The Feshbach-Schur Map and Perturbation Theory Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

The Feshbach-Schur map and perturbation theory 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

HAL Id: hal-03216856 https://hal.archives-ouvertes.fr/hal-03216856 Submitted on 4 May 2021

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Feshbach-Schur map and perturbation 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 operators. 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: [email protected] . 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: [email protected] . Supported in part by NSERC Grant No. NA7901. B. Stamm: Center for Computational Engineering Science, RWTH Aachen University, 52062 Aachen, Germany; email: [email protected].

1 2 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm 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

퐻 = 퐻0 + 푊, (1.1) where 퐻0 is an operator with some isolated eigenvalues and 푊 is an operator, small relative to 퐻0 in an appropriate norm. The goal is to show that 퐻 has eigenvalues near those of 퐻0 and estimate these eigenvalues. Speciﬁcally, with k · k standing for the vector and operator norms in the underlying Hilbert space, we assume that (A) 퐻0 is a self-adjoint, non-negative operator (퐻0 ≥ 훽 > 0); (B) 푊 is symmetric and form-bounded with respect to 퐻0, in the sense that

− 1 − 1 k 2 2 k ∞ 퐻0 푊퐻0 < .

(C) 퐻0 has an isolated eigenvalue 휆0 > 0 of a ﬁnite multiplicity, 푚. k · k −푠 Here is the operator norm and 퐻0 , 0 < 푠 < 1, is deﬁned either by the spectral theory or by the explicit formula

∫ ∞ 푑휔 −푠 = ( + )−1 퐻0 : 푐 퐻0 휔 푠 , 0 휔

h ∞ i−1 = ∫ ( + )−1 푑휔 where 푐 : 0 1 휔 휔푠 . It turns out to be useful in the proofs below to use the following form-norm

− 1 − 1 k k = k 2 2 k 푊 퐻0 : 퐻0 푊퐻0 ,

Let 푃 be the orthogonal projection onto the span of the eigenfunctions of 퐻0 ⊥ corresponding to the eigenvalue 휆0 and let 푃 := 1 − 푃. Let 훾0 be the distance of 휆0 to The FS-map and perturbation theory 3 the rest of the spectrum of 퐻0, and 휆∗ := 휆0 + 훾0. In what follows, we often deal with the expression 휆 휆 Φ(푊) := 0 ∗ k푃⊥푊푃k2 . (1.2) 퐻0 훾0 The following theorem proven in Section2 is the main result of this paper. Main Theorem 1.1. Let Assumptions (A)-(C) be satisﬁed and assume that, for some 0 <푏< 1 and 0 <푎< 1 − 푏,

⊥ ⊥ 푏훾0 k푃 푊푃 k퐻0 ≤ , (1.3) 휆∗ k푃푊푃k + 푘Φ(푊) < 푎훾0, (1.4) 1 푘Φ(푊) < 2 (푎훾0 − k푃푊푃k), (1.5)

1 where 푘 := 1−푎−푏 . Then the spectrum of the operator 퐻 near 휆0 consists of isolated eigenvalues 휆푖 of the total multiplicity 푚 satisfying, together with their normalized eigenfunctions 휑푖, the following estimates

|휆푖 − 휆0| ≤ k푃푊푃k + 푘Φ(푊), (1.6) √︄ Φ(푊) k휑푖 − 휑0푖 k ≤ 푘 , (1.7) 훾0 where 휑0푖 are appropriate eigenfunctions of 퐻0 corresponding to the eigenvalue 휆0. 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 on 푊) The ﬁne tuning the conditions (1.3)-(1.5) on 푊 is used in application of this theorem to atomic systems in Section3. Note that because of the elementary estimate 1 Φ(푊) ≤ k푃푊푃⊥푊푃k, 훾0 see (3.11) below, the computation of k푃푊푃k and Φ(푊) reduces to computing the largest eigenvalues of the simple 푚 × 푚-matrices 푃푊푃, −푃푊푃 and 푃푊푃⊥푊푃.

Remark 3 (Higher-order estimates and non-degenerate 휆0). In fact, one can estimate ⊥ 휆푖 (as well as the eigenfunctions 휑푖) to an arbitrary order in k푃 푊푃k퐻0 /훾0. As a demonstration, we derive (after the proof of Theorem 1.1) the second-order estimate of the eigenvalue in the rank-one (푚 = 1) case

|휆1 − 휆0 − h푊i| ≤ 푘Φ(푊), (1.8) 4 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm where 휑0 is the eigenfunction of 퐻0 corresponding to 휆0, h푊i := h휑0, 푊휑0i. For degenerate 휆0, we would like to prove a similar bound on |휆푖 − 휆0 − 휇푖 |, where 휇푖 is the corresponding eigenvalue of the 푚 × 푚-matrix 푃푊푃. Here, we have a partial result (proven at the end of the next section) for the lowest eigenvalue 휆1 of 퐻:

휆0 + 휇min − 푘Φ(푊) ≤ 휆1 ≤휆0 + 휇min, (1.9) where 휇min denotes the smallest eigenvalue of the matrix 푃푊푃. Remark 4 (Non-self-adjoint 퐻). With the sacriﬁce of the explicit constants in (1.6) and (1.7) (mostly coming from (2.5)), the self-adjointness assumption on 퐻 can be removed. However, for the problem of quantum resonances, one can still obtain explicit estimates. In the rest of this section 퐻 is 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. Let 퐻 be an operator on a Hilbert space and 푃 and 푃⊥, a pair of projections such that 푃 + 푃⊥ = 1. Assume 퐻⊥ := 푃⊥퐻푃⊥ is invertible on Ran 푃⊥ and the expression

⊥ 퐹푃 (퐻) := 푃(퐻 − 퐻푅 퐻)푃, (1.10)

⊥ ⊥ ⊥ −1 ⊥ where 푅 := 푃 (퐻 ) 푃 , deﬁnes a bounded operator. Then 퐹푃, considered as a map on the space of operators, is isospectral in the following sense:

(a) 휆 ∈ 휎(퐻) if and only if 0 ∈ 휎(퐹푃 (퐻 − 휆));

(b) 퐻휓 = 휆휓 if and only if 퐹푃 (퐻 − 휆) 휑 = 0;

(d) 휓 and 휑 in (b) are related as 휑 = 푃휓 and 휓 = 푄푃 (휆)휑, where

⊥ ⊥ −1 ⊥ 푄푃 (휆) := 푃 − 푃 (퐻 − 휆) 푃 퐻푃. (1.11)

∗ ∗ Finally, 퐹푃 (퐻) = 퐹푃∗ (퐻 ) and therefore, if 퐻 and 푃 are self-adjoint, then so is 퐹푃 (퐻).

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 퐹푃 on the space of operators, is called the Feshbach-Schur map. The relation 휓 = 푄푃 (휆)휑 allows us to reconstruct the full eigenfunction from the projected one. By statement (a), we have The FS-map and perturbation theory 5

Corollary 1.3. Assume there is an open set Λ ⊂ C such that 퐻 − 휆, 휆 ∈ Λ, is in the domain of the map 퐹푃, i.e. 퐹푃 (퐻 − 휆) is well deﬁned. Deﬁne the operator-family

퐻(휆) := 퐹푃 (퐻 − 휆) + 휆푃, and let 휈푖 (휆), for 휆 in Λ, denote its eigenvalues counted with multiplicities. Then the eigenvalues of 퐻 in Λ are in one-to-one correspondence with the solutions of the equations

휈푖 (휆) = 휆. (1.12) Concentrating on the eigenvalue problem, Corollary 1.3 shows that the original problem 퐻휓 = 휆휓, (1.13) is mapped into the equivalent eigenvalue problem, 퐻(휆)휑 = 휆휑, (1.14) nonlinear in the spectral parameter 휆, but on the smaller space Ran 푃. Since the projection 푃 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 Ran 푃⊥. We call this approach the Feshbach-Schur map method, or FSM method, for short. It is rather compact, as one easily see skimming through this paper and entirely elementary. We call 퐻(휆) the effective Hamiltonian (matrix) and write as

