Existence of ground states for negative ions at the binding threshold

0.2 MB·English

by  Jacopo Bellazzini

EXISTENCE OF GROUND STATES FOR NEGATIVE IONS
AT THE BINDING THRESHOLD
3 JACOPO BELLAZZINI, RUPERT L. FRANK, ELLIOTT H. LIEB,
1
AND ROBERT SEIRINGER
0
2
n Abstract. As the nuclear charge Z is continuously decreased an N-electron
a atom undergoes a binding-unbinding transition at some critical Zc. We in-
J vestigate whether the electrons remain bound when Z = Zc and whether the
3 radius of the system stays finite as Zc is approached. Existence of a ground
2 stateatZc isshownunderthe conditionZc <N−K,whereK isthe maximal

] number of electrons that can be removed at Zc without changing the ground
h state energy.
p
-
h
t
a
1. Introduction and main result
m
[ The energy of a quantum-mechanical system composed of N electrons and one

1 fixed nucleus of charge Z > 0 is described by the Hamiltonian
v
N
0 Z 1
7 (1.1) p2 − + .
3 i=1 (cid:18) i |xi|(cid:19) i<j |xi −xj|
5 X X
. (Here we use units in which the electron mass m = 1/2, the electron charge
1
0 e = −1 and Planck’s constant ~ = 1.) The terms p2, where p = −i∇ , and
i i i
3
−Z/|x | describe the kinetic energy of the i-th electron and its potential energy
1 i
: due to the attraction to the nucleus, respectively. The term |xi−xj|−1 stands for
v
the potential energy due to the repulsion between the i-th and the j-th electron.
i
X
The Pauli principle dictates that the Hamiltonian is considered as acting in the
r subspace L2(R3N) of anti-symmetric functions in L2(R3N), that is, ψ’s satisfying
a a
ψ(...,x ,...,x ,...) = −ψ(...,x ,...,x ,...) for i 6= j.
i j j i
(For the sake of simplicity, we ignore the electron spin. It can be included easily.)
The ground state energy of the system is given by the bottom of the spectrum
of the Hamiltonian (1.1). If this bottom of the spectrum is an eigenvalue, that
Date: January 22, 2013.
(cid:13)c2013 by the authors. This paper may be reproduced, in its entirety, for non-commercial
purposes.
Partial financial support from PRIN 2009 ‘Metodi Variazionali e Topologici nello Studio di
Fenomeni non Lineari’ (J.B.), the U.S. National Science Foundation through grants PHY-
1068285(R.F.),PHY-0965859(E.L.),theSimonsFoundation(#230207,E.L.)andtheNSERC
(R.S.) is acknowledged.
1
2 J. BELLAZZINI,R. L. FRANK,E. H. LIEB, AND R.SEIRINGER
is, if there is an eigenfunction in L2(R3N), then the system is said to be bound
a
and the eigenfunction describes its ground state.
It is intuitively clear that a nucleus of charge Z can bind N electrons if Z is
large compared to N, and that it cannot if Z is small compared to N. This fact
canalso be shown mathematically: Zhislin [14] proved that the system is bound if
Z > N−1andNam[9]showed thatthesystemisnotboundifN ≥ 1.22Z+3Z1/3,
improving the earlier condition N ≥ 2Z + 1 of [7]. For asymptotic results as
Z → ∞, see, for instance, [10, 12, 8, 11]. In these results, and also in our paper,
we shall consider Z as an arbitrary positive (not necessarily integer) parameter.
Then the cited results imply that for fixed N there is (at least one) critical value
of Z where a binding-unbinding transition occurs.
In this paper we are interested in this binding-unbinding transition. More
precisely, we investigate whether the system is bound at this critical Z value,
that is, whether a ground state exists. Intuitively, this question is related to
the size of the system. As the continuous parameter Z moves from the binding
regime across the critical value into the unbinding regime, there are two possible
scenarios: Either the size of the system increases indefinitely and becomes infinite
as Z reaches the critical value Z , or else the size of the system approaches a finite
c
value at the critical value and then jumps discontinuously to infinity. The first
scenario corresponds to the case where no groundstate exists at the critical value,
and the second one to where it does exist.
