SOME MATHEMATICAL AND NUMERICAL PROBLEMS
IN RELATIVISTIC QUANTUM MECHANICS
Maria J. Esteban
Ceremade (UMR CNRS no. 7534), Universit´e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris C´edex 16, France
E-mail: esteban@ceremade.dauphine.fr
Internet: http://www.ceremade.dauphine.fr/ eesteban/
May 3, 2008
Reporting on various works done in collaboration with
J. Dolbeault, E. S´er´e and M. Loss.
Abstract
This note is a written version of the talk given at the French-Italian conference which took place in Torino in July 2006. In this talk I presented an overwiew of various results about the characterization of eigenvalues of self-adjoint operators in a spectral gap and its application to relativistic quantum mechanics, in order to define and compute the eigenvalues of Coulomb-Dirac type operators, with or without external magnetif field. The talk contained information about rigorous mathematical results and also about numerical computations done for some model problems in atomic and molecular physics. All the results described here can be found in [2, 4, 3, 1] and a very complete review about all this kind of problems for strongly indefinite operators, in particular the Dirac operator, can be found in [5].
1
Introduction.
The basic mathematical question that lies behind the works described here is the study of eigenvalues of a self-adjoint operator A in a spectral gap of the spectrum of A : an interval which does not contain any point of the essential spectrum, and such that there is continuous spectrum at the right and at the left of it. Of course, computing eigenvalues in a spectral gap is equivalent, at least formally, to finding critical values of the Rayleigh quotient, (Ax, x)
||x||2 ,
corresponding to critical points with an infinite Morse index. This is in contrast what which happens in the case of a semi-bounded self-adjoint operator. In this case, the eigenvalues of finite multiplicity that are below the continuous spectrum correspond to critical points with
finite Morse index and their variational characterization is well known. For instance, the lowest one is given as λ1= min x6=0 (Ax, x) ||x||2 ,
and the next ones can be characterized either by a min-max problem or by a minimization problem in the orthogonal space to the space generated by the previous eigenfunctions.
In the second case, for eigenvalues in spectral gaps, if there is for instance a lowest eigenvalue inside the gap, it should be defined as
λ1= min
? max?
(Ax, x) ||x||2
and the question is what to write below the min and the max, but in any case it is clear that it would have to be infinite dimensional spaces.
This question can arise in many problems in physics. For instance in solid state physics, where one is dealing with electronic behaviour in a regular cristal in which some impurities are included. The periodic potential created by the periodic cristal lattice will often open gaps in the otherwise only continuous spectrum of the Schr¨odinger operator. The impurities can create eigenvalues in those spectral gaps and their computation is important to understand the properties of such cristals.
Another field in which computing eigenvalues in spectral gaps is important is relativistic quantum mechanics, where one is interested in understanding the dynamics of an electron in the presence of an external electrostatic field created by the nucleii around and maybe also some external magnetic field. The basic operator in this theory is the free Dirac operator which in well chosen units can be written as :
H = −i α · ∇ + β , α1, α2, α3, β ∈ M4×4(CI ) ,
where the Dirac-Pauli matrices can be written as follows:
β = I 0 0 −I , αk = 0 σk σk 0 (k = 1, 2, 3) σ1= 0 1 1 0 , σ2= 0 −i i 0 , σ3= 1 0 0 −1 Two of the main properties of H are:
H2 = − ∆ + 1I, σ(H) = (−∞, −1] ∪ [1, +∞)
Remark. Note that since the Dirac-Pauli matrices are 4×4, H acts on functions ψ : IR3→ CI4 The main question now is what happens if we add an external local potential V to H. Does it create eigenvalues in the spectral gap (−1, 1)? Let us consider the particular case of Coulomb potentials Vν:= −|x|ν , ν > 0. These potentials play an important role in atomic and molecular
physics since they are used as basic description for electrostatics description. In the units that we used to write the Dirac operator, Vν would describe the potential created by a nucleus of
charge ν/α, where α = 1/127, 027... is the so-called fine structure constant.
As it is well known (see [6] for references) that the operator Hν := H − |x|ν can be defined
as a self-adjoint operator keeping its physical sense if 0 < ν < 1. Its spectrum is given by
0 < λν1=
p
1 − ν2≤ · · · ≤ λν
k· · · < 1 . (1)
But what happens if we consider a potential V which is not exactly a Coulomb potential? The explicit computations used to compute the eigenvalues and the eigenfunctions for the Dirac-Coulomb operator cannot be performed anymore and then a more general theory is necessary to characterize (and count) the possible eigenvalues and also good, robust and efficient methods are necessary to compute them approximately.
In the case of an external magnetic field B, the question is still the same and the only change to be performed is to replace ∇ by ∇B = ∇ − iAB in the operator expression, where AB is
a potential associated to B, that is, it satisfies curl AB = B.