퐻(휆) = 푃퐻푃 + 푈(휆), (1.15) with the self-adjoint effective interaction, or a Schur complement, 푈(휆), defined as 푈(휆) := −푃퐻푃⊥ (퐻⊥ − 휆)−1푃⊥퐻푃. (1.16) It is shown in Lemma 2.1 below that (1.16) defines a bounded operator family. We mention here some additional properties discussed in [3]. Proposition 1.4. Let 퐻 be self-adjoint, let Λ be the same as in Corollary 1.3, let 푚 be the rank of 푃 and let the eigenvalues of 퐻(휆), 휆 ∈ Λ ∩ R, be labeled in the order of their increase counting their multiplicities so that

휈1 (휆) ≤ ... ≤ 휈푚 (휆). (1.17)

Then we have that (a) a solution of the equation 휈푖 (휆) = 휆, for 휆 ∈ Λ ∩ R, is the 0 푖-th eigenvalue, 휆푖, of 퐻 and vice versa; (b) 휈푖 is diﬀerentiable in 휆 and 휈푖 (휆)≤0, for 휆 ∈ Λ ∩ R. 6 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

Proof. (a) By (1.17), 휆푖 = 휈푖 (휆푖) < 휈푖+1 (휆) = 휆푖+1 (except for the level crossings), which proves the result. For (b), by the explicit formula

푈0(휆) := −푃퐻푃⊥ (퐻⊥ − 휆)−2푃⊥퐻푃 ≤ 0, we have 푈0(휆) ≤ 0, which by the Hellmann-Feynman theorem (cf. (2.19)) implies 0 휈푖 (휆) ≤ 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, 푃 is the orthogonal projection onto Null(퐻0 − 휆0), 푃⊥ := 1 − 푃 and 푈(휆) can be written as

푈(휆) := −푃푊푃⊥ (퐻⊥ − 휆)−1푃⊥푊푃.

⊥ ⊥ ⊥ Now, using the notation 퐴 := 푃 퐴푃 |Ran 푃⊥ and expanding

⊥ −1 ⊥ ⊥ −1 (퐻 − 휆) = (퐻0 + 푊 − 휆) in 푊⊥ and at the same time iterating ﬁxed point equation (1.12), we generate a perturbation expansion for eigenvalues 퐻 to an arbitrary order (see also Remark 2 above). The paper is organized as follows. In Section2, we present the proof of the main result, Theorem 1.1. Section3 uses this result to obtain bounds of the ground-state energy of Helium-type ions.

2 Perturbation estimates

We want to use Theorem 1.2(b, c) to reduce the original eigenvalue problem to a simpler one. In this section, we assume that Conditions (A)-(C) of Section1 are satisfied. Recall that 훾0 is the distance of 휆0 to the rest of the spectrum of 퐻0 and the expression Φ(푊) is defined in (1.2). First, we prove that the operator 퐹푃 (퐻 − 휆) is well-defined for 휆 ∈ Ω, where, with 푎 the same as in Theorem 1.1,

Ω := {푧 ∈ C : |푧 − 휆0| ≤ 푎훾0}. (2.1)

⊥ Recall that 푃 denotes the orthogonal projection onto Null(퐻0 − 휆0) and 푃 := 1 − 푃. Denote

⊥ ⊥ ⊥ ⊥ ⊥ ⊥ −1 ⊥ 퐻 := 푃 퐻푃 |Ran 푃⊥ and 푅 (휆) := 푃 (퐻 − 휆) 푃

1 and recall 푘 = 1−푎−푏 . We have

Lemma 2.1. Recall 휆∗ := 휆0 + 훾0 and assume (1.3). Then, for 휆 ∈ Ω, the following statements hold (a) The operator 퐻⊥ − 휆 is invertible on Ran 푃⊥; The FS-map and perturbation theory 7

(b) The inverse 푅⊥ (휆) := 푃⊥ (퐻⊥ − 휆)−1푃⊥ defines a bounded, analytic operator- family; (c) The expression 푈(휆) := −푃퐻푅⊥ (휆)퐻푃 (2.2) defines a finite-rank, analytic operator-family, bounded as

k푈(휆)k ≤ 푘Φ(푊). (2.3)

(d) 푈(휆) is symmetric for any 휆 ∈ Ω ∩ R and therefore 퐻(휆) is self-adjoint as well.

⊥ ⊥ ⊥ Proof. With the notation 퐴 := 푃 퐴푃 |Ran 푃⊥ , we write ⊥ ⊥ ⊥ 퐻 = 퐻0 + 푊 .

1 1 ⊥ 2 2 To prove (a), we let 퐻휆 := 퐻0 − 휆 and use the factorization 퐻휆 = |퐻휆| 푉휆|퐻휆| , where 푉휆 is a unitary operator, and use that, for 휆 ∈ Ω, the operator 퐻휆 is invertible and therefore we have the identity

⊥ 1 1 퐻 − 휆 = |퐻휆| 2 [푉휆 + 퐾휆]|퐻휆| 2 , (2.4)

1 1 − 2 ⊥ − 2 ⊥ −1 ⊥ where 퐾휆 := |퐻휆| 푊 |퐻휆| . Next, for 휆 ∈ Ω, we have k퐻0 |퐻휆| 푃 k = 푓 (휆), where 푠 푓 (휆) := sup : 푠 ≥ 0, |푠 − 휆 | ≥ 훾 . 푠 − 휆 0 0

Assuming |휆 − 휆0| ≤ 푎훾0 and using

푠 휆 휆0 + 푎훾0 휆∗ ≤ 1 + ≤ 1 + = , 푠 − 휆 푠 − 휆 (1 − 푎)훾0 (1 − 푎)훾0 we obtain 휆 푓 (휆) ≤ ∗ . (1 − 푎)훾0

1 1 1 2 − ⊥ −1 ⊥ | | 2 ( | | | ⊥ ) 2 ∈ Since 퐻0 퐻휆 푃 = 퐻0 퐻휆 Ran 푃 푃 , we have for 휆 Ω,

1 1 2 − 1 ⊥ 휆∗ k 2 | | 2 k ≤ 퐻0 퐻휆 푃 , (2.5) (1 − 푎)훾0 which implies in particular that k퐾 k ≤ 휆∗ k푊⊥ k . By the assumption (1.3), i.e., 휆 (1−푎)훾0 퐻0 k푊⊥ k ≤ 푏훾0 , we have 퐻0 휆∗ 푏 k퐾 k ≤ . (2.6) 휆 1 − 푎 8 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

Since 1 − 푎 > 푏, by (2.4), the operator 퐻⊥ − 휆 is invertible and its inverse is analytic in 휆 ∈ Ω, which proves (a) and (b). ⊥ We show that statement (c) is also satisﬁed. Since 푃퐻0 = 퐻0푃 and 푃푃 = 0, we have

푃퐻푃⊥ = 푃푊푃⊥, 푃⊥퐻푃 = 푃⊥푊푃.

These relations and deﬁnition (2.2) yield

푈(휆) = −푃푊 푅⊥ (휆)푊푃. (2.7)

Inverting (2.4) on Ran 푃⊥ and recalling the notation 푅⊥ (휆) := 푃⊥ (퐻⊥ − 휆)−1푃⊥ gives

⊥ − 1 ⊥ −1 ⊥ − 1 푅 (휆) = |퐻휆| 2 푃 [푉휆 + 퐾휆] 푃 |퐻휆| 2 . (2.8)

Now, using identity (2.8), estimate (2.5) and (2.6), we ﬁnd, for 휆 ∈ Ω,

1 1 푘휆 k 2 ⊥ ( ) 2 k ≤ ∗ 퐻0 푅 휆 퐻0 . (2.9) 훾0

Furthermore, by the eigen-equation 퐻0푃 = 휆0푃, we have

1 k 2 k2 퐻0 푃 = 휆0. (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 of 푅⊥ (휆). For (d), since 퐻0, 푊 and 푃 are self-adjoint, then so are 푈(휆), for any 휆 ∈ Ω ∩ R, and, since 푈(휆) is bounded, 퐻(휆) is self-adjoint as well. Proof of Theorem 1.1. Let Ω be given by equation (2.1). Recall that, by Lemma 2.1 and Theorem 1.2, the 푚 × 푚 matrix-family 퐻(휆) := 퐹푃 (퐻 − 휆) + 휆푃, with 퐹푃 given in (1.10), is well deﬁned, for each 휆 ∈ Ω, and can be written as (1.15). Since 푃퐻푃 = 휆0푃 + 푃푊푃, Eq. (1.15) can be rewritten as

퐻(휆) = 휆0푃 + 푃푊푃 + 푈(휆). (2.11)

Eqs (2.3) and (2.11) imply the inequality

k퐻(휆) − 휆0푃 − 푃푊푃k ≤ 푘Φ(푊). (2.12)

By a fact from Linear Algebra, for each 휆 ∈ Ω ∩ R, the total multiplicity of the eigenvalues of the 푚 × 푚 self-adjoint matrix 퐻(휆) is 푚. Denote by 휈푖 (휆), 푖 = 1, 2, . . . , 푚, the eigenvalues of 퐻(휆), repeated according to their multiplicities. Eq (2.12) yields

|휈푖 (휆) − 휆0| ≤ k푃푊푃k + 푘Φ(푊). (2.13) The FS-map and perturbation theory 9

Indeed, let 푃푖 (휆) be the orthogonal projection onto Null(퐻(휆) − 휈푖 (휆)). Then

퐻(휆)푃푖 (휆) = 휈푖 (휆)푃푖 (휆), which, due to (2.11) and 푃푖 (휆)푃 = 푃푖 (휆), can be rewritten as

(휈푖 (휆) − 휆0)푃푖 (휆) = (푃푊푃 + 푈(휆))푃푖 (휆).

Equating the operator norms of both sides of this equation and using (2.12) and k푃푖 (휆)k = 1 gives (2.13). By Corollary 1.3, the eigenvalues of 퐻 in the interval Ω ∩ R are in one-to-one correspondence with the solutions of the equations

휈푖 (휆) = 휆 (2.14) in Ω ∩ R. If this equation has a solution, then, due to (2.13), this solution would satisfy (1.6). Thus, we address (2.14). Let

0 Ω := {푧 ∈ R : |푧 − 휆0| ≤ 푟훾0}, (2.15) with 1 푟 := k푃푊푃k + 푘Φ(푊). 훾0 By our assumption (1.4), 푟 < 푎 < 1 − 푏. 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 of 퐻(휆) could be chosen in a diﬀerentiable way and estimates their derivatives.

Proposition 2.2. The following statements hold.

i) The eigenvalues 휈푖 (휆) of 퐻(휆) and the corresponding eigenfunctions can be chosen to be diﬀerentiable for 휆 ∈ Ω0.

0 0 ii) The derivatives 휈푖 (휆), 휆 ∈ Ω , are bounded as

0 푘 Φ(푊) |휈푖 (휆)| ≤ . (2.16) 푎 − 푟 훾0

0 iii) 휈푖 (휆) maps the interval Ω into itself.

Consequently, since by (1.5) the r.h.s. of (2.16) is < 1, the equations 휈푖 (휆) = 휆 have unique solutions in Ω0. 10 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

Proof. Proof of (i) for simple 휆0. For a simple eigenvalue 휆0, 푃 is a rank-one projection on the space spanned by the eigenvector 휑0 of 퐻0 corresponding to the eigenvalue 휆0 and therefore Eq. (2.11) implies that 퐻(휆) = 휈(휆)푃, with

휈(휆) := 휆0 + h휑0, (푊 + 푈(휆))휑0i. This and Lemma 2.1 show that the eigenvalue 휈(휆) is analytic. 0 Proof of (i) for degenerate 휆0. We pick an arbitrary point 휇 in Ω and let 푃휇푖 be the orthogonal projections onto the eigenspaces of 퐻(휇) corresponding to the eigenvalues 휈푖 (휇), i.e. for a ﬁxed 푖, 푃휇푖 projects on the spans of all eigenvectors with the same eigenvalue 휈푖 (휇). We now show that, in a neighbourhood of 휇, the eigenfunctions 휒푖 (휆) can be chosen in a diﬀerentiable way. We introduce the system of equations for the eigenvalues 휈푖 (휆) and corresponding eigenfunctions 휒푖 (휆): h휒 (휆), 퐻(휆)휒 (휆)i 휈 (휆) = 푖 푖 , (2.17) 푖 2 k휒푖 (휆)k ⊥ 휒푖 (휆) = 휒푖 (휇) − 푅 (휈푖 (휆), 휆)푊휆 휒푖 (휇), (2.18) where

⊥ ⊥ ⊥ ⊥ −1 ⊥ 푅 (휈, 휆) := 푃휇푖 (푃휇푖 퐻(휆)푃휇푖 − 휈) 푃휇푖, 푊휆 := 푈(휆) − 푈(휇).

The expression on the right side of (2.18) is the 푄-operator for 퐻(휆) and 푃휇푖 defined according to (1.11) and applied to 휒푖 (휇). Note that systems (2.17)-(2.18) for different ⊥ indices 푖 are not coupled. Furthermore, since 휒푖 (휇) and 푅 (휈푖 (휆), 휆)푊휆 휒푖 (휇) are orthogonal, we have k휒푖 (휆)k ≥ k 휒푖 (휇)k; and 휒푖 (휆) and 휒푗 (휆) are almost orthogonal for 푖 ≠ 푗. Assuming this system has a solution (휈푖 (휆), 휒푖 (휆)), we see that, by Theorem 1.2(d), with 푃 = 푃휇푖, 휒푖 (휆) are eigenfunctions of 퐻(휆) with the eigenvalues 휈푖 (휆) and (2.17) follows from the eigen-equation 퐻(휆)휒푖 (휆) = 휈푖 (휆) 휒푖 (휆). For each 푖, we can reduce system (2.17)-(2.18) to the single equation for 휈푖 (휆), by treating 휒푖 (휆) in (2.17) as given by (2.18). This leads to the fixed point problem for the functions 휈푖 (휆):

휈 = 퐹푖 (휈, 휆), where h휒 (휈, 휆), 퐻(휆)휒 (휈, 휆)i 퐹 (휈, 휆) := 푖 푖 , 푖 2 k 휒푖 (휈, 휆)k ⊥ 휒푖 (휈, 휆) := 휒푖 (휇) − 푅 (휈, 휆)푊휆 휒푖 (휇). ⊥ ⊥ ⊥ Notice that since 푃휇푖 퐻(휆)푃휇푖 are self-adjoint, the resolvent 푅 (휈, 휆) and its derivatives in 휈 are uniformly bounded in a neighbourhood of (휈푖 (휇), 휇), which does not contain branch points, except, possibly, for 휇. To be more speciﬁc, the resolvent 푅⊥ (휈, 휆) is uniformly bounded in a neighbourhood Ω휇푖 of (휈푖 (휇), 휇) whether 휇 is a branch point or not, as long as Ω휇푖 does not contain any other branch point than 휇. The FS-map and perturbation theory 11

Hence 퐹푖 (휈, 휆) is diﬀerentiable in 휈 and 휆 and |휕휈 퐹푖 (휈, 휆)| → 0, as 휆 → 휇 (since ⊥ 2 휕휈 휒푖 (휈, 휆) = 푅 (휈, 휆) 푊휆 휒푖 (휇) and therefore

1 1 ⊥ 2 2 2 k휕휈 휒푖 (휈, 휆)k ≤ k푅 (휈, 휆) (퐻0 + 훼) kk푊휆 k퐻0,훼 k(퐻0 + 훼) 휒푖 (휇)k → 0, as 휆 → 휇). Moreover, 휈푖 (휇) − 퐹푖 (휈푖 (휇), 휇) = 0. Let 푓푖 (휈, 휆) := 휈 − 퐹푖 (휈, 휆). Then, by the above, 푓푖 (휈푖 (휇), 휇) = 0 and |휕휈 푓푖 (휈, 휆) − 1| → 0, as 휆 → 휇. Thus, the implicit function theorem is applicable to the equation 푓푖 (휈, 휆) = 0 and shows that there is a unique solution 휈푖 (휆) in a neighbourhood of (휈푖 (휇), 휇) and that this solution is diﬀerentiable in 휆. Next, we have Lemma 2.3. We have, for 휆 ∈ Ω0, h휒 (휆), 푈0(휆)휒 (휆)i 휈0(휆) = 푖 푖 . (2.19) 푖 2 k휒푖 (휆)k (Eq. (2.19) is closely related to the widely used Hellmann-Feynman theorem.)