This question has been discussed in the physics literature (see, e.g., [13]) and
it was proved by Th. and M. Hoffmann-Ostenhof and Simon [5] that for two-
electron atoms in the spin singlet state the second scenario occurs, that is, there
is a ground state at the critical coupling value. This corresponds to the case
N = 2, but without the anti-symmetry assumption. On the other hand, Th. and
M. Hoffmann-Ostenhof [6] showed that in the triplet S-sector the first scenario
occurs. This corresponds to the N = 2 case with anti-symmetry, but the ad-
missible functions are further restricted to depend only on |x |, |x | and x ·x .
1 2 1 2
Very recently Gridnev [4] has, among other things, generalized the Hoffmann-
Ostenhof–Simon existence result to arbitrary N, but with the additional assump-
tion that Z lies in the interval (N −2,N −1). He has also conjectured that the
c
assumption of a lower bound N −2 on the critical Z is not necessary.
Our goal here in this paper is to reprove Gridnev’s result by completely dif-
ferent means and to replace his assumption by a weaker one which we believe
to be optimal. (In contrast to [4], however, we only consider an infinitely heavy
nucleus.) Our method extends the one introduced in [3], where an alternative
proof of the Hoffmann-Ostenhof–Simon existence result was given. In contrast to
[5] no positivity of the ground state is needed and the proof in [3], as well as the
proof in this paper, extend to the case where the particles move in an external
magnetic field, for instance.
EXISTENCE OF GROUND STATES AT THE BINDING THRESHOLD 3
We proceed to formulate our results precisely. It is physically equivalent and
mathematically convenient to rescale the x in (1.1) by U = 1/Z. We then
i
find that, apart from an overall factor of Z2, the Hamiltonian (1.1) is unitarily
equivalent to the Hamiltonian
N
1 U
(1.2) H(N) := p2 − + .
U i |x | |x −x |
i=1 (cid:18) i (cid:19) i<j i j
X X
We consider H(N) as a self-adjoint operator in the Hilbert space L2(R3N) of anti-
U a
symmetric functions and denote its ground state energy by
(N) (N) (N)
E = inf spec H = inf hψ|H |ψi.
U U U
kψk=1
(N) (N)
The ground state energy E of H is a non-decreasing, concave function of U
U U
and one has the ordering
(N) (N−1) (N−2)
E ≤ E ≤ E ≤ ...
U U U
with respect to N. We define U(N) = {U > 0 : E(N) < E(N−1)}. The set of
U U
critical coupling constants is given by
U(N) = ∂U(N)
c
(N) (N−1) (N) (N−1)
= U > 0 : E = E and E < E for some U → U .
U U Un Un n
n o
(N) (N−1)
As mentioned before, by Zhislin’s theorem E < E for U < 1/(N−1) and,
U U
(N) (N−1) (N)
for instance by [7], E = E for all large U. Thus, U is non-empty. It is
U U c
natural to believe that U(N) is a single interval and that U(N) contains only one
c
element, but we do not know how to prove this.
(N)
According to the HVZ theorem (see, e.g., [2]) the strict inequality E <
U
(N−1) (N) (N)
E implies that E is an eigenvalue of H and, consequently, a ground
U U U
state exists. On the other hand, it is clear that if U ∈ R \ U(N) ∪U(N) , then
+ c
E(N) is not an eigenvalue of H(N). (cid:16) (cid:17)
U U
The following theorem is our main result. It gives a sufficient condition for
(N) (N) (N)
E to be an eigenvalue of H for U ∈ U . The sufficient condition (for
Uc Uc c c
(N)
fixed U ∈ U ) depends on an integer 1 ≤ K ≤ N − 1, which is the largest
c c
(N) (N−K)
integer such that E = E . Thus, we have
Uc Uc
(N) (N−1) (N−K) (N−K−1)
E = E = ... = E < E ,
Uc Uc Uc Uc
(0)
where we interpret E = 0. Physically, K denotes the maximal number of
Uc
electrons that can be removed from the system at U = U without changing the
c
energy.
4 J. BELLAZZINI,R. L. FRANK,E. H. LIEB, AND R.SEIRINGER
Our main result is as follows.
Theorem 1.1 (Binding for N electrons at threshold). Let U ∈ U(N) and assume
c c
that U > 1 , where K is the largest integer such that E(N) = E(N−K). Then
c N−K Uc Uc
H(N) has a ground state eigenfunction 0 6≡ ψ ∈ L2(R3N).