So, the free magnetic Dirac operator can be written as
HB= −i α · ∇B+ β .
If we consider HB+ V , if this operator is self-adjoint and if its spectrum is the union of
(−∞, −1] ∪ [1, +∞)
plus a sequence of eigenvalues in the gap (−1, 1), a question of physical interest is to understand for which values of B, λ1(B, V ) approaches the limits of the gap, that is, either −1 or 1.
2
Abstract min-max theorem (Dolbeault, Esteban, S´
er´
e,
[2]) and consequences.
In [2], with Jean Dolbeault and Eric S´er´e we proved a general theorem to deal with this kind of question. The proof lies upon the “elimination” of half of the functional space in an optimal way, and then apply more classical spectral theory to the reduced operator.
Theorem 1 Let H be a Hilbert space and A : D(A) ⊂ H → H a self-adjoint operator. Let us denote by D(A) A’s domain. Let H+, H− be two orthogonal Hilbert subspaces of H such that
H = H+⊕H− and let denote by Λ± the corresponding projectors. Assume the existence of a
core F (i.e. a subspace of D(A) which is dense for the norm . D(A)), such that : (i) F+= Λ+F and F−= Λ−F are two subspaces of A’s form domain F(A).
(ii) a−:= sup x−∈F−\{0}
(x−, Ax−)
kx−k2H
< +∞. Let us define ck= inf
V subspace of F+ dim V =k sup x∈(V ⊕F−)\{0} (x, Ax) ||x||2 H , k ≥ 1.
If (iii) c1> a− , then ck is either the k-th eigenvalue of A in the interval (a−, b), where
b = inf (σess(A) ∩ (a−, +∞)) or it is equal to b.
In the case where the operator A is built on the Dirac operator, we can consider the following orthogonal decomposition of the functional space :
ψ : IR3→ C4, ψ = ϕ χ = ϕ 0 + 0 χ , ϕ, χ : IR3→ CI2 Assume now for instance that the potential V satisfies
lim
|x|→+∞V (x) = 0 , −
ν
This assumption is far from being necessary, but helps to understand how the method works. For better assumptions and more discussion about it, see [2]. We will assume it from now on, in the sequel of this paper.
Then, for all k ≥ 1, the k-th eigenvalue of H + V in the spectral gap (−1, 1) is defined by: λk(V ) = inf Y subspace of C∞o (IR3,CI 2 ) dimY =k sup ϕ∈Y \{0} sup ψ=(ϕχ) χ∈C∞0 (IR3 ,C2) (ψ, (H + V )ψ) (ψ, ψ) .
In particular, one can verify that the first eigenvalue of H + V in the gap (−1, 1) is given by λ1(V ) := inf ϕ6=0 supχ (ψ, (H + V )ψ) (ψ, ψ) , ψ = ϕ χ and λ(ϕ) := sup ψ=(ϕχ) χ∈C∞0 (IR3 ,C2) (ψ, (H + V )ψ) (ψ, ψ)
is the unique number λ such that Z IR3 |σ · ∇ϕ|2 1 − V + λ+ (1 + V )|ϕ| 2dx = λZ IR3|ϕ| 2dx . (3)
A very nice corollary of the abstract theorem and the above comments is that the lowest eigenvalue in the spectral gap (−1, 1) can be defined as the infimum of some “energy” functional, which is very satisfactory for the definition of a number that is supposed to describe the ground state of an electron:
λ1(V ) = inf
ϕ∈C∞0 (IR3 ,C2) ϕ6=0
λ(ϕ) (4)
where the functional λ(·) is defined in (3).
A straightforward corollary of (3)-(4) is the inequality Z IR3 |σ · ∇ϕ|2 1 − V + λ1(V )+ (1 + V )|ϕ| 2dx ≥ λ 1(V ) Z IR3|ϕ| 2dx (5)
where λ1(V ) is the best constant in the above inequality.
As we see below, easy consequences of the above inequalities are: • Easy to implement algorithms, which are robust.
• A new type of Hardy-like inequality for Dirac operators.
• The possibility to study rather deeply the case with an external magnetic field.
3
Hardy-like inequalities for Dirac operators.
When N = 3, the classical Hardy inequality is written as −∆ ≥ 1
4|x|2 , or what is equivalent, Z R3|∇u(x)| 2 dx ≥ Z R3 |u(x)|2 4|x|2 dx , ∀u ∈ H 1(R3) .
In the case of the Dirac operator, the “maximal” singularity is |x|1 . and the optimal inequality is: Z R3 |σ · ∇ϕ|2 1 + 1 |x| + |ϕ|2 ! dx ≥ Z R3 |ϕ|2 |x| dx , for all ϕ ∈ H 1(R3, C2) .
