1
Semiclassical Eigenvalue Asymptotics
for a Schrödinger Operator with a Degenerate Potential
Abderemane MORAME1 and Françoise TRUC2
1
Université de Nantes, Faculté des Sciences, Dpt. Mathématiques, UMR 6629 du CNRS,
B.P. 99208, 44322 Nantes Cedex 3, (FRANCE), E.Mail: morame@math.univ-nantes.fr
2
Université de Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, B.P. 74,
38402 St Martin d’Hères Cedex, (France), E.Mail: Francoise.Truc@ujf-grenoble.fr
Abstract
We give the asymptotic behaviour, when h tends to zero, of the number of eigenb h = −h2 ∆ +
values less then a fixed energy λ, of the Schrödinger operator: H
d
n
m
f (x)g(y), (x, y) ∈ R = R × R . f and g are continuous and positive functions
tending to infinity at infinity, g(y) is homogeneous and f (x) > 0.
1
Introduction
c = −h2 ∆ + V on L2 (Rd )
We are interested in the spectrum of the Schrödinger operator H
h
for a nonnegative, real and continuous potential V on Rd . h ∈]0, 1] is a small parameter.
c is essentially selfadjoint on L2 (Rd ) and has discret
It is well known, (see [Re-Si], that H
h
spectrum when V (X) 7→ +∞ as |X| 7→ ∞. Then for any fixed λ ∈ R,
c ) = rk(E (] − ∞, λ[)
N (λ; H
h
bh
H
(1.1)
is finite, (for un operator P , rk(P ) denotes its rank, and when P is selfadjoint,
EP (I) denotes its spectral projection on the borel subset I of R).
c ), (when h 7→ 0), is
The asymptotic behaviour of N (λ; H
h
c ) ∼ ω h−d
N (λ; H
h
d
Z
d/2
Rd
[λ − V (X)]+ dX, (ωd = (2π)−d
Z
{X∈Rd ; |X|<1}
dX),
(1.2)
see [Re-Si] for the particular case λ − V (X) ∈ Ld/2 (Rd ), [He-Ro-2] for a proof with a
remainder estimate, in the C ∞ case, and the book [Ivr] for the history of this asymptotic
formula, (see also [Khu] for up to date C ∞ semiclassical results).
c ) when the
In this paper, we are interested in the asymptotic behaviour of N (λ; H
h
c
potential V (X) is degenerated, (does not tend to infinity with |X|), but Hh is still with
compact resolvent. For such a potential, there are some results on the asymptotic behaviour
c ) when λ tends to infinity, see [Rob], [Sim], [Sol] and [Mo-No]. Our model is
of N (λ; H
1
closer to the one of [Rob], it is a model of the ones studied in [He-Sj] of a potential V such
that {X; V (X) = 0} is a submanifold.
Here we consider the special case, (for X = (x, y) ∈ Rn × Rm = Rd ),
V (X) = f (x)g(y), f ∈ C(Rn ; R?+ ) and g ∈ C(Rm ; R+ ).
(1.3)
with g homogeneous
∃ a > 0 s.t. g(µy) = µa g(y), ∀ (µ, y) ∈ R?+ × Rm , and g(y) > 0 if y 6= 0.
(1.4)
2
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
If we consider the Schrödinger operator −∆ + g(y) = Dy2 + g(y) on L2 (Rm ), then its
spectrum is discret and positive:
σ(Dy2 + g(y)) = {µj ; j ∈ N}, 0 < µj ≤ µj+1 ∀j ∈ N and
(1.5)
rk(E{Dy2 +g(y)} (] − ∞, λ[) = ]{j; µj < λ} < ∞ ∀λ ∈ R.
(Dy = −i( ∂y∂ 1 , . . . , ∂y∂m ) and σ(P ) denotes the spectrum of the operator P.)
From the homogeneity of g we get that, for any fixed x ∈ Rn ,
Z
Rm
[h2 |Dy u(y)|2 +f (x)g(y)|u(y)|2 ]dy ≥ h2a/(2+a) f 2/(2+a) (x)µ0
Z
Rm
|u(y)|2 dy, ∀u ∈ C0∞ (Rm ),
c ≥ h2a/(2+a) f 2/(2+a) (x)µ , and then
so H
h
0
c ≥ ²f (x)g(y) + (1 − ²)h2a/(2+a) f 2/(2+a) (x)µ , ∀ ² ∈ [0, 1].