Uc Uc a
Remark 1.1. Thetheoremisalsovalidiftheanti-symmetryassumptionisdropped
or, which is the same, replaced by a symmetry assumption. The proof in this
unconstrained case follows along the same line and is actually somewhat simpler,
see Remark 3.1. The method also goes through essentially without modification
in the case of spin.
Remark 1.2. In the case N = 2 and without the anti-symmetry assumption, the
strict inequality U > 1 is a classical result of Bethe [1]. Thus, (the symmetric
c
version of) Theorem 1.1 extends the results of [5] and [3].
Remark 1.3. In the case N = 2 and with the anti-symmetry assumption enforced,
one has equality U = 1 and no ground state exists (at least no ground state
c
depending only on |x |, |x | and x · x ) [6]. This suggests that, in general, the
1 2 1 2
assumption U > 1 may not be dropped.
c N−K
Remark 1.4. Our theorem generalizes Gridnev’s result (in the case of infinite
nuclear mass). Indeed, Gridnev’s assumption U < 1 implies, by Zhislin’s
c N−2
(N−1) (N−2)
theorem, that E < E and thus K = 1. Gridnev’s second assumption
Uc Uc
U > 1 coincides with our assumption. We emphasize that the main difficulty
c N−1
that we overcome in this paper is the case U ≥ 1 or, more generally, the case
c N−2
where K ≥ 2.
2. Strategy of the proof
A standard way to prove the existence of ground states at threshold is to
prove that the weak limit of a ground state sequence ψ when U → U is
Un n c
not zero. Once one knows that the weak limit is non-zero it is easy to see that
this weak limit has to be a ground state. In the spirit of [3] we give an upper
bound on the ‘radius’ of ψ which is independent on U . Hereafter we denote
Un n
|x| := max{|x |,...,|x |}. We interpret this as the maximal distance of the
∞ 1 N
electrons from the nucleus.
Theorem 2.1 (Uniformupperboundontheradiusoftheatom). Forany δ,θ > 0
and any 1 ≤ K ≤ N −1 there are constants m,R > 0 such that for all U ∈ U(N)
satisfying
1
(2.1) U ≥ +δ and E(N−1) ≤ E(N−K−1) −θ
N −K U U
EXISTENCE OF GROUND STATES AT THE BINDING THRESHOLD 5
and for all normalized ground states ψ of H(N) one has
U U
(2.2) hψ ||x|−1|ψ i ≥ R−1 and hψ |χ |ψ i ≥ m.
U ∞ U U {|x|∞<R} U
(N)
This theorem immediately implies Theorem 1.1. Indeed, let U ∈ U , let K
c c
bethelargestinteger suchthatE(N) = E(N−K) andassumethatU > 1 . Then
Uc Uc c N−K
there is a non-empty subset of U(N), containing U in its closure, with elements
c
satisfying (2.1) for
1 1 1
(N−K−1) (N−1)
δ = U − > 0 and θ = E −E > 0.
2 c N −K 2 Uc Uc
(cid:18) (cid:19)
(cid:16) (cid:17)
Then Theorem 2.1 implies that any sequence ψ of normalized ground states of
Un
(N)
H with U from this subset and with U → U satisfies (2.2). Since ψ is
Un n n c Un
bounded in H1(R3N) it has a weak limit in this space which, according to (2.2)
is not identically zero. One easily verifies that the weak limit is a ground state of
(N)
H , thus proving Theorem 1.1. This observation reduces the proof of Theorem
Uc
1.1 to the proof of Theorem 2.1.
Theorem 2.1, in turn, can be deduced from two lemmas that we state next.
The first one, and this is the novelty of this paper, is an operator inequality for
(N) (N) (N)
H . It estimates H −E from below by a potential well which is attractive
U U U
of order l−2 for |x| < l, but has a repulsive Coulomb tail for |x| > l. The
∞ ∞
parameter l can be chosen arbitrary large and the constants are uniform in U.
The precise statement is
Lemma 2.1 (Operator inequality). For any δ,θ > 0 and 1 ≤ K ≤ N −1 there
are three constants c,C,l > 0 such that for all U ∈ U(N) satisfying (2.1) and all
0
l ≥ l one has
0
C c
(N) (N) (N−1) (N)
(2.3) H −E ≥ − χ + E −E + χ .