Indeed, we know that for Vν= −|x|ν , ν ∈ (0, 1) , λ1(ν/|x|) =
√
1 − ν2 (see (1)). So,
from (5), for all ν ∈ (0, 1), Z R3 |σ · ∇ϕ|2 1 +√1 − ν2+ ν |x| + 1 −p1 − ν2|ϕ|2 ! dx ≥ ν Z R3 |ϕ|2 |x| dx . Now, passing to the limit as ν goes to 1 in this functional inequality, we get
Z R3 |σ · ∇ϕ|2 1 + 1 |x| + |ϕ|2 ! dx ≥ Z R3 |ϕ|2 |x| dx , for all ϕ ∈ H 1(R3, C2) ,
where all the constants are optimal and cannot be improved. We called this inequality Hardy-Dirac inequality, because of its relation with the usual Hardy inequality (see below) and also because w.r.t. the Dirac operator it plays the same part as the classical Hardy inequality for the Schr¨odinger operator. To explain the relation between the two inequalities, replace ϕ(·) by ε−1ϕ(ε−1·) and take the limit as ε → 0, we obtain
Z R3|x| |σ · ∇ϕ| 2 dx ≥ Z R3 |ϕ|2 |x| dx , for all ϕ ∈ H 1(R3, C2) .
By taking ϕ = (f, 0) with f purely real, we end up with Z R3|x| |∇f| 2 ≥ Z R3 |f|2 |x| , for all f ∈ H 1(R3, C) ,
which is itself equivalent to Z R3|∇u| 2 dx ≥ 14 Z R3 |u|2 |x|2dx for all u ∈ H 1(R3, C) .
4
Towards an algorithm to compute the eigenvalues.
Taking into account (3)-(4), for every λ, let A(λ) be the symmetric bounded from below operator defined by the quadratic form acting on 2-spinors:
(ϕ, A(λ)ϕ) := Z IR3 |σ · ∇ ϕ|2 λ + 1 − V + (V + 1 − λ) |ϕ| 2dx
and consider its lowest eigenvalue, µ1(λ). Because of the monotonicity of A(λ) with respect
to λ , there exists a unique λ for which µ1(λ) = 0. This λ is actually the lowest eigenvalue of
A in the gap (−1, 1).
Let us now consider a n-dimensional space of functions from IR3into C2spanned by ϕ
1, ϕ2, . . . , ϕn.
Define the n × n matrix An(λ) whose entries are
Ai,jn (λ) = Z IR3 (σ · ∇ ϕi, σ · ∇ ϕj) λ + 1 − V + (V + 1 − λ) (ϕi, ϕj) dx Define µn
1(λ), the smallest eigenvalue of An(λ)
Then, the unique zero of the map λ 7→ µn
1(λ), λn1, is an approximation of the lowest
1 2 3 4 5 c1 c2 c3 c4 c5 ¯ µ( )λ λ λ λ λ λ λ
More concretely, if for every i, the curve Ci corresponds to the set { (λ, µi(λ) } where µi(λ)
denotes the i-th eigenvalue of the matrix A(λ), The point λi at which the curve crosses the
level µ = 0, that is, the point λi such that µi(λi) = 0 is the i-th eigenvalue of H + V in the
gap (−1, 1). We can do it at the continuous and also at the discrete level : Ai,jn (λ) = Z IR3 (σ · ∇ ϕi, σ · ∇ ϕj) λ + 1 − V + (V + 1 − λ) (ϕi, ϕj) dx
We have implemented this method on atomic computations (basically 1-d problem), for H, He+, Cr23+ and T h89+, using Hermite polynomial basis sets or B-splines and in diatomic
molecular computations ( 2-d problem in cylindrical coordinates), for H2+and T h179+2 , using
B-splines discretization.
0
z
s Ground state of T h
89+
corresponding to Z = 90, one atom
J. Dolbeault, M.J. Esteban, E. S´er´e, M. Vanbreugel. Phys. Rev. Letters (2000)
0 s
z
Ground state of H+
2 corresponding to Z = 1, two atoms
0 z
s Ground state of T h
179+
2 corresponding to Z = 90, two atoms
5
Magnetic case.
As in the non-magnetic case, if we consider a potential V satisfying (2) and a constant magnetic potential B and if the conditions of the abstract theorem in Section 2 are satisfied for the operator HB+ V , then its lowest eigenvalue in the spectral gap (−1, 1) can be again written as
λ1(B, V ) = inf ϕ∈C∞0 (IR3 ,C2) ϕ6=0 λB(ϕ) , where λB(ϕ) is defined by Z IR3 |σ · ∇Bϕ|2 1 − V + λB(ϕ) + (1 − λ B(ϕ))|ϕ|2dx = Z IR3V |ϕ| 2dx .
From the physical viewpoint, a relevant question here is for which values of B do we have λ1(B, V ) ∈ (−1, 1) and when does it approach the limits of the spectral gap.
The tool that we used in [2] to answer this question is inequality (4):
Z IR3 |(σ · ∇B)ϕ|2 1 + λ1(B, V ) − V dx + (1 − λ1 (B, V )) Z IR3|ϕ| 2 dx ≥ Z IR3V |ϕ| 2dx .
Indeed, since we know that λ1(B, V ) is the best constant in (4), to check that λ1(B, V ) < C
for some C, it is enough to find a ¯ϕ such that
(∗) Z IR3 |(σ · ∇B) ¯ϕ|2 1 + C − V dx + (1 − C) Z IR3| ¯ϕ| 2 dx < − Z IR3V | ¯ϕ| 2dx
And to check that λ1(B, V ) > C one has to prove that for all ϕ
(∗∗) Z IR3 |(σ · ∇B)ϕ|2 1 + C − V dx + (1 − C) Z IR3|ϕ| 2dx > −Z IR3V |ϕ| 2dx
and of course we are interested in the case C = 1 in (*), and C = −1 in (**).
In the case of the Coulomb potential and constant external magnetic field : Vν = −|x|ν ,
AB(x) := B2 −x2 x1 0 ! , B(x) := 00 B ! ; ν ∈ (0, 1), B ≥ 0 we prove the following results (see [2]) :
• For B = 0, λ1(0, Vν) =
√
1 − ν2 ∈ (−1, 1) .
• For all B ≥ 0, λ1(B, Vν) < 1 .
• For all ν ∈ (0, 1) there exists a critical magnetic strength Bν such that
λ1(B, Vν) ≤ −1 if B ≥ Bν.
• limν→1Bν> 0 , and lim
ν→0 ν log Bν = π .
The first three above results are obtained by using approppriate test functions and also by some functional inequalities related to (4). For ν small, the asymptotics for Bν is given by a
very simple problem corresponding to the lowest relativistic Landau level. In order to prove the asymptotics ν ∼ 0+, we introduce a new problem :
c0(B, ν) := inf ϕ∈X0 ϕ6=0 sup χ∈X0 ψ=(ϕ χ) ((HB+ Vν)ψ, ψ) ||ψ||2 ,
where X0 is the space of functions in the lowest Landau level :
ϕ ∈ X0 iff ϕ(x1, x2, x3) = √ B √ 2π e −Bs2 4 0 f (x3) , s2= x21+ x22
and this leads us to work with the 1d problem
c0(B, ν) = inf f ∈C∞ 0 (IR,C)\{0} λB0(f ) , where λB 0(f ) is defined by λB0(f ) Z +∞ −∞ |f(z)| 2dz =Z +∞ −∞ |f′(z)|2 1 + λB 0(f ) + ν aB0(z) + (1 − ν aB0(z)) |f(z)|2 dz, aB
0 being the Coulomb potential reduced to the class X0 :
aB0(z) = B Z +∞ 0 s e−Bs2 2 √ s2+ z2ds .
In [2] we proved the following important result
Theorem 2 for every ν ∈ (0, 1) such that ν + ν3/2< 1, there exists a constant d
ν > 0 such
that for all dν ≤ B < Bν,
c0(B, ν + ν3/2) ≤ λ1(B, ν) ≤ c0(B, ν − ν3/2) . (6)
Moreover, since as ν → 0+, ν3/2<< ν,
c0(B, ν + ν3/2) ∼ c0(B, ν − ν3/2) . (7)
And from (6) and (7) we find:
lim
ν→0 ν log Bν = π ,
by looking for the smallest B > 0 s.t. there is a sequence {Bn}n, such that
Bnր B and c0(Bn, ν ± ν3/2) ց −1 ;
Remark. The two problems c0(B, ν ±ν3/2) in (7) are both 1d ! Hence they are much easier to
analyze mathematically and it is also much easier to build approximate solutions by numerical methods.
References
[1] J. Dolbeault, M.J. Esteban, M. Loss. Relativistic hydrogenic atoms in strong magnetic fields. Ann. H. Poincar´e 8(4) (2007), p. 749–779.
[2] J. Dolbeault, M.J. Esteban, E. S´er´e. On the eigenvalues of operators with Dolbeault-Esteban-Sere-00B. Application to Dirac operators. J. Funct. Anal. 174 (2000), p. 208–226.
[3] J. Dolbeault, M.J. Esteban, E. S´er´e. A variational method for relativistic compu-tations in atomic and molecular physics. Int. J. Quantum. Chemistry 93 (2003), p. 149–155.
[4] J. Dolbeault, M.J. Esteban, E. S´er´e, M. Vanbreugel. Minimization methods for the one-particle Dirac equation. Phys. Rev. Letters 85(19) (2000), p. 4020–4023. [5] M. J. Esteban, M. Lewin, E. S´er´e. Variational methods in relativistic quantum
me-chanics. To appear in Bull. A.M.S.