H
0
h
(1.6)
f (x) 7→ +∞ as |x| 7→ +∞.
(1.7)
c has compact resolvent if
This shows that H
h
c ) = N (h−2(a+b)/(2+a+b) λ; H
c)
If f is also homogenuous of degree b > 0, then N (λ; H
h
1
and its asymptotic estimate is given in [Sol]. Here, in our case f is strictly positive, more
over we assume some uniformly locally regularity on f :
0
0
∃ b, c > 0 s.t. c−1 ≤ f (x) and |f (x) − f (x )| ≤ cf (x)|x − x |b
0
(1.8)
0
∀ x, x ∈ Rm , |x − x | ≤ 1.
Theorem 1.1 Under the assumptions (1.3), (1.4), (1.7) and (1.8),there exists τ 1 , τ2 ∈]0, 1[
such that, for any given λ > 0, one can find h0 ∈ ]0, 1[, C1 , C2 > 0 to get
c ) ≤ (1 + hτ1 C )n (λ + hτ2 C ), ∀ h ∈ ]0, h ]. (1.9)
(1 − hτ1 C1 )nh,f (λ − hτ2 C2 ) ≤ N (λ; H
h
1
h,f
2
0
if
nh,f (λ) = h
with ωn = (2π)−n
R
{x∈Rn ; |x|<1}
−n
ωn
Z
X
Rn j∈N
2a
2
n/2
[λ − h 2+a f 2+a (x)µj ]+ dx
(1.10)
dx.
Corollary 1.2 Under the assumptions of Theorem (1.1), and the one below
∃ C3 > 0 s.t.
Z
{x; f (x)<2µ}
f −m/a (x)dx ≤ C3
Z
{x; f (x)<µ}
f −m/a (x)dx, ∀ µ > 1,
(1.11)
then one can take C2 = 0 in (1.9):
c ) ≤ (1 + hτ1 C )n (λ), ∀ h ∈ ]0, h ].
(1 − hτ1 C1 )nh,f (λ) ≤ N (λ; H
h
1
h,f
0
(1.12)
Moreover, if f −m/a ∈ L1 (Rm ), (or equivalently
d/2
[λ − V (X)]+ ∈ L1 (Rd )), and if g ∈ C 1 (Rm \ {0}, then
c ) ∼ h−d ω
N (λ; H
h
d
Z
d/2
Rd
[λ − V (X)]+ dX, as h 7→ 0.
(1.13)
3
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
Similar formulas, as (1.10), where a trace of some operator is related to the asymptotic
of the number of eigenvalues are current for operators on L2 (Rd ) with “principal symbol”
which can degenerate on some non bounded submanifold of T ? (Rd ), see for example [Sol]
and [Rob]. The well-known case is the Schrödinger with a magnetic field (DX − A(X))2
which degenerates on {(X, Ξ) ∈ T ? (Rd ); Ξ = A(X)}. Its asymptotic formula of the number
of eigenvalues was established by Y. Colin de Verdière [Col]. Similar formulas as (1.12) and
c ) when λ 7→ ∞ for the case n = m = 1
(1.13) are still valid for the asymptotic of N (λ; H
h
established in [Rob].
Colin de Verdière work [Col] shows that the famous method of Courant and Hilbert, the
so called min-max method (see chapter XIII.15 of [Re-Si] ), is still adequate for Schrödinger
operator with degenerated symbol, (see also [Tru] for more recent developments), so we will
perform it to get Theorem (1.1) and its Corollary.
Let us remark the well-known asymptotic formula, (see [Roz] or [Re-Si]),
N (µ;
Dy2
+ g(y)) ∼ ωm µ
m(2+a)/(2a)
Z
m/2
Rm
[1 − g(y)]+ dy as µ 7→ +∞.
(1.14)
From (1.10) and (1.14) we get ∀ λ > 0, ² ∈]0, λ/2[, ∃ C² > 1 s.t.
h
−d
C²−1
Z
{x;
−d
(2+a)/2 }
f (x)<h−a (µ−1
0 (λ−²))
h C²
Z
f −m/a (x)dx ≤ nh,f (λ) ≤
(2+a)/2 }
{x; f (x)<h−a (µ−1
0 (λ+²))
(1.15)
f −m/a (x)dx.
From this estimate we get that, if there exists k > 0 and C4 > 0 s.t.