Proof. Let 휒ˆ (휆) := 휒푖 (휆) . Writing equation (2.17) as 푖 k휒푖 (휆) k

휈푖 (휆) = h휒ˆ푖 (휆), 퐻(휆)휒ˆ푖 (휆)i and diﬀerentiating this with respect to 휆, we obtain 0 0 휈푖 (휆) = h휒ˆ푖 (휆), 퐻 (휆)휒ˆ푖 (휆)i + 휂(휆), 0 0 where 휂(휆) := h휒ˆ푖 (휆), 퐻(휆)휒ˆ푖 (휆)i + h휒ˆ푖 (휆), 퐻(휆)휒ˆ푖 (휆)i. Now, moving 퐻(휆) in the last term to the l.h.s. and using that 퐻(휆)휒ˆ푖 (휆) = 휈푖 (휆) 휒ˆ푖 (휆), we ﬁnd 0 0 휂(휆) = 휈푖 (휆)[h휒ˆ푖 (휆), 휒ˆ푖 (휆)i + h휒ˆ푖 (휆), 휒ˆ푖 (휆)i] = 휈푖 (휆)휕휆h휒ˆ푖 (휆), 휒ˆ푖 (휆)i = 0, which implies (2.19).

To prove Eq. (2.16), we use formula (2.19) above and the normalization of 휒ˆ푖 (휆) 0 to estimate 휈푖 (휆) as 0 0 |휈푖 (휆)| ≤ k푈 (휆)k. (2.20) To bound the r.h.s. of (2.20), we use the analyticity of 푈(휆) in Ω and estimate (2.3). Indeed, by the Cauchy integral formula, we have 1 k푈0(휆)k ≤ sup k푈(휆0)k, 0 푅훾0 |휆 −휆|=푅훾0 0 0 0 with 푅 ≤ 푎 − 푟, so that {휆 ∈ C : |휆 − 휆| < 푅훾0} ⊂ Ω, for 휆 ∈ Ω . This, together with (2.3) and under the conditions of Lemma 2.1, gives the estimate 푘 Φ(푊) k푈0(휆)k ≤ , (2.21) 푎 − 푟 훾0 12 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm for 휆 ∈ Ω0. Combining Eqs. (2.20) and (2.21), we arrive at (2.16). Due to the deﬁnition of 푟 after (2.15), we can rewrite estimate (2.13) as

|휈푖 (휆) − 휆0| ≤ 푟훾0.

0 By (2.15), this shows that 휈푖 (휆) maps the interval Ω into itself which proves (iii). Hence, under condition (1.4), the ﬁxed point equations 휆 = 휈푖 (휆) have unique solutions on the interval Ω0, proving Proposition 2.2.

By Corollary 1.3, the eigenvalues of 퐻 in Ω0 are in one-to-one correspondence with the solutions of the equations 휈푖 (휆) = 휆. By Proposition 2.2, these equations have unique solutions, say, 휆푖. Then, estimate (2.13) implies inequality (1.6). To obtain estimate (1.7), we recall from Theorem 1.2 that 푄(휆 푗 )휑0 푗 = 휑 푗 , where the operator 푄(휆) is given by

푄(휆) := 1 − 푅⊥ (휆)푃⊥푊푃,

휆 푗 are the eigenvalues of 퐻 and 휑0 푗 are eigenfunctions of 퐻0 corresponding to 휆0. This gives

⊥ ⊥ 휑0 푗 − 휑 푗 = 휑0 푗 − 푄(휆 푗 )휑0 푗 = 푅 (휆 푗 )푃 푊푃. (2.22)

Now, as in the derivation of (2.9), using identity (2.8), estimates (2.5) and

− 1 1 k| | 2 k ≤ 퐻휆 √︁ (1 − 푟)훾0

푏 0 and the estimate k퐾휆 k ≤ 1−푎 , see (2.6), we ﬁnd, for 휆 ∈ Ω ,

1 1 푘휆 2 k ⊥ ( ) 2 k ≤ ∗ 푅 휆 퐻0 , (2.23) 훾0 noting that 푟 < 푎. sing inequalities (2.23) and (2.10), we estimate the r.h.s. of (2.22) as √︄ ⊥ ⊥ Φ(푊) k푅 (휆 푗 )푃 푊푃k ≤ 푘 . 훾0

This, together with (2.22), gives (1.7). This proves Theorem 1.1.

Remark 6. Diﬀerentiating (2.18) with respect to 휆, setting 휆 = 휇, using 푊휆=휇 = 0 and 0 0 0 푊휆 = 푈 (휆) and changing the notation 휇 to 휆, we ﬁnd the following formula for 휒푖 (휆): 0 ⊥ 0 휒푖 (휆) = −푅 (휈푖 (휆), 휆)푈 (휆)휒푖 (휆).

Finally we prove the relations (1.8) and (1.9) stated in the introduction. The FS-map and perturbation theory 13

Proof of (1.8). Eq. (2.11) gives

퐻(휆) = (휆0 + h푊i + h푈(휆)i)푃, where we use the notation h퐴i := h휑0, 퐴휑0i. Since 푃 is a rank one projection, it follows that 퐻(휆) has only one eigenvalue and this eigenvalue is

휈1 (휆) = 휆0 + h푊i + h푈(휆)i.

This formula and estimate (2.3) show that 휈1 (휆) obeys the estimate

|휈1 (휆) − 휆0 − h푊i| ≤ 푘Φ(푊).

By Proposition 2.2, the eigenvalue 휆1 of 퐻 satisﬁes the equation 휆1 = 휈1 (휆1). Then the estimate above implies inequality (1.8).

Proof of Eq (1.9). Let 푊 be symmetric and denote by 휇min the smallest eigenvalue of the matrix 푃푊푃. Then 푈(휆) ≤ 0 and (2.11) implies 퐻(휆) ≤ 휆0푃 + 푃푊푃. This, together with inf 퐴 ≤ inf 퐵 for 퐴 ≤ 퐵, gives the upper bound 휈1 (휆) ≤ 휆0 + 휇min on the smallest eigenvalue-branch 휈1 (휆) of 퐻(휆). For the lower bound, Eqs (2.3) and (2.11) imply the following estimate

휆0 + 휇min − 푘Φ(푊) ≤ 휈1 (휆).

The last two estimates and the equation 휈푖 (휆) = 휆 (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 charge 푧푒, 푧 ≥ 1. Then the corresponding Schrödinger operator (describing 2 electrons of mass 푚 and charge −푒, and the nucleus of inﬁnite mass and charge 푧푒) is given by

2 ∑︁ ℏ2 휅푧푒2 휅푒2 퐻 = − Δ − + , (3.1) 푧,2 2푚 푥 푗 |푥 | |푥 − 푥 | 푗=1 푗 1 2

2 6 2 acting on the space 퐿푠푦푚 (R ) of 퐿 -functions symmetric (or antisymmetric) w.r.t. the permutation of 푥 and 푥 2. Here 휅 = 1 is Coulomb’s constant and 휀 is the vacuum 1 2 4휋 휀0 0 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. 14 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm permittivity. For 푧 = 2, 퐻푧,2 describes the Helium atom, for 푧 = 1, the negative ion of the Hydrogen and for 2 < 푧 ≤ 94 (or ≤ 103, depending on what one counts as stable elements), Helium-type positive ion. (We can call (3.1) with 푧 > 103 a 푧-ion.) It is well known that 퐻푧,2 has eigenvalues below its continuum. Variational tech- niques give excellent upper bounds on the eigenvalues of 퐻푧,2, but good lower bounds are hard to come by. Thus, we formulate

Problem 3.1. Estimate the ground state energy of 퐻푧,2. The most diﬃcult case is of 푧 = 1, the negative ion of the hydrogen, and the problem simpliﬁes as 푧 increases. Here we present fairly precise bounds on the ground state energy of 퐻푧,2 implied by our actual estimates. However, the conditions under which these estimates are valid impose rather sever restrictions in 푧. We introduce the reference energy

휅2푒4푚 퐸ref := = 훼2푚푐2 ℏ2 (twice the ground state energy of the hydrogen, or 2Ry), where 푐 is the speed of light in vacuum and 푒2 훼 = 휅 ℏ푐 (푧) is the ﬁne structure constant, whose numerical value, approximately 1/137. Let 퐸# sym/ ref as / ref stand for either 퐸푧,2 퐸 or 퐸푧,2 퐸 , the ground state energy of 퐻푧,2 on either symmetric or anti-symmetric functions. We have Proposition 3.2. Assume 푧 ≥ 31.25 for the symmetric space and 푧 ≥ 170 for the anti-symmetric one. Then the ground state energy of 퐻푧,2 is bounded as

10푤# −푐#푧2 + 푤#푧 − 2 ≤ 퐸 (푧) ≤ −푐#푧2 + 푤#푧, 1 # # 1 (3.2) 훾0

3 where, for symmetric functions, 푐 = 1, 훾0 = 8 and 푤1 ≈ 0.6 and 푤2 ≈ 0.27, defined in as 5 as 5 equations (3.13) and (3.15), and, for anti-symmetric functions, 푐 = 8 , 훾0 = 72 and as ≈ as ≈ 푤1 0.20 and 푤2 0.01, defined in (3.12). as as The approximate values of 푤1, 푤2, 푤1 and 푤2 are computed numerically in AppendixB. Here we report the stable digits of our computations. The inequalities 푧 ≥ 31.25 and 푧 ≥ 170 come from condition (1.3), while estimates (3.2), from (1.8), with 푏 = 0.8 and 푎 = 0.1 (which give 푘 = 10). Table1 compares the result for symmetric functions with computations in quantum chemistry. (We did not find results for the antisymmetric space.)

We observe that, except for 푧 = 50, the results of [16, 17, 18] lie in the interval provided by the estimation (3.2). The FS-map and perturbation theory 15

푧 10 20 30 40 50

−퐸exact (from [16, 17, 18]) 93.9 387.7 881.4 1575.2 2468.9 − sym,lead main part 퐸푧,2 in (3.2) 94 388 882 1576 2470 relative diﬀerence err푧 0.11% 0.077% 0.068% 0.051% 0.045% relative error term Δ푧 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 − sym,lead 2 − main part 퐸푧,2 := 푧 푤1푧 in equation (3.2), its relative contribution of the /(− sym,lead ) error estimation Δ푧 := 10푤2 퐸푧,2 훾0 and the relative diﬀerence deﬁned by | − sym,lead|/(− sym,lead) err푧 := 퐸exact 퐸푧,2 퐸푧,2 .

For computations of the relativistic and QED contributions, see [10, 18, 19]. Now, we derive some consequences of estimates (3.2). Let 푧∗ be the smallest 푧 for which the error bound in the symmetric case is less than or equal to the smallest explicit (subleading) term 푤1. Inequality (3.2) shows that the 10푤2 10푤2 latter bound is satisﬁed for 푧 such that 푤 푧 ≥ , which shows that 푧∗ = and 1 훾0 푤1 훾0 consequently, 푧∗ ≈ 12. According to equation (3.2), the symmetric ground state energy is lower than − 2 + − 5 2 + as − as the anti-symmetric one, if 푧 푤1푧 < 8 푧 푤1 푧 160 푤2 . Using the values ≈ ≈ as ≈ as ≈ 푤1 0.6, 푤2 0.3, 푤1 0.20 and 푤2 0.01, we ﬁnd that, based on (3.2), the ground state is symmetric and therefore its spin is 0 for √︂ ! 4 2 4 96 8 푧 ≥ + + = ≈ 2.6. 3 5 25 5 3 We conjecture that the symmetric ground state energy is lower than the anti-symmetric one for all 푧’s. Finally, note that on symmetric functions, the eigenvalue of the unperturbed operator is simple and on anti-symmetric ones, has the degeneracy 4, see below. Proof of Proposition 3.2. First, we rescale the Hamiltonian (3.1) as 푥 → 푥/휇, with 푧휅푒2푚 푧 휇 := = , ℏ2 the Bohr radius to obtain 푧2휅2푒4푚 푈−1퐻 푈 = 퐻 (푧) , 휇 푧,2 휇 ℏ2 3 (푧) where 푈휇 : 휓(푥) → 휇 휓(휇푥) and 퐻 is the rescaled Hamiltonian given by 2 ∑︁ 1 1 1/푧 퐻 (푧) = − Δ − + . 2 푥 푗 |푥 | |푥 − 푥 | 푗=1 푗 1 2 16 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

Thus, it suﬃces to estimate the ground state energy of 퐻 (푧) . We consider 1/푧 푊 = |푥1 − 푥2| as the perturbation and Í2 (− 1 Δ − 1 ) as the unperturbed operator. 푗=1 2 푥 푗 |푥푖 | First, we consider the rescaled Hamiltonian 퐻 (푧) on the symmetric functions subspace. On symmetric functions, the ground state energy of Í2 (− 1 Δ − 1 ) is −1 푗=1 2 푥 푗 |푥 푗 | (see below), so we shift the operator 퐻 (푧) by 1 + 훽, for some 훽 > 0, so that (푧) 퐻 + 1 + 훽 = 퐻0 + 푊, with 2 ∑︁ 1 1 1/푧 퐻 := − Δ − + 1 + 훽 and 푊 = . 0 2 푥 푗 |푥 | |푥 − 푥 | 푗=1 푗 1 2

Now, the ground state energy of 퐻0 is 훽 and we can use inequality (1.8) and Proposition 1.4(c) to estimate the ground state energy of 퐻 (푧) + 1 + 훽. By the HVZ theorem, the spectrum of 퐻0 on symmetric functions consists of the continuum [푒1 + 1 + 훽, ∞) and the eigenvalues 푒푚 + 푒푛 + 1 + 훽, 푚, 푛 ≥ 1, where 푒푛, 푛 = 1 1 1, 2,..., denote the discrete eigenvalues of the Hydrogen Hamiltonian − 2 Δ푥 − |푥 | . The eigenvalues 푒푛, 푛 = 1, 2,..., are known explicitly: 1 푒 = − , (3.3) 푛 2푛2 with the multiplicities of 푛2 and the corresponding eigenfunctions are given by Eq (A.1) below. Then the ground state energy of 퐻0 is 휆0 = 2푒1 + 1 + 훽 = 훽, as claimed, and the gap, 훾0 = 푒1 + 푒2 − 2푒1 = 푒2 − 푒1 = 3/8. Next, we show that the condition (1.3) is satisﬁed for 푧 ≥ 31.25. We begin with the really rough estimate: 1 ≤ ℎ푥1 + ℎ푥2 + 10, (3.4) |푥1 − 푥2| = − 1 Δ − 1 −Δ ≥ 1 where ℎ푥 : 2 푥 |푥 | . First, we use Hardy’s inequality, 4|푥 |2 , and the estimate 1 − 1 ≥ − 4푚|푥 |2 |푥 | 푚 to obtain 1 1 − Δ − ≥ −푚. (3.5) 푚 |푥| 1 Next, passing to the relative and centre-of-mass coordinates, 푥 − 푦 and 2 (푥 + 푦), we ﬁnd 1 − Δ푥 − Δ푦 = −2Δ푥−푦 − Δ 1 (푥+푦) ≥ −2Δ푥−푦 . 2 2 The FS-map and perturbation theory 17 which, together with (3.5), yields 1 1 1 ≤ − Δ푥1−푥2 + 2 ≤ − (Δ푥1 + Δ푥2 ) + 2. |푥1 − 푥2| 2 4 1 1 1 Now, we use − 2 Δ푥 = ℎ푥 + |푥 | ≤ ℎ푥 − 4 Δ푥 + 4 to obtain 1 − Δ ≤ ℎ + 4. 4 푥 푥 The last two inequalities yield (3.4). Eq. (3.4), together with the relation

퐻0 = ℎ푥1 + ℎ푥2 + 1 + 훽, implies the estimate: 1 ≤ 퐻0 + 9 − 훽. (3.6) |푥1 − 푥2| Now, we check the condition (1.3):

⊥ ⊥ 푏훾0 k푃 푊푃 k퐻0 ≤ . 휆∗ Using the last relation, we estimate

1 1 ⊥ − 2 ⊥ − 2 ⊥ −1 ⊥ 푧(퐻0 ) 푊 (퐻0 ) ≤ (퐻0 ) (퐻0 + 9 − 훽) ⊥ −1 = 1 + (9 − 훽)(퐻0 ) . ⊥ Recalling that 휆0 = 훽, we see that 퐻0 ≥ 훽 + 훾0, so that the last inequality gives

1 1 1 9 + 훾0 ⊥ − 2 ⊥ − 2 0 ≤ (퐻0 ) 푊 (퐻0 ) ≤ , 푧 훽 + 훾0 provided 훽 < 9. This implies

⊥ ⊥ 1 9 + 훾0 k푃 푊푃 k퐻0 ≤ . (3.7) 푧 훽 + 훾0

9+훾0 9+훾0 Since 휆∗ = 훽 + 훾 , condition (1.3) is satisﬁed, if ≤ 푏훾 , which gives 푧 ≥ . 0 푧 0 푏훾0 Since 훾0 = 3/8 and 푏 = 0.8, this implies 푧 ≥ 31.25. We will need the following lemma, whose proof is given after the proof of the proposition.

Lemma 3.3. Recall that 푃 is the orthogonal projection onto the eigenspace of 퐻0 corresponding to the lowest eigenvalue 훽. We have 푤 푤 h푊i = k푃푊푃k = 1 and Φ(푊) ≤ 2 , (3.8) 2 푧 푧 훾0 with, recall, h푊i := h휑0, 푊휑0i, where 휑0 is the eigenfunction of 퐻0 corresponding to 휆0. 18 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

We check now the second condition of Theorem 1.1,(1.4). Recall the deﬁnition k푃푊푃k + 푘Φ(푊) 푟 := . 훾0 Then condition (1.4) can be written as 푟 < 푎. By Lemma 3.3, 1 푤 푘푤 푟 ≤ 1 + 2 . 2 훾0 푧 푧 훾0

Recall the condition 푧 ≥ 31.25 and 푘 = 10, 푤1 ≈ 0.6 and 푤2 ≈ 0.3. Thus 푘푤 푤 푘 푤 푤 2 ≤ 1 2 ≤ 0.5 1 . 2 푧 훾0 푧 31.25 훾0 푤1 푧 Then we find 푟 ≤ 1.5 푤1 and therefore condition (1.4) is satisfied if 푎 > 1.5 푤1 , or 푧훾0 푧훾0 (for 푎 = 0.1) 3 8 0.6 푧 > = 24. 2 3 푎 This is less restrictive than 푧 ≥ 31.25. For the third condition (1.5), recalling that 휆0 = 훽, so that 휆∗ = 훾0 + 훽 (= 휆1, the second eigenvalue of 퐻0) and using both relations in (3.8), we obtain that condition 2 1 (1.5) is satisfied if 푘푤2/(푧 훾0) < 2 (푎훾0 − 푤1/푧), which is equivalent to 1 √︃ + 2 + 푧 > 푤1 푤1 8푎푘푤2 . 2푎훾0

3 Using the values 훾0 = 8 , 푤1 ≈ 0.6 and 푤2 ≈ 0.3, and 푏 = 0.8 and 푎 = 0.1, giving 푘 = 10, this shows that the latter inequality, and therefore (1.5), holds if 푧 > 30, which is less restrictive than 푧 ≥ 31.25. Thus, all three conditions are satisﬁed for 푧 ≥ 31.25. (푧) (푧) Next, we use (1.8) to estimate the ground state energy, 퐸sym of 퐻 . The ﬁrst and second relations in (3.8), together with (1.8) and the fact that 푈(휆) ≤ 0, gives the following bounds 푘푤 푤 − 2 ≤ 퐸 (푧) + 1 − 1 ≤ 0. 2 sym 훾0푧 푧 which after rescaling gives (3.2). Now, we consider the ground state of 퐻 (푧) on anti-symmetric functions. In this case, the ground state energy of Í2 (− 1 Δ − 1 ) is 푒 + 푒 = − 5 of multiplicity 4, 푗=1 2 푥 푗 |푥푖 | 1 2 8 with the ground states 1 휙2ℓ푘 (푥1, 푥2) := √ (휓100 (푥1)휓2ℓ푘 (푥2) − 휓2ℓ푘 (푥1)휓100 (푥2)), (3.9) 2 for (ℓ, 푘) = (0, 0), (1, −1), (1, 0), (1, 1), where 휓푛ℓ푘 (푥) for ℓ = 0, 1, 2, . . . , 푛 − 1, 푘 = −ℓ, −ℓ+1, . . . , ℓ, 푛 = 1, 2,..., are the eigenfunctions of the Hydrogen-like Hamiltonian The FS-map and perturbation theory 19

1 1 − 2 Δ푥 − |푥 | corresponding to eigenvalues 푒푛, so that 휓100 (푥) = 휓1 (푥), see AppendixA. Now, we deﬁne the unperturbed operator as

2 ∑︁ 1 1 5 퐻 := − Δ − + + 훽, 0 2 푥 푗 |푥 | 8 푗=1 푗 to have 훽 as the lowest eigenvalue of 퐻0. For the gap, since the first two eigenvalues are 5 5 5 푒 + 푒 + + 훽 = 훽 and 푒 + 푒 + + 훽(< 푒 + 푒 + + 훽), 1 2 8 1 3 8 2 2 8 as + − ( + ) − as 5 we have 훾0 = 푒1 푒3 푒1 푒2 = 푒3 푒2 giving 훾0 = 72 . + + 5 + For condition (1.3), note first that since now 퐻0 = ℎ푥1 ℎ푥2 8 훽, Eq. (3.4) implies 1 3 ≤ 퐻0 + 9 + − 훽, |푥1 − 푥2| 8 instead of (3.6), which shows that (3.7) should be modified as

1 9 + 3 + 훾as k푃⊥푊푃⊥ k ≤ 8 0 , 퐻0 + as 푧 훽 훾0 as + as 1 ( + 3 + in the anti-symmetric case. Since 휆∗ = 훽 훾0 , condition (1.3) is satisfied, if 푧 9 8 as) ≤ as 훾0 푏훾0 , which gives + 3 + as 9 8 훾0 푧 ≥ as . 푏훾0 as 5 ≥ Since 훾0 = 72 and 푏 = 0.8, this implies 푧 170 for the anti-symmetric case. We skip the verification of conditions (1.4) and (1.5), which are simpler and similar to that of the symmetric case. (푧) (푧) To estimate the ground state energy, 퐸as , of 퐻 on anti-symmetric functions, we use bound (1.9). First, we claim that Φ(푊) defined in (1.2) satisfies (cf. Remark 2)

휇max Φ(푊) ≤ as , (3.10) 훾0 ⊥ where 휇max is the largest eigenvalue of the positive, 4 × 4-matrix 푃푊푃 푊푃. Indeed, recall that ⊥ 2 휆 휆∗ k푃 푊푃k 0 퐻0 Φ(푊) := as 훾0 − 1 − 1 = k ⊥ k = k k = 2 ⊥ 2 with 휆0 훽 and observe that 푃푊푃 퐻0 퐴 , with 퐴 : 퐻0 푃푊푃 퐻0 . Using now the relations

1 1 ∗ − 2 ⊥ −1 − 2 퐴퐴 = 퐻0 푃푊푃 퐻0 푊푃퐻0 , 20 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

− 1 − 1 2 2 −1 ⊥ ≤ −1 ⊥ 퐻0 푃 = 휆0 푃 (since 퐻0푃 = 휆0푃), and 퐻0 푃 휆∗ 푃 , we ﬁnd

∗ −1 ⊥ 퐴퐴 ≤ (휆0휆∗) 푃푊푃 푊푃,

∗ ∗ 1 which, together with k퐴k = k퐴 k = k퐴퐴 k 2 , gives

k푃푊푃⊥ k2 ≤ (휆 휆 )−1 k푃푊푃⊥푊푃k. (3.11) 퐻0 0 ∗ ⊥ ⊥ Since 푃푊푃 푊푃 ≥ 0, we have k푃푊푃 푊푃k = 휇max, which gives (3.10). Hence, we have to compute the smallest eigenvalue 휇min of the 4 × 4 matrix 푃푊푃 ⊥ and the largest eigenvalue, 휇max, of the positive, 4 × 4-matrix 푃푊푃 푊푃, see (1.9). By scaling, it is easy to see that the matrices 푃푊푃 and 푃푊푃⊥푊푃 are of the form 푃푊푃 = 푧−1 퐴 and 푃푊푃⊥푊푃 = 푧−2퐵, where 퐴 and 퐵 are 푧-independent 4×4 matrices. Hence 푤as 푤as 휇 = 1 and 휇 = 2 , (3.12) min 푧 max 푧2 as as for some positive 푧-independent constants 푤1 and 푤2 . These constants are computed in AppendixB. Now, we see that (1.9) and (3.10) give

푤as 푘푤as 5 푤as 1 − 2 ≤ 퐸 (푧) + ≤ 1 , as 2 as 푧 훾0 푧 8 푧 which after rescaling and setting 푘 = 10 gives (3.2) in the antisymmetric case.

1 1 Proof of Lemma 3.3. The ground state energy, 푒1, of − 2 Δ푥 − |푥 | is non-degenerate, with the normalized ground state (known as the 1s wavefunction) given by (see e.g. [7], or Wiki, with 푎0 replaced by 1 due to the rescaling above)

1 −|푥 | 휓1 (푥) = √ 푒 . 휋

Hence, the ground state energy, 0, of 퐻0 is also non-degenerate, with the normalized ground state 휓1 (푥1)휓1 (푥2) (in the symmetric subspace). First, we compute h푊i and k푃푊푃k. Let (휓 ⊗ 휙)(푥, 푦) := 휓(푥) 휙(푦) and, for an 2 6 operator 퐴 on 퐿 (R ), we denote h퐴i := h휓1 ⊗ 휓1, 퐴휓1 ⊗ 휓1i. In the symmetric case, 푃푊푃 = h푊i푃, where

∫ 2 1 |휓1 (푥)휓1 (푦)| 1 h푊i := h휓1 ⊗ 휓1, 푊휓1 ⊗ 휓1i = 푑푥 푑푦 =: 푤1. (3.13) 푧 R6 |푥 − 푦| 푧

푤1 Hence k푃푊푃k = 푧 . This gives the ﬁrst relation in (3.8). Now, using that 푃⊥ = 1 − 푃, we compute 푤 푃푊푃⊥푊푃 = 푃푊2푃 − 푃푊푃푊푃 = 2 푃, (3.14) 푧2 The FS-map and perturbation theory 21 and 2 2 2 푤2 := 푧 h푊 i − h푊i ∫ |휓 (푥)휓 (푦)|2 ∫ |휓 (푥)휓 (푦)|2 2 = 1 1 푑푥 푑푦 − 1 1 푑푥 푑푦 . (3.15) 2 R6 |푥 − 푦| R6 |푥 − 푦|

By the Schwarz inequality and the normalization of 휓1, we have 푤2 ≥ 0, which, together with (3.14), gives 푤 k푃푊푃⊥푊푃k = 2 . (3.16) 푧2 Eqs. (3.11) and (3.16) imply the second relation in (3.8).

Acknowledgments. The second author is grateful to Volker Bach and Jürg Fröhlich for enjoyable collaboration. The authors are indebted to the anonymous referee for many useful suggestions and remarks.

A The eigenfunctions of the Hydrogen-like Hamilto- nian

The eigenfunctions 휓푛ℓ푘 (푥) are given by (see Wiki, “Hydrogen atom” with 푎0 replaced by 1) √︄ 2푧 3 (푛 − ℓ − 1)! 휓 (푥) = 푒−휌/2 휌ℓ 퐿2ℓ+1 (휌)푌 푘 (휃, 휙), (A.1) 푛ℓ푘 푛 2푛(푛 + ℓ)! 푛−ℓ−1 ℓ for ℓ = 0, 1, 2, . . . , 푛 − 1, 푘 = −ℓ, −ℓ + 1, . . . , ℓ, 푛 = 1, 2,.... Here (푟, 휃, 휙) are the polar 2푧푟 coordinates of 푥, 휌 = , 퐿2ℓ+1 (휌) is a generalized Laguerre polynomial of degree 푛 푛−ℓ−1 푘 3 푛 − ℓ − 1, and 푌ℓ (휃, 휙) is a spherical harmonic function of the degree ℓ and order 푘.

B The numerical approximation of the constants 푤1, 푤 푤푎푠 푤푎푠 2, 1 and 2

Let us first focus on 푤1 and 푤2. From the definition ∫ 2 |휓1 (푥)휓1 (푦)| 푤1 := 푑푦 푑푥, R6 |푥 − 푦| 3Quoting from Wikipedia (https://en.wikipedia.org/wiki/Hydrogen_atom, 30.10.2020): “... the generalized Laguerre polynomials are defined differently by different authors. The usage here is consistent with the definitions used by [7], p. 1136, and Wolfram Mathematica. In other places, the Laguerre polynomial includes a factor of (푛 + ℓ)!. ... or the generalized Laguerre polynomial appearing 2ℓ+1 in the hydrogen wave function is 퐿푛+ℓ (휌) instead.” 22 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

∫ |휓 (푥)휓 (푦)|2 푤 := 1 1 푑푦 푑푥 − 푤2, 2 2 1 R6 |푥 − 푦| it becomes clear that we need to compute terms of the form ∫ |휓 (푥)휓 (푦)|2 = 1 1 퐶훼 훼 푑푦 푑푥 R6 |푥 − 푦| ∫ ∫ |휓 (푥 − 푧)|2 = | ( )|2 1 휓1 푥 훼 푑푧 푑푥, R3 R3 |푧| with 훼 = 1, 2. We thus introduce a numerical quadrature in order to approximate the values of these integrals. We do not aim within this work to obtain the most eﬃcient implementation. We deﬁne a numerical quadrature grid in terms of spherical coordinates with integration points 푥푖,푛 and weights 휔푖,푛 given by

3 leb 푥푖,푛 = 푟푖 푠푛, 휔푖,푛 = ℎ 푟푖 휔푛 with

2 ln(푅max) 푟푖 = exp(− ln(푅max) + (푖 − 1)ℎ), 푖 = 1, . . . , 푁푟 , ℎ = 푁푟 − 1 leb and where {푠푛, 휔푛 } denote Lebedev integration points on the unit sphere with 푁leb points. Thus, 푁푟 denotes the number of points along the 푟-coordinate, 푅max the radius of the furthest points away from the origin. We thus approximate 퐶훼 by

푁푟 푁leb ∑︁ ∑︁ 2 퐶훼 ≈ 휔푖,푛 |휓1 (푥푖,푛)| Φ훼 (푥푖,푛) 푖=1 푛=1 with 푁푟 푁leb 2 ∑︁ ∑︁ |휓1 (푥푖,푛 − 푥 푗,푚)| Φ (푥 ) = 휔 . 훼 푖,푛 푗,푚 |푥 | 훼 푗=1 푚=1 푗,푚 2 Then, we approximate 푤1 ≈ 퐶1 and 푊2 ≈ 퐶2 − 퐶1 . as as We now shed our attention to the constants 푤1 and 푤2 and we ﬁrst introduce the bi-electronic integrals, for (ℓ, 푘), (ℓ0, 푘 0) = (0, 0), (1, −1), (1, 0), (1, 1), by ∫ ∫ 2 ℓ0,푘0 |휓1 (푦)| ( ) 0 0 ( ) Aℓ,푘 = 휓2ℓ푘 푥 휓2ℓ 푘 푥 푑푦 푑푥, R3 R3 |푥 − 푦| ∫ ∫ ℓ0,푘0 휓1 (푦)휓2ℓ푘 (푦) ( ) 0 0 ( ) Bℓ,푘 = 휓1 푥 휓2ℓ 푘 푥 푑푦 푑푥. R3 R3 |푥 − 푦| Then, using the deﬁnition (3.9) and symmetry properties, we derive the following expressions for the matrix elements

ℓ0,푘0 ℓ0,푘0 ℓ0,푘0 h | | 0 0 i − Mℓ,푘 = 푧 휓2ℓ푘 푊 휓2ℓ 푘 = Aℓ,푘 Bℓ,푘 . The FS-map and perturbation theory 23

as as Figure 1: Value of the approximate coeﬃcients 푤1, 푤2, 푤1 and 푤2 depending on the quadrature rule reported in Table2.

We then employ the same quadrature rule as above to approximate the matrix elements ℓ0,푘0 ℓ0,푘0 as Aℓ,푘 , Bℓ,푘 and compute the smallest eigenvalue of M as an approximation of 푤1 . as Finally, the approximation of 푤2 involves the matrix

푃푊푃⊥푊푃 = 푃푊2푃 − 푃푊푃푊푃, whose elements are given by Q = N − M2 with

ℓ0,푘0 2 2 ℓ0,푘0 ℓ0,푘0 h | | 0 0 i − Nℓ,푘 = 푧 휓2ℓ푘 푊 휓2ℓ 푘 = Cℓ,푘 Dℓ,푘 and ∫ ∫ 2 ℓ0,푘0 |휓1 (푦)| C = 휓 (푥)휓 0 0 (푥) 푑푦 푑푥, ℓ,푘 2ℓ푘 2ℓ 푘 2 R3 R3 |푥 − 푦| ∫ ∫ ℓ0,푘0 휓1 (푦)휓2ℓ푘 (푦) D = 휓 (푥)휓 0 0 (푥) 푑푦 푑푥. ℓ,푘 1 2ℓ 푘 2 R3 R3 |푥 − 푦| We then again, approximate the different integrals with the above introduced quadrature as rule and find the maximal eigenvalue of Q in order to approximate 푤2 . as as In Figure1, we plot the the approximate value of 푤1, 푤2, 푤1 and 푤2 for different values of the parameters 푅max, 푁푟 and 푁leb given in Table2. We obtain approximate values

as as 푤1 ≈ 0.625, 푤2 ≈ 0.2628, 푤1 ≈ 0.3, 푤2 ≈ 0.01 where the last reported digit is stable over the last three quadrature rules. 24 Geneviève Dusson, Israel Michael Sigal, Benjamin Stamm

index 1 2 3 4 5 6 7 8 9 10 11 12

푅max 16 18 20 22 24 26 28 30 32 34 34 34 푁푟 10 12 14 16 18 20 22 24 26 28 30 32 푁leb 194 266 350 590 974 1454 2030 2702 3470 4334 5294 5810

Table 2: Parameters for the diﬀerent quadrature rules.

References

[1] V. Bach, J. Fröhlich and I.M. Sigal, Quantum electrodynamics of conﬁned non-relativistic particles. Adv. Math., 137(2):299–395, 1998.

[2] H. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators (with Applications to Quantum Mechanics and Global Geometry. Springer, 1987.

[3] G. Dusson, I.M. Sigal and B. Stamm, Analysis of the Feshbach-Schur method for the planewave discretizations of Schrödinger operators. arXiv:2008.10871.

[4] S. Gustafson and I.M. Sigal, Mathematical Concepts of Quantum Mechanics. Springer Science & Business Media, Sept. 2011.

[5] P. Hislop and I.M. Sigal, Introduction to Spectral Theory. With Applications to Schrödinger Operators. Springer-Verlag, 1996.

[6] T. Kato, Perturbation theory for linear operators. Springer-Verlag, 1976.

[7] A. Messiah, Quantum Mechanics. New York: Dover. (1999).

[8] H. Nakashima, H. Nakatsuji, Solving the Schrödinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ICI) method, J. Chem. Phys. 127, 224104 (2007).

[9] H. Nakashima, H. Nakatsuji, Solving the electron-nuclear Schrödinger equation of helium atom and its isoelectronic ions with the free iterative- complement-interaction method. J. Chem. Phys. 128, 154107 (2008).

[10] K. Pachucki, V. Patkos and V.A. Yerokhin, Testing fundamental interactions on the helium atom. Phys. Rev. A95 (2017) 062510.

[11] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II. Fourier Analysis and Self-Adjointness. Academic Press, 1975.

[12] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV: Anal- ysis of Operators. Academic Press, 1979.

[13] F. Rellich, Perturbation Theory of Eigenvalue Problems. Gordon and Breach: New York, 1969. The FS-map and perturbation theory 25

[14] B. Simon, A Comprehensive Course in Analysis, Part 4: Operator Theory (American Mathematical Society, Providence, RI, 2015). [15] B. Simon, Tosio Kato’s work on non-relativistic quantum mechanics: Part 1 and 2, Bulletin of Mathematical Sciences, Vol 8 (2018), 121–232 and Vol 9 (2019) 1950005 (105 pages). [16] A. V. Turbiner and J. C. L. Vieyra, Helium-like and Lithium-like ions: Ground state energy. arXiv:1707.07547v3, 2017. [17] A. V. Turbiner, J. C. L. Vieyra, J.C. del Valle, and DJ Nader, Ultra-compact accurate wave functions for He-like and Li-like iso?electronic sequences and variational calculus: I. Ground state. International Journal of Quantum Chem- istry, e26586, 2020; A. V. Turbiner, J. C. L. Vieyra and J.C. del Valle, Ultra- compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. I. Ground state. arXiv:2007.11745v, 2020. [18] A.V. Turbiner, J.C. L. Vieyra, H. Olivares-Pilón, Few-electron atomic ions in non-relativistic QED: The ground state. Annals of Physics 409 (2019) 167908 [19] V.A. Yerokhin and K. Pachucki, Theoretical energies of low-lying states of light helium-like ions. Phys Rev A81 (2010) 022507.