U U l2 {|x|∞<l} U U |x| {|x|∞≥l}
(cid:18) ∞(cid:19)
The second ingredient in the proof of Theorem 2.1 is the following elementary
lemma, which we cite from [3].
Lemma 2.2 (Calculuslemma). Letρ ∈ L1(R+) benon-negativewith ∞ρ(r)dr =
0
1. Assume that there are constants b > 0 and l ≥ 0 such that
0
R
b l ∞ ρ(r)
(2.4) ρ(r)dr ≥ dr
l2 r
Z0 Zl
for all l ≥ l . Then
0
∞ ρ(r) 1
(2.5) dr ≥
r 2(b+l )
Z0 0
6 J. BELLAZZINI,R. L. FRANK,E. H. LIEB, AND R.SEIRINGER
and
l0 l2
(2.6) ρ(r)dr ≥ 0 .
(b+l )2
Z0 0
We now use Lemmas 2.1 and 2.2 to give the
Proof of Theorem 2.1. Fix δ,θ > 0 and 1 ≤ K ≤ N − 1 and consider U’s as in
(N)
the statement of the theorem. Any normalized ground state ψ of H satisfies,
U U
according to the operator inequality 2.3,
C |ψ |2
|ψ |2dx ≥ c U dx
l2 U |x|
Z{|x|∞<l} Z{|x|∞≥l} ∞
for all l ≥ l and some constants c,C,l > 0. (Here we also used the fact that
0 0
(N−1) (N)
E ≥ E .) Lemma 2.2 now implies that
U U
|ψ |2 c c2l2
U dx ≥ and |ψ |2dx ≥ 0 .
|x| 2(C +l c) U (C +l c)2
ZR3N ∞ 0 Z{|x|∞<l0} 0
(cid:3)
This proves (2.2).
To summarize the content of this section, we have reduced the proof of our
main result to the proof of the operator inequality in Lemma 2.1. This will be
accomplished in Section 4 after some preparations in Section 3.
3. A partition of unity
Our goal in this section is to construct a partition of unity in R3N with aneffec-
tive control on the Coulomb interaction between (subsets of) the ‘particles’ (elec-
trons) x ,...,x ∈ R3 and another ‘particle’ (nucleus) at the origin. Throughout
1 N
this section we fix an integer N ≥ 2.
In order to formulate the properties of our partition of unity we need to intro-
duce some notation. For every integer 1 ≤ k ≤ N we denote by J the collection
k
of all integer sequences (J ,...,J ), where 1 ≤ J ≤ N for all l and where all the
1 k l
J ’s are mutually distinct. For given J ∈ J and x ∈ R3N we denote by xˆ the
l k J
vector in R3(N−k) which coincides with x, but where the entries x ,...,x have
J1 Jk
been erased. Thus, if we denote as usual |x| := max{|x |,...,|x |}, then
∞ 1 N
|xˆ | = max{|x | : j 6= J ,...,J }.
J ∞ j 1 k
One can think of |xˆ | as the maximum distance of the N−k electrons from the
J ∞
nucleus after having removed the k electrons with indices in J.
Finally, if π ∈ S is a permutation and x ∈ R3N,J ∈ J , we set
N k
x = (x ,...,x ) and π(J) = (π(J ),...,π(J )).
π π(1) π(N) 1 k
Here is the description of our partition of unity.
EXISTENCE OF GROUND STATES AT THE BINDING THRESHOLD 7
Proposition 3.1 (Partition of unity). For any integer 1 ≤ K ≤ N −1 and any
0 < ǫ < 1/2 there is a constant c > 0 with the following property. For any l > 0
ǫ
there is a quadratic partition of unity,
N
Λ (x)2 + Λ (x)2 + Λ (x)2 = 1 for all x ∈ R3N ,
0 j J
Xj=1 J∈XJK+1
with
Λ (x) = 0 unless |x| ≤ l
0 ∞
and, for 1 ≤ j ≤ N,
1 1 1−2ǫ
Λ (x) = 0 unless ≥
j
N −K |x −x | |x |
m j j
m6=j
X
and, for J = (J ,...,J ) ∈ J ,
1 K+1 K+1
l
Λ (x) = 0 unless |x| ≥ ,
J ∞
2
1
|x | ≥ |xˆ | for all 1 ≤ n ≤ K +1,
Jn 2 J1,J2,...,Jn−1 ∞
1 1 1−ǫ
≤ for all 1 ≤ n ≤ K.
N −n |x −x | |x |
m6=XJ1,...,Jn m Jn Jn
Moreover, we have
N
c
|∇Λ (x)|2 + |∇Λ (x)|2 + |∇Λ (x)|2 ≤ ǫ if Λ (x) > 0
0 j J l2 0
Xj=1 J∈XJK+1
and
N
c
|∇Λ (x)|2 + |∇Λ (x)|2 + |∇Λ (x)|2 ≤ ǫ if Λ (x) < 1.
0 j J 0
l|x|
∞
Xj=1 J∈XJK+1
Finally, the Λ’s behave as follows under permutations π ∈ S .
N
Λ (x ) = Λ (x),
0 π 0
for 1 ≤ j ≤ N : Λ (x ) = Λ (x) if π(j) = j,
j π j
for J ∈ J : Λ (x ) = Λ (x) if π(J) = J .
K+1 J π J
Here, for n = 1 we set xˆ = x, that is, the condition |x | ≥
J1,J2,...,Jn−1 Jn
1|xˆ | for n = 1 means |x | ≥ 1|x| .
2 J1,J2,...,Jn−1 ∞ J1 2 ∞
Before proving this proposition we record some useful geometric facts.
8 J. BELLAZZINI,R. L. FRANK,E. H. LIEB, AND R.SEIRINGER
Lemma 3.1. Let 1 ≤ n ≤ N − 1, (J ,...,J ) ∈ J and assume that x ∈ R3N
1 n n
satisfies
1 1 1−ǫ
≤ .
N −n |x −x | |x |
m6=XJ1,...,Jn m Jn Jn
Then
|x |
(3.1) min |x −x | ≥ Jn
m6=J1,...,Jn m Jn (N −n)(1−ǫ)
and
ǫ
(3.2) |xˆ | ≥ |x |.
J1,J2,..,Jn ∞ 1−ǫ Jn
Proof. To prove the first claim it suffices to notice that
1 1 (N −n)(1−ǫ)
max ≤ ≤ .
m6=J1,...Jn |xm −xJn| m6=XJ1,...Jn |xm −xJn| |xJn|
To prove the second claim we shall show that |xˆ | < ǫ |x | implies
J1,J2,..,Jn ∞ 1−ǫ Jn
1 1 > 1−ǫ . Thus, assume that for all m 6= J ,...,J one
N−n m6=J1,...,Jn |xm−xJn| |xJn| 1 n
has |x | < ǫ |x |. By the triangle inequality we get
Pm 1−ǫ Jn
ǫ 1
|x −x | ≤ |x |+|x | < +1 |x | = |x |,
m Jn m Jn 1−ǫ Jn 1−ǫ Jn
(cid:18) (cid:19)
and therefore
1 1 1−ǫ
> ,
N −n |x −x | |x |
m6=XJ1,...,Jn m Jn Jn
(cid:3)
as claimed.
The following result is a consequence of the construction of the partition of
unity in Proposition 3.1 and of Lemma 3.1. It says that on the support of Λ
J
with J ∈ J , all K + 1 particles x ,...,x are ‘far out’, that is, at a
K+1 J1 JK+1
distance comparable to the distance of the particle that is farthest out. This will
be useful in the proof of Lemma 2.1.
Lemma 3.2. Let 1 ≤ k ≤ N and (J ,...,J ) ∈ J . Assume that x ∈ R3N
1 k k
satisfies
1
|x | ≥ |xˆ | for all 1 ≤ n ≤ k,
Jn 2 J1,J2,...,Jn−1 ∞
1 1 1−ǫ
≤ for all 1 ≤ n ≤ k −1.
N −n |x −x | |x |
m6=XJ1,...,Jn m Jn Jn
Then
n−1
1 ǫ
(3.3) |x | ≥ |x|
Jn 2n 1−ǫ ∞
(cid:18) (cid:19)
EXISTENCE OF GROUND STATES AT THE BINDING THRESHOLD 9
for every 1 ≤ n ≤ k.
In particular, with the notation of Proposition 3.1, let J ∈ J and let x ∈
K+1
R3N such that Λ (x) 6= 0. Then (3.3) holds for every 1 ≤ n ≤ K +1.
J
Proof. By (3.2) the second assumption yields |xˆ | ≥ ǫ |x | for all 1 ≤
J1,J2,..,Jn ∞ 1−ǫ Jn
n ≤ k −1. Combining this with the first assumption, we get
1 ǫ
|x | ≥ |xˆ | ≥ |x | for all 2 ≤ n ≤ k.
Jn 2 J1,J2,..,Jn−1 ∞ 2(1−ǫ) Jn−1
Iteratingthisboundandrecallingthat|x | ≥ 1|x| weobtainclaimedinequality.
J1 2 ∞
If Λ (x) 6= 0 for some J ∈ J , then the above assumptions are satisfied by
J K+1
(cid:3)
the construction of Λ .
J
We now turn to the
Proof of Proposition 3.1. We begin by constructing a finer partition of unity,
K+1
Λ (x)2 + Λ (x)2 = 1 for all x ∈ R3N .
0 J
Xk=1 JX∈Jk
The terms Λ and Λ with J ∈ J are as in the statement of the proposition.
0 J K+1
For J = (J ,...,J ) ∈ J with 1 ≤ k ≤ K, we will have
1 k k
l
Λ (x) = 0 unless |x| ≥ ,
J ∞
2
1
|x | ≥ |xˆ | for all 1 ≤ n ≤ k,
Jn 2 J1,J2,...,Jn−1 ∞
1 1 1−ǫ
≤ for all 1 ≤ n ≤ k −1,
N −n |x −x | |x |
m6=XJ1,...,Jn m Jn Jn
1 1 1−2ǫ
≥ .
N −k |x −x | |x |
m6=XJ1,...,Jk m Jk Jk
We will also prove that
K+1
c
(3.4) |∇Λ (x)|2 + |∇Λ (x)|2 ≤ ǫ if Λ (x) > 0
0 J l2 0
Xk=1 JX∈Jk
and
K+1
c
(3.5) |∇Λ (x)|2 + |∇Λ (x)|2 ≤ ǫ if Λ (x) < 1.
0 J 0
l|x|
∞
Xk=1 JX∈Jk
Moreover, the refined partition of unity has the symmetry property
(3.6) Λ (x ) = Λ (x)
J π π(J)
for all J ∈ J , 1 ≤ k ≤ K +1, and Λ (x ) = Λ (x).
k 0 π 0
10 J. BELLAZZINI,R. L. FRANK,E. H. LIEB, AND R.SEIRINGER
We now explain how to obtain the asserted partition of unity from the refined
one. We keep the terms Λ and Λ with J ∈ J . Moreover, for 1 ≤ j ≤ N we
0 J K+1
define
K
Λ (x) = Λ (x)2
j J
v
uuXk=1J∈JXk,Jk=j
t
Thus, if Λ (x) 6= 0, then there is a 1 ≤ k ≤ K and a J ∈ J with J = j such
j k k
that
1 1 1−2ǫ
≥ .
N −k |x −x | |x |
m j j
m6=XJ1,...,Jk
But this inequality implies that also
1 1 1−2ǫ
≥ ,
N −K |x −x | |x |
m j j
m6=j
X
which is the asserted condition for Λ . Moreover, by the Schwarz inequality,
j
K
|∇Λ | ≤ |∇Λ |2.
j v J
uuXk=1J∈JXk,Jk=j
t
Therefore, the gradient boundsfor thecoarser partitionof unity followfromthose
for the refined one.
The reason why we pass from the refined partition to the one stated in the
proposition is that the latter satisfies the required symmetry assumptions. This
is an immediate consequence of (3.6).
Thus, it remains to construct the refined partition of unity. Let us start by
introducing the following functions f,g : R+ → R+,
1 if 0 < s ≤ 1 ,
2
f(s) = cos(π(s− 1)) if 1 < s ≤ 1,

2 2
0 if s ≥ 1,
0 if 0 < s ≤ 1−2ǫ,

g(s) = sin(π(s−(1−2ǫ))) if 1−2ǫ < s < 1−ǫ,

2ǫ
1 if s ≥ 1−ǫ,
and the corresponding f˜,g˜ : R+ → R+ fulfilling f2(s)+f˜2(s) = g2(s)+g˜2(s) = 1
for all s ≥ 0. With this notation we define
|x|
∞
Λ (x) = f ,
0
l
(cid:18) (cid:19)