C4−1 |x|k ≤ f (x) ≤ C4 |x|k , if |x| > 1, then
and
2
c ) ≈ h−n−ma/k if k < a, N (λ; H
c ) ≈ h−d log(h−1 ) if k = a
N (λ; H
h
h
c ) ≈ h−d if k > a.
N (λ; H
h
(1.16)
(1.17)
Proof of the Theorem
If Ω ⊂ Rn is open and bounded, we will denote by −∆Ω,D and −∆Ω,N the Dirichlet and
Newman self adjoint operator on L2 (Ω) (associated to the Laplace operator −∆ = Dx2 ).
We will denote Q the unit cube of Rn , Q = (] − 21 , 21 [)n , and Qr (α) = α + rQ the cube
with center α and side r > 0.
Lemma 2.1 There exists a constant Cn > 0 such that,
(1 − ²)n/2 [µ −
∀ (µ, ²) ∈ R+ ×]0, 12 [,
Cn n/2
Cn n/2
]+ ωn ≤ N (µ; −∆Q,D ) ≤ N (µ; −∆Q,N ) ≤ (1 + ²)n/2 (µ +
) ωn , (2.1)
²
²
The estimate comes easily from the well known one, ∀ µ > 1,
µn/2 ωn (1 − Cn,1 µ−1/2 ) ≤ N (µ; −∆Q,D ) ≤ N (µ; −∆Q,N ) ≤ µn/2 ωn (1 + Cn,1 µ−1/2 );
(Cn,1 is a constant). For the proof of (2.2), see for example chapter XIII.15 of [Re-Si].
(2.2)
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
4
An elementary proof of (2.1) is given in [Roz], let us sketch it.
As N (µ; −∆Q,D ) = ]ID (µ) and N (µ; −∆Q,N ) = ]IN (µ)
with ID (µ) = {α ∈ (NR? )n , π 2 α2 < µ} and IN (µ) = {αR∈ Nn , π 2 α2 < µ},
then N (µ; −∆Q,D ) = ΩD (µ) dx and N (µ; −∆Q,N ) = ΩN (µ) dx
1
1
1
1
if ΩD (µ) = ∪α∈ID (µ) Q1 (α − ( , . . . , )) and ΩN (µ) = ∪α∈IN (µ) Q1 (α + ( , . . . , )).
2√
2
2
2
√
µ
But {x ∈ (R+ )n ; |x| < [ π − n]+√} ⊂ ΩD (µ), (if this last one is not empty),
√
µ
and ΩN (µ) ⊂ {x ∈ (R+ )n ; |x| < π + n}.
√
√
√
√
Then using that (1 − ²)[µ − Cn /²]+ ≤ ( µ − π n)2 and ( µ + π n)2 ≤ (1 + ²)(µ + Cn /²)
if Cn ≥ 2nπ 2 , ( and for any ² ∈]0, 1[), we get (2.1) from the last inclusions •
From now on, C will denote any positive constant independent of h, (depending only
on n, d, a in (1.4), b, c of (1.8) and λ ).
Let now r ∈]0, 1] be fixed. As in chapter VIII.15 of [Re-Si], we use the min-max principle
to get
X
c )≤
N (λ; −h2 ∆Qr (rγ),D + h2 Dy2 + f (x)g(y)) ≤ N (λ; H
(2.3)
h
γ∈Zn
X
γ∈Zn
N (λ; −h2 ∆Qr (rγ),N + h2 Dy2 + f (x)g(y)).
Using once more the min-max principle and (1.4), (1.5) and (2.3), then we get
X X
γ∈Zn j∈N
c )≤
N (λ − (ha f (x+ (γ)))2/(2+a) µj ; −h2 ∆Qr (rγ),D ) ≤ N (λ; H
h
X X
γ∈Zn
j∈N
(2.4)
N (λ − (ha f (x− (γ)))2/(2+a) µj ; −h2 ∆Qr (rγ),N );
if x± (γ) ∈ Qr (rγ) and f (x+ (γ)) = M ax{f (x); x ∈ Qr (rγ)},
f (x− (γ)) = min{f (x); x ∈ Qr (rγ)}.
The properties of the Laplace operator and (2.4) entail
X X
γ∈Zn j∈N
c )≤
N (h−2 r2 [λ − (ha f (x+ (γ)))2/(2+a) µj ]; −∆Q,D ) ≤ N (λ; H
h
X X
γ∈Zn j∈N
(2.5)
N (h−2 r2 [λ − (ha f (x− (γ)))2/(2+a) µj ]; −∆Q,N ).
Let ² ∈]0, 1/2[ be given and let us use (2.1) of Lemma (2.1) in (2.5), then we get
ωn (1 − ²)n/2 h−n
X X
γ∈Zn j∈N
rn [λ − h2 r−2
≤ ωn (1 + ²)n/2 h−n
X X
Cn
n/2
c )
− (ha f (x+ (γ)))2/(2+a) µj ]+ ≤ N (λ; H
h
²
rn [λ + h2 r−2
γ∈Zn j∈N
(2.6)
Cn
n/2
− (ha f (x− (γ)))2/(2+a) µj ]+ .
²
We will choose r and ² such that
h2 r−2
Cn
≤ 1.
²
(2.7)
5
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
Then, from the hypothesis (1.8) on f (x), from our choice of r ∈]0, 1] and from the fact that
(2.7) entails (ha f (x± (γ)))2/(2+a) µj ≤ C if 0 ≤ λ ± h2 r−2 C²n − (ha f (x∓ (γ)))2/(2+a) µj , we get
Z
Qr (rγ)
[λ − Crb − h2 r−2
rn [λ − h2 r−2
and
Cn
n/2
− (ha f (x))2/(2+a) µj ]+ dx ≤
²
(2.8)
Cn
n/2
− (ha f (x+ (γ)))2/(2+a) µj ]+
²
Cn
n/2
− (ha f (x− (γ)))2/(2+a) µj ]+ ≤
²
Z
Cn
n/2
− (ha f (x))2/(2+a) µj ]+ dx.
[λ + rb C + h2 r−2
²
Qr (rγ)
rn [λ + h2 r−2
(2.9)
Then we choose
² = hτ1 , r = hτ3 , τ1 , τ3 > 0 s.t. τ2 := min{2 − τ1 − 2τ3 , bτ3 } > 0,
(2.10)
so (2.7) is satisfied if h is small enough; then (1.9) comes easily from (2.6), (2.8), (2.9) and
(2.10) •
3
Proof of the Corollary
As in the proof of Theorem (1.1), C will denote any constant independent of h.
Proof of (1.12)
By the definitions (1.5) and (1.10), we get
≤h
−n
C
≤ h−n+τn,2 C
Z
Z
|nh,f (λ ± hτ2 C2 ) − nh,f (λ)|
Rn
X
j
n/2
n/2
|[λ ± hτ2 C2 − µj (ha f (x))2/(2+a) ]+ − [λ − µj (ha f (x))2/(2+a) ]+ |dx
τ2
(2+a)/2 }
{x∈Rn ; f (x)<h−a µ−1
0 (λ+h C2 )
2
N ((λ+hτ2 C2 )(ha f (x))−2/(2+a) µ−1
0 ; Dy +g(y))dx,
with τn,2 = τ2 if n > 1 and τ1,2 = τ2 /2. Then using (1.14), we have the estimate
h−d+τn,2 C
Z
|nh,f (λ ± hτ2 C2 ) − nh,f (λ)| ≤
τ2
(2+a)/2 }
{x∈Rn ; f (x)<h−a (µ−1
0 (λ+h C2 ))
(3.1)
f −m/a (x)dx.
The hypothesis (1.11) and the estimates (1.9), (1.15) and (3.1) entail (1.12), (with a new
τ1 , the minimum of the old one and τn,2 •
Proof of the second part of the Corollary (1.13)
We will need the asymptotic formula (1.14) with a remainder estimate.
6
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
Lemma 3.1 As µ 7→ +∞,
with θ(g) = ωm
Z
N (µ; Dy2 + g(y)) = µm(2+a)/(2a) θ(g)(1 + O(µ−1/4 )).
(3.2)
m/2
Rm
[1 − g(y)]+ dy.
A sharp remainder estimate (with µ−(2+a)/(2a) if m > 1 and µ−(2+a)/(2a) log(µ) if m = 1, in
place of µ−1/4 ) has been established in [He-Ro-1], (see also in [Moh]),
but with g ∈ C ∞ (Rm \ {0}), and the proof is not elementary. For our purpose we need only
a remainder estimate O(µ−δ ), with δ > 0, but the best one we can easily get is δ = 1/4.
We give a simple proof of (3.2) using min-max principle.
We use the same strategy as in the proof of Theorem (1.1).
The homogeneity of g and its regularity, g ∈ C 1 (Rm \ {0}), entails that
| g(y) − g(z) |≤| y − z | g(y)C, ∀ y, z ∈ Rm , | y − z |≤ 1 and | y |> 1, | z |> 1. (3.3)
For any µ > 1, we take a partition of Rm into cubes of side r = µ−1/4 :
Rm = ∪α∈Zm Qr (rα).
Let y ± (α) ∈ Qr (rα) such that
g(y + (α)) = M ax{g(y); y ∈ Qr (rα)},
then
X
α∈Zm
g(y − (α)) = min{g(y); y ∈ Qr (rα)};
N (µ; −∆Qr (rα),D + g(y + (α))) ≤ N (µ; Dy2 + g(y))
≤
X
(3.4)
N (µ; −∆Qr (rα),N + g(y − (α))).
α∈Zm
It comes from (2.1) of Lemma (2.1), with ² = µ−1/4 , and from (3.4) that
(1 − µ−1/4 )m/2 rm ωm
X
m/2
α∈Zm
[µ − µ3/4 Cm − g(y + (α))]+
≤ (1 + µ−1/4 )m/2 rm ωm
X
α∈Zm
≤ N (µ; Dy2 + g(y))
(3.5)
m/2
[µ + µ3/4 Cm − g(y − (α))]+ .
Then we use (3.3), for the Qr (rα) ⊂ Rm \ 2Q, (let us remind that r = µ−1/4 so on such
Qr (rα), | g(y) − g(y ± (α)) |≤ µ−1/4 Cg(y ± (α))), and if Qr (rα) ∩ (2Q) 6= ∅ we use 0 ≤ g ≤ C
on 2Q to get from (3.5) that
(1 − µ−1/4 )m/2 ωm
Z
m/2
Rm
[µ − µ3/4 C − g(y)]+ dy ≤ N (µ; Dy2 + g(y))
≤ (1 + µ−1/4 )m/2 ωm
Z
(3.6)
m/2
Rm
[µ + µ3/4 C − g(y)]+ dy.
So the homogeneity and (3.6) give
(1 − µ
−1/4 m/2
)
(1 − µ
−1/4
C)
m(2+a)/(2a) m(2+a)/(2a)
µ
ωm
Z
m/2
Rm
[1 − g(y)]+ dy
(3.7)
7
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
≤ N (µ; Dy2 + g(y)) ≤
(1 + µ
−1/4 m/2
)
(1 + µ
−1/4
C)
m(2+a)/(2a) m(2+a)/(2a)
µ
ωm
Z
m/2
Rm
[1 − g(y)]+ dy.
The Lemma (3.1) follows from (3.7) •
We will need a precise estimate of the semiclassical asymptotic of the moment of the
eigenvalues of ²(2+a)/a Dy2 + g(y), (see [Li-Th]),
X
j
n/2
[λ − ²µj ]+ ∼ ²−m(2+a)/(2a) λd/2+m/a Θ(g), as ² 7→ 0;
(3.8)
m m
m(2 + a) n
B( + 1, + )
2a
2
2
a
m(2 + a)
m
m
n
m m Z
so Θ(g) =
g −m/a (ω)dω,
ωm B( + 1, )B( + 1, + )
2a2R
2
a
2
2
a S m−1
with B(u, v) = 01 su−1 (1 − s)v−1 ds = Γ(u)Γ(v)/Γ(u + v) is the Beta function.
if µj are the eigenvalues of Dy2 + g(y) and if Θ(g) = θ(g)
> 0 such that, for any ² ∈]0, ²0 ],
Lemma 3.2 There exist constants C and ²0
d/2+m/a −m(2+a)/(2a)
[λ − C²1/4 ]+
²
Θ(g) ≤
X
²
(3.9)
j∈N
d/2+m/a −m(2+a)/(2a)
≤ [λ + C²1/4 ]+
n/2
[λ − ²µj ]+
Θ(g).
Proof of the Lemma From (3.2) of Lemma (3.1) with µ = µj , we get that the multiplicity
m/2+m/a−1/4
mj of µj is bounded by Cµj
and
2a
3a
|µj − ((θ(g))−1 j) m(2+a) | ≤ Cj 2m(2+a) .
(3.10)
So if 0 ≤ λ − ²µj , then
2a
2a
λ − C²1/4 − ²((θ(g))−1 j) m(2+a) ≤ λ − ²µj ≤ λ + C²1/4 − ²((θ(g))−1 j) m(2+a) ,
which proves
2a
n/2
2a
n/2
n/2
[λ − C²1/4 − ²((θ(g))−1 j) m(2+a) ]+ ≤ [λ − ²µj ]+ ≤ [λ + C²1/4 − ²((θ(g))−1 j) m(2+a) ]+ . (3.11)
It comes from (3.11)
²−m(2+a)/(2a) θ(g)
Z
R
≤²
2a
n/2
[λ − C²1/4 − C² − s m(2+a) ]+ ds ≤
−m(2+a)/(2a)
θ(g)
Z
R
2a
X
n/2
[λ − ²µj ]+
(3.12)
j∈N
n/2
[λ + C²1/4 + C² − s m(2+a) ]+ ds.
The Lemma (3.2) follows from (3.12) •
End of the proof of (1.13) Let τ ∈]0, 1[ be fixed.
For any x ∈ Rn such that (ha f (x))2/(2+a) ≤ hτ , let us use (3.9) with ² = (ha f (x))2/(2+a) ,
then we get
d/2+m/a
h−m f −m/a (x)Θ(g)[λ − Chτ /4 ]+
≤
X
j∈N
n/2
[λ − (ha f (x))2/(2+a) µj ]+
(3.13)
8
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
d/2+m/a
≤ h−m f −m/a (x)Θ(g)[λ + Chτ /4 ]+
.
Let Ω(h, τ ) = {x ∈ Rn ; (ha f (x))2/(2+a) > hτ }. Then
Z
X
a
[λ − (h f (x))
2/(2+a)
Ω(h,τ ) j∈N
n/2
µj ]+ dx
≤C
Z
Ω(h,τ )
N (λ(ha f (x))−2/(2+a) ; Dy2 + g(y))dx,
so we can use (3.2) with µ = λ(ha f (x))−2/(2+a) > λh−τ to get
Z
{x∈Rn ;
f (x)>h−a+τ (2+a)/2 }
≤ h−m C
Z
X
n/2
[λ − (ha f (x))2/(2+a) µj ]+ dx
(3.14)
j∈N
{x∈Rn ; f (x)>h−a+τ (2+a)/2 }
f −m/a (x)dx.
Then, from (1.12), (3.13) and (3.14), (with for exemple τ = a/(2 + a)), we get that
d
m
c ) ∼ h−d λ 2 + a Θ(f, g), when h 7→ 0,
N (λ; H
h
(3.15)
Z
if f −m/a ∈ L1 (Rn ), with Θ(f, g) = Θ(g) × ωn ×
f −m/a (x)dx, (Θ(g) is defined in (3.8)).
n
R Z
Z
Z
d
m
1 m d
d/2
+
−m/a
But ωd
[λ−V (X)]+ = λ 2 a ωd B( , +1)(
g
(ω)dω)×(
f −m/a (x)dx).
a
a 2
Rd
S m−1
Rn
So (1.13) is exactly (3.15) if we check
ωn ωm (
n
m m
m m
m d
m m
+ )B( + 1, + )B( , + 1) = ωd B( , + 1)
2
a
2
2
a
a 2
a 2
(3.16)
or equivalently, (after writing the Beta functions into the Gamma ones and after simplifications),
n
m
n m
|Bn | × |Bm | × Γ( + 1) × Γ( + 1) = |Bn+m | × Γ( +
+ 1);
(3.17)
2
2
2
2
if Bk is the unit ball of Rk and |Bk | itsZ volume. The last equality (3.17) can be checked
easily by writing that |Bn+m | = |Bm | × (
m/2
Rn
[1 − |x|2 ]+ dx) •
Remark 3.3 If f −m/a (x) ∈ L1 (Rn ) and if there exist C and ρ > 0 such that
Z
{x∈Rn ; f (x)>µ}
f −m/a (x)dx ≤ Cµ−ρ , ∀ µ > 1,
then, for some τ > 0,
c ) = (1 + O(hτ ))ω
N (λ; H
h
d
Z
d/2
Rd
[λ − V (X)]+ dX, as h 7→ 0.
This comes easily from the proof of (1.13).
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(3.18)
A. Morame and F. Truc: Semiclassical Eigenvalue Asymptotics
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