### BibTeX

@MISC{Lewin09contents,

author = {Mathieu Lewin and Éric S Éré},

title = {Contents},

year = {2009}

}

### OpenURL

### Abstract

This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some self-adjoint operators with a gap in their essential spectrum, occuring in Quantum Mechanics. First we consider Galerkin bases which preserve the decomposition of the ambient Hilbert space into a direct sum H = PH ⊕ (1 − P)H given by a fixed orthogonal projector P, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schrödinger operators (we show that pollution is absent in a Wannier-type basis), and to Dirac operators (several natural decompositions are considered). In the second part, we add the constraint that within the Galerkin basis there is a certain relation between vectors in PH and vectors in (1 − P)H. Abstract results are proved and applied to several practical methods like the famous kinetic balance condition of relativistic Quantum Mechanics.

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Citation Context ...nds to a compact operator on H. Then we have Spu(A + B, Π) ∩ (a, b) = ∅. Proof. Under our assumption we have that (A + B − i) −1 − (A − i) −1 is compact, hence σess(A + B) = σess(A) by Weyl’s Theorem =-=[30, 8]-=- and A + B is also bounded from below. Changing γ if necessary we may assume that γ /∈ σ(A + B) ∪ σ(A). Next we take a curve C in the complex plane enclosing the whole spectrum of A and A + B below γ ... |

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Citation Context ... V | 1/2 = K(V n 1 ) + K(V n 2 ) + K(V n 3 ) K(W) := 1 ∫ ∞ 1 2π −∞ D0 + iη W |D0 + V | 1/2 D0 + V + iη dη and estimate each term in an appropriate trace norm. We denote by Sp the usual Schatten class =-=[33, 30]-=- of operators A having a finite p-trace, |A|| Sp = Tr(|A| p ) 1/p < ∞. Let us recall the Kato-Seiler-Simon inequality (see [31] and Thm 4.1 in [33]) The term K(V n 1 where we have used that ∀p ≥ 2, |f... |

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Citation Context ... on nonlinear Schrödinger equations suggests that, in order to remove this smallness assumption, one would need some assumption on w like w � 0, as well as some additional (almost) conservation laws =-=[1]-=-. Our study of the linear response operator however indicates that the situation is involved and more information on the momentum distribution f is certainly also necessary. We finally note that in ou... |

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Citation Context ...nds to a compact operator on H. Then we have Spu(A + B, Π) ∩ (a, b) = ∅. Proof. Under our assumption we have that (A + B − i) −1 − (A − i) −1 is compact, hence σess(A + B) = σess(A) by Weyl’s Theorem =-=[30, 8]-=- and A + B is also bounded from below. Changing γ if necessary we may assume that γ /∈ σ(A + B) ∪ σ(A). Next we take a curve C in the complex plane enclosing the whole spectrum of A and A + B below γ ... |

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Citation Context ... explain our strategy for proving Theorem 1. The idea of the proof relies on a fixed point argument, in the spirit of [11, Sec. 5]. If we can prove that ργ −ργf ∈ L2 t,x(R+×R 2 ), then we deduce from =-=[19, 4]-=- that there exists a family of unitary operators UV(t) ∈ C 0 t (R+,B) on L 2 (R 2 ) such that for all t ∈ R+. We furthermore have γ(t) = UV(t)γ0UV(t) ∗ , UV(t) = e it∆ WV(t), where WV(t) is the wave o... |

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Citation Context ...κ |x| , P 0 ) + ∩ (−1, 1) = ∅. Proof. The operators P 0 +(D 0 + κ|x| −1 )P 0 + and P 0 −(D 0 + κ|x| −1 )P 0 − are known to have a self-adjoint Friedrichs extension as soon as |κ| < 2/(π/2 + 2/π), see =-=[17]-=-. Furthermore one has σess(D0 + κ|x| −1 )|P0 + L2 = [1, ∞) and σess(D0 + κ|x| −1 )|P0 −L2 = (−∞, −1], see Theorem 2 in [17]. As √ 3/2 < 2/(π/2 + 2/π), the result immediately follows from Theorem 2.1 a... |

