Stabilit´e en m´ecanique quantique

Stabilit´ e en m´ ecanique quantique

Jean DOLBEAULT

Ceremade, Universit´e Paris Dauphine, Place de Lattre de Tassigny,

75775 Paris C´edex 16, France

E-mail: