Δ𝑥1 +Δ𝑥2) +2. Now, we use−1
2Δ𝑥=ℎ𝑥+ |1𝑥| ≤ℎ𝑥−1
4Δ𝑥+4 to obtain
−1
4Δ𝑥 ≤ℎ𝑥+4.
The last two inequalities yield (3.4). Eq. (3.4), together with the relation 𝐻0=ℎ𝑥
1+ℎ𝑥
2+1+𝛽, implies the estimate:
1
|𝑥1−𝑥2| ≤𝐻0+9−𝛽. (3.6)
Now, we check the condition (1.3):
k𝑃⊥𝑊 𝑃⊥k𝐻0 ≤ 𝑏 𝛾0 𝜆∗
. Using the last relation, we estimate
𝑧(𝐻⊥
0)−12𝑊(𝐻⊥
0)−12 ≤ (𝐻⊥
0)−1(𝐻⊥
0 +9−𝛽)
=1+ (9−𝛽) (𝐻⊥
0)−1. Recalling that𝜆0= 𝛽, we see that𝐻⊥
0 ≥ 𝛽+𝛾0, so that the last inequality gives 0≤ (𝐻⊥0)−12𝑊(𝐻0⊥)−12 ≤ 1
𝑧 9+𝛾0 𝛽+𝛾0 , provided𝛽 <9. This implies
k𝑃⊥𝑊 𝑃⊥k𝐻0 ≤ 1 𝑧
9+𝛾0 𝛽+𝛾0
. (3.7)
Since𝜆∗ = 𝛽+𝛾0, condition (1.3) is satisfied, if 9+𝑧𝛾0 ≤ 𝑏 𝛾0, which gives 𝑧 ≥ 9+𝑏 𝛾𝛾0
0. Since𝛾0=3/8 and𝑏=0.8, this implies𝑧≥31.25. We will need the following lemma, whose proof is given after the proof of the proposition.
Lemma 3.3. Recall that 𝑃 is the orthogonal projection onto the eigenspace of 𝐻0 corresponding to the lowest eigenvalue𝛽. We have
h𝑊i=k𝑃𝑊 𝑃k = 𝑤1
𝑧 and Φ(𝑊) ≤ 𝑤2 𝑧2𝛾0
, (3.8)
with, recall, h𝑊i := h𝜑0, 𝑊 𝜑0i, where𝜑0is the eigenfunction of 𝐻0 corresponding to𝜆0.
We check now the second condition of Theorem1.1, (1.4). Recall the definition 𝑟 := k𝑃𝑊 𝑃k +𝑘Φ(𝑊)
𝛾0
. Then condition (1.4) can be written as𝑟 < 𝑎. By Lemma3.3,
𝑟 ≤ 1 𝛾0
𝑤1 𝑧
+ 𝑘 𝑤2 𝑧2𝛾0
.
Recall the condition𝑧≥31.25 and𝑘=10,𝑤1≈0.6 and𝑤2≈0.3. Thus 𝑘 𝑤2
𝑧2𝛾0
≤ 𝑤1 𝑧
𝑘 𝑤2 31.25𝛾0𝑤1
≤0.5𝑤1 𝑧
. Then we find𝑟 ≤ 1.5𝑧 𝛾𝑤1
0 and therefore condition (1.4) is satisfied if𝑎 > 1.5𝑧 𝛾𝑤1
0, or (for𝑎=0.1)
𝑧 >
3 2 8 3
0.6 𝑎
=24. This is less restrictive than𝑧 ≥31.25.
For the third condition (1.5), recalling that𝜆0= 𝛽, so that𝜆∗ = 𝛾0+𝛽(=𝜆1, the second eigenvalue of𝐻0) and using both relations in (3.8), we obtain that condition (1.5) is satisfied if𝑘 𝑤2/(𝑧2𝛾0) < 1
2(𝑎 𝛾0−𝑤1/𝑧), which is equivalent to 𝑧 >
1 2𝑎 𝛾0
𝑤1+
√︃
𝑤2
1+8𝑎 𝑘 𝑤2
.
Using the values𝛾0 = 38,𝑤1 ≈0.6 and 𝑤2 ≈ 0.3, and 𝑏 = 0.8 and𝑎 = 0.1, giving 𝑘 =10, this shows that the latter inequality, and therefore (1.5), holds if𝑧 >30, which is less restrictive than 𝑧≥31.25. Thus, all three conditions are satisfied for𝑧≥31.25.
Next, we use (1.8) to estimate the ground state energy,𝐸(
𝑧)
symof𝐻(𝑧). The first and second relations in (3.8), together with (1.8) and the fact that𝑈(𝜆) ≤ 0, gives the following bounds
−𝑘 𝑤2 𝛾0𝑧2
≤𝐸(
𝑧)
sym+1− 𝑤1 𝑧
≤0. which after rescaling gives (3.2).
Now, we consider the ground state of 𝐻(𝑧) onanti-symmetric functions. In this case, the ground state energy ofÍ2
𝑗=1(−12Δ𝑥𝑗− |𝑥1
𝑖|)is𝑒1+𝑒2=−58of multiplicity 4, with the ground states
𝜙2ℓ 𝑘(𝑥1, 𝑥2):= 1
√ 2
(𝜓100(𝑥1)𝜓2ℓ 𝑘(𝑥2) −𝜓2ℓ 𝑘(𝑥1)𝜓100(𝑥2)), (3.9) for(ℓ, 𝑘) = (0,0),(1,−1),(1,0),(1,1), where𝜓𝑛ℓ 𝑘(𝑥) forℓ = 0,1,2, . . . , 𝑛−1, 𝑘 =
−ℓ,−ℓ+1, . . . , ℓ, 𝑛=1,2, . . ., are the eigenfunctions of the Hydrogen-like Hamiltonian