34 | Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation
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Citation Context ... the square root of the above inequality, this proves that |D0| −1/2 |D0 + V | 1/2 and its inverse are both bounded operators. Next we use the resolvent formula together with Cauchy’s formula like in =-=[19]-=- to infer (P 0 + − Π)|D 0 + V | 1/2 = − 1 2π = 1 2π ∫ ∞ −∞ ∫ ∞ −∞ ( 1 D 0 + V + iη − 1 D 0 + iη 1 D0 + iη V |D0 + V | 1/2 D0 + V + iη dη. ) |D 0 + V | 1/2 dη Let us now write V = V n 1 + V n 2 + V n 3... |

31 | On the eigenvalues of operators with gaps. Application to Dirac operators
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Citation Context ...3.8) holds true for b = +∞ > d when A |PH is a bounded operator. Theorem 3.2 has many similarities with the characterization of eigenvalues in a gap which was proved by Dolbeault, Esteban and Séré in =-=[12]-=- (where our number d = supσ(A (1−P)H) was denoted by ‘a’). In particular the reader should compare the assumptions d < m ′ 2 and d < m ′′ 2 with (iii) at the bottom of p. 209 in [12]. The proof indeed... |

29 |
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Citation Context ...n (1) reduces to the well-known Hartree equation for one function { i∂tu = (−∆+w∗|u| 2 )u, (6) u(0) = u0. There is a large literature about scattering for the nonlinear equation (6), see for instance =-=[5, 18, 10, 14, 6, 15]-=-. The intuitive picture is that the nonlinear term is negligible for small u, since w ∗|u| 2u is formally of order 3. It is important to realize that this intuition does not apply in the case f = 0 c... |

24 |
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Citation Context ...e electrons of the crystal set to m = e = 1. Also we neglect their spin for simplicity (reinserting the spin in our model is straightforward). By Bloch-Floquet theory (see Chapter XIII, Section 16 of =-=[25]-=-), the spectrum of the L-periodic Schrödinger operator H 0 per = − 1 2 ∆+V0 per(x) is composed of bands. When there is a gap between the Zth and the (Z +1)st bands, the crystal is an insulator and εF ... |

23 |
Scattering theory in the energy space for a class of hartree equations
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(Show Context)
Citation Context ...n (1) reduces to the well-known Hartree equation for one function { i∂tu = (−∆+w∗|u| 2 )u, (6) u(0) = u0. There is a large literature about scattering for the nonlinear equation (6), see for instance =-=[5, 18, 10, 14, 6, 15]-=-. The intuitive picture is that the nonlinear term is negligible for small u, since w ∗|u| 2u is formally of order 3. It is important to realize that this intuition does not apply in the case f = 0 c... |

22 | Spectral pollution
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(Show Context)
Citation Context ...7, 32]. It also appears in elasticity, electromagnetism and hydrodynamics; see, e.g. the references in [2]. Eventually, it has raised as well a huge interest in the mathematical community, see, e.g., =-=[23, 9, 4, 21, 10, 28, 29]-=-. In this article we will study spectral pollution from a rather new perspective. Although many works focus on how to determine whether an approximate eigenvalue is spurious or not (see, e.g., the suc... |

22 |
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Citation Context ...n (1) reduces to the well-known Hartree equation for one function { i∂tu = (−∆+w∗|u| 2 )u, (6) u(0) = u0. There is a large literature about scattering for the nonlinear equation (6), see for instance =-=[5, 18, 10, 14, 6, 15]-=-. The intuitive picture is that the nonlinear term is negligible for small u, since w ∗|u| 2u is formally of order 3. It is important to realize that this intuition does not apply in the case f = 0 c... |

21 | On the thermodynamic limit for Hartree-Fock type models
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Citation Context ...hat the system must be locally neutral (third equation in (2.1)). The unique solution to the self-consistent equation (2.1) is found by minimizing the so-called reduced Hartree-Fock energy functional =-=[5, 2]-=-. We are working in atomic units with the mass m and the charge e of the electrons of the crystal set to m = e = 1. Also we neglect their spin for simplicity (reinserting the spin in our model is stra... |

19 |
Energy scattering for Hartree equations
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19 |
Geometric methods in multiparticle quantum systems
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Citation Context ...is ‖A‖. We use the notation S p (H) for the Schatten space of all the compact operators A on H such that Tr|A| p < ∞, with |A| = √ A ∗ A, and use the norm |A|| S p (H) := (Tr|A| p ) 1/p . We refer to =-=[16]-=- for the properties of Schatten spaces. The spaces S 2 (H) and S 1 (H) correspond to Hilbert-Schmidt and4 M. LEWIN AND J. SABIN trace-class operators. We often use the shorthand notation B and S p wh... |

18 | An analytical proof of Hardy-like inequalities related to the Dirac operator
- Esteban, Vega
(Show Context)
Citation Context ...〉 + ≥ 0 (3.15) 〈(2 − V )Lϕ, Lϕ〉 where the denominator of the second term does not vanish due to (3.13). Note that (3.15) takes the form of a Hardy-type inequality similar to those which were found in =-=[12, 11]-=-. In the following we will have to study this kind of inequalities for sequences ϕn which converge weakly to 0. The Hardy inequalities of [12, 11] will indeed be an important tool as we will see below... |

17 | A new approach to the modeling of local defects in crystals: the reduced Hartree-Fock case
- Cancès, Deleurence, et al.
(Show Context)
Citation Context ...s of large polarons. In [14] we have introduced a new polaron model by coupling the energy functional for electrons in the vacuum to a microscopic model of quantum crystals with defects introduced in =-=[2, 3]-=-. Unlike in Fröhlich and Pekar theories we take the crystal explicitly into account and make no assumption on the size of the electron. Our approach thus qualifies for the description of both small an... |

16 |
On the spectra of Schrödinger multiparticle Hamiltonians
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- 1966
(Show Context)
Citation Context ...rt spaces. In the case of atoms and molecules, the fact that inequalities of the form (1.6) imply the existence of bound states is the content of the famous HVZ theorem, first proved independently in =-=[12, 27, 28]-=-. In this paper we prove an HVZ-type theorem for our polaron functional (1.5) when N ≥ 2. We have to face two difficulties. First the functional is invariant under the action of arbitrarily large tran... |

15 |
Discussion of the spectrum of Schrödinger operators for systems of many particles (in Russian), Trudy Moskovskogo matematiceskogo obscestva 9
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- 1960
(Show Context)
Citation Context ...rt spaces. In the case of atoms and molecules, the fact that inequalities of the form (1.6) imply the existence of bound states is the content of the famous HVZ theorem, first proved independently in =-=[12, 27, 28]-=-. In this paper we prove an HVZ-type theorem for our polaron functional (1.5) when N ≥ 2. We have to face two difficulties. First the functional is invariant under the action of arbitrarily large tran... |

12 |
On the approximation of spectra of linear operators on Hilbert spaces
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(Show Context)
Citation Context ...7, 32]. It also appears in elasticity, electromagnetism and hydrodynamics; see, e.g. the references in [2]. Eventually, it has raised as well a huge interest in the mathematical community, see, e.g., =-=[23, 9, 4, 21, 10, 28, 29]-=-. In this article we will study spectral pollution from a rather new perspective. Although many works focus on how to determine whether an approximate eigenvalue is spurious or not (see, e.g., the suc... |

11 | Non-variational approximation of discrete eigenvalues of self-adjoint operators
- Boulton
(Show Context)
Citation Context ...um Chemistry and Physics in particular regarding relativistic computations [13, 18, 22, 34, 14, 27, 32]. It also appears in elasticity, electromagnetism and hydrodynamics; see, e.g. the references in =-=[2]-=-. Eventually, it has raised as well a huge interest in the mathematical community, see, e.g., [23, 9, 4, 21, 10, 28, 29]. In this article we will study spectral pollution from a rather new perspective... |

11 |
Scattering theory for Hartree type equations
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- 1987
(Show Context)
Citation Context |

10 |
Maximally localized generalized wannier functions for composite energy bands, Phys
- Marzari, Vanderbilt
- 1997
(Show Context)
Citation Context ...ressively the radius R. Of course in practice exponentially localized Wannier functions are not simple to calculate. But some authors have defined the concept of maximally localized Wannier functions =-=[24]-=- and proposed efficient methods to find these functions numerically. The efficiency of the computation of the eigenvalues of A in the gap using a Wannier basis (compared to that of the so-called super... |

10 |
Bounds in the Yukawa2 quantum field theory: upper bound on the pressure, Hamiltonian bound and linear lower bound
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- 1975
(Show Context)
Citation Context ... an appropriate trace norm. We denote by Sp the usual Schatten class [33, 30] of operators A having a finite p-trace, |A|| Sp = Tr(|A| p ) 1/p < ∞. Let us recall the Kato-Seiler-Simon inequality (see =-=[31]-=- and Thm 4.1 in [33]) The term K(V n 1 where we have used that ∀p ≥ 2, |f(−i∇)g(x)|| Sp ≤ (2π) −3/p |f | L p (R 3 ) ||g | L p (R 3 ) . (2.37) ) is treated as follows: |K(V n 1 )|| 1 ≤ Sp 2π ∫ ∞ −∞ |D ... |

9 | Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics
- Hainzl, Lewin, et al.
- 2009
(Show Context)
Citation Context ...l energy of two distant clusters of mass decouples we will need a localization procedure. Due to the constraint (2.5), it is convenient to use a specific localization method for Qn, as noted first in =-=[11, 2]-=-. We here provide several new facts about this procedure that will be useful in the next section. We introduce a smooth partition of unity χ2 + η2 = 1 such that χ = 1 on the ball B(0,1) and χ = 0 outs... |

9 | Exact ground state energy of the strong-coupling polaron
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- 1997
(Show Context)
Citation Context ...is described as an homogeneous quantized polarization field with which the electrons interact. In the limit of strong coupling between the electrons and the field, the model reduces to Pekar’s theory =-=[21, 22, 23, 17, 20]-=-. There the crystal is a classical continuous polarizable model, leading to an effective attractive Coulomb interaction in the energy functional of the theory: E P 1 [ψ] = εM 2 ∫ R 3 |∇ψ(x)| 2 dx+ (εM... |

8 |
Spectral pollution and second-order relative spectra for self-adjoint operators
- Levitin, Shargorodsky
(Show Context)
Citation Context ...7, 32]. It also appears in elasticity, electromagnetism and hydrodynamics; see, e.g. the references in [2]. Eventually, it has raised as well a huge interest in the mathematical community, see, e.g., =-=[23, 9, 4, 21, 10, 28, 29]-=-. In this article we will study spectral pollution from a rather new perspective. Although many works focus on how to determine whether an approximate eigenvalue is spurious or not (see, e.g., the suc... |

8 | Triviality of bloch and bloch-dirac bundles
- Panati
(Show Context)
Citation Context ...k(ξ, x)), one can use K different bands for which it is possible to construct K exponentially localized Wannier functions as soon as the union of the K bands is isolated from the rest of the spectrum =-=[26, 5]-=-. The union of the K bands is called a composite band. In our case we typically have a natural composite band corresponding to the spectrum of Aper which is below γ, and another one corresponding to t... |

8 | The dielectric permittivity of crystals in the reduced hartree-fock approximation
- Cancès, Lewin
- 2010
(Show Context)
Citation Context ...∩L 2 (R 3 ), this minimization problem has at least one solution in Q. The corresponding density ρQ is in L 2 (R 3 ) but in general it has long range oscillations which are not integrable at infinity =-=[4]-=-.M. Lewin, N. Rougerie, Binding of small polarons 7 hal-00673045, version 3 - 14 Nov 2012 2.2. The small polaron To our crystal we now add N quantum particles, which are by assumption distinguishable... |

8 |
Quantum Theory of the Electron Liquid
- Giuliani, Vignale
- 2005
(Show Context)
Citation Context ...the linear operator L1 in detail, and we prove that it is a space-time Fourier multiplier of the form w(k)mf(ω,k) where mf is a famous function in the physics literature called the Lindhard function =-=[12, 13, 7]-=-), which only depends on f and d. We particularly investigate when L1 is bounded on L p t,x (R×R2 ) and we show it is the case when w and f are sufficiently smooth. For the Fermi sea (2), we prove tha... |

7 | On approximation of the eigenvalues of perturbed periodic Schrödinger operators
- Boulton, Levitin
(Show Context)
Citation Context ...f A. Spectral pollution is an important issue which arises in many different practical situations. It is encountered when approximating the spectrum of perturbations of periodic Schrödinger operators =-=[4]-=- or Sturm-Liouville operators [35, 36, 1]. It is a well-documented difficulty in Quantum Chemistry and Physics in particular regarding relativistic computations [13, 18, 22, 34, 14, 27, 32]. It also a... |

7 | M.: Non-perturbative embedding of local defects in crystalline materials
- Cancès, Deleurence, et al.
- 2008
(Show Context)
Citation Context ...d by a potential vanishing at infinity. We prove that a splitting associated with the unperturbed (periodic) Hamiltonian allows to avoid spectral pollution, as was already demonstrated numerically in =-=[6]-=- using Wannier functions. In Section 3, we study a general method inspired by the ones used in quantum Physics and Chemistry. Namely, in an abstract Hilbert space H we consider bases which preserve a ... |

7 |
Application of discrete-basis-set methods to the dirac equation, Phys
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- 1981
(Show Context)
Citation Context ...e effect but we have not studied this question more deeply. In the following we again work in units for which m = c = 1. 3.2.1. Kinetic Balance The most common method is the so-called kinetic balance =-=[13, 18, 22, 34]-=-. It consists in choosing as balanced operator LKB = −iσ · ∇ (3.17) We can for instance define LKB on the domain D(LKB) = C ∞ 0 (R 3 , C 2 ), in which case LKB satisfies all the assumptions of Section... |

7 |
Existence of the exponentially localised Wannier functions
- Nenciu
- 1983
(Show Context)
Citation Context ...fine wk uniquely since the uk(ξ, x) are in the best case only known up to a phase. Choosing the right phase, one can prove that when the kth band is isolated from other bands, wk decays exponentially =-=[25]-=-. 3 We use the notation of [30]: X + L ∞ ǫ = {f ∈ X + L ∞ | ∀ǫ > 0, ∃fǫ ∈ X such that ||f − fǫ|| L ∞ ≤ ǫ}.16 Mathieu LEWIN & Éric SÉRÉ More generally, instead of using only one band (i.e. one eigenfu... |

7 | Tail posterior probability for inference in pairwise and multiclass gene expression data - Bochkina, Richardson - 2007 |

6 | Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials. arXiv:0808.0228
- Boulton, Boussaid
- 2008
(Show Context)
Citation Context ... is dealt with in Quantum Physics and Chemistry by means of several different methods, the most widely used being the so-called kinetic balance which we will study later in Section 3.2.1. We refer to =-=[3]-=- for a recent numerical study based on the so-called second-order method for the radial Dirac operator. We now present a heuristic argument which can be made mathematically rigorous in many cases [38,... |

6 | Variational methods in relativistic quantum mechanics
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(Show Context)
Citation Context ...thod) were illustrated for a nonlinear model in [6]. 2.3.2. Dirac operators in upper/lower spinor basis The Dirac operator is a differential operator of order 1 acting on L 2 (R 3 , C 4 ), defined as =-=[38, 15]-=- D 0 = −ic 3∑ k=1 αk∂xk + mc2 β := cα · p + mc 2 β. (2.19) Here α1, α2, α3 and β are the so-called Pauli 4 × 4 matrices [38] which are chosen to ensure that (D 0 ) 2 = −c 2 ∆ + m 2 c 4 . The usual rep... |

5 |
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5 | Self-adjointness for Dirac operators via Hardy-Dirac inequalities
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Citation Context ...ed domain satisfies H 1 (R 3 , C 4 ) � D(D 0 +V ) ⊂ H 1/2 (R 3 , C 4 ). Furthermore this domain is not invariant by the projector P on the upper spinor (a characterization of this domain was given in =-=[16]-=-). The generalization to this case is possible but it is outside the scope of this paper. Remark 2.7. By Theorem 2.4, we see that Spu(D 0 , P) = ∅ but Spu(D 0 + V, P) ̸= ∅ for all smooth potentials V ... |

5 |
Basis set expansion of the dirac operator without variational collapse
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Citation Context ...s of periodic Schrödinger operators [4] or Sturm-Liouville operators [35, 36, 1]. It is a well-documented difficulty in Quantum Chemistry and Physics in particular regarding relativistic computations =-=[13, 18, 22, 34, 14, 27, 32]-=-. It also appears in elasticity, electromagnetism and hydrodynamics; see, e.g. the references in [2]. Eventually, it has raised as well a huge interest in the mathematical community, see, e.g., [23, 9... |

5 | BGX: a fully Bayesian gene expression index for Affymetrix GeneChip data - Hein, Richardson, et al. - 2005 |

5 |
Theory of finite systems of particles
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Citation Context ...rt spaces. In the case of atoms and molecules, the fact that inequalities of the form (1.6) imply the existence of bound states is the content of the famous HVZ theorem, first proved independently in =-=[12, 27, 28]-=-. In this paper we prove an HVZ-type theorem for our polaron functional (1.5) when N ≥ 2. We have to face two difficulties. First the functional is invariant under the action of arbitrarily large tran... |

4 |
Conditions for convergence of variational solutions of Dirac’s equation in a finite basis, Phys
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- 1982
(Show Context)
Citation Context ...s of periodic Schrödinger operators [4] or Sturm-Liouville operators [35, 36, 1]. It is a well-documented difficulty in Quantum Chemistry and Physics in particular regarding relativistic computations =-=[13, 18, 22, 34, 14, 27, 32]-=-. It also appears in elasticity, electromagnetism and hydrodynamics; see, e.g. the references in [2]. Eventually, it has raised as well a huge interest in the mathematical community, see, e.g., [23, 9... |

4 |
Method of orthogonal projections and approximation of the spectrum of a bounded operator
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4 |
Dual kinetic balance approach to basis-set expansions for the Dirac equation, Phys
- Shabaev, Tupitsyn, et al.
(Show Context)
Citation Context ...s of periodic Schrödinger operators [4] or Sturm-Liouville operators [35, 36, 1]. It is a well-documented difficulty in Quantum Chemistry and Physics in particular regarding relativistic computations =-=[13, 18, 22, 34, 14, 27, 32]-=-. It also appears in elasticity, electromagnetism and hydrodynamics; see, e.g. the references in [2]. Eventually, it has raised as well a huge interest in the mathematical community, see, e.g., [23, 9... |

4 |
Geometric methods for nonlinear many-body quantum systems
- Lewin
(Show Context)
Citation Context ...th of the attractive Coulomb term as compared to the natural repulsion between the electrons, one can get binding or not. It is an important issue to determine in which parameter range binding occurs =-=[10, 6, 7, 13]-=-. The approximations made in the construction of Fröhlich’s and Pekar’s models reduce their applicability to situations where the N-polaron is spread over a region of space much larger than the charac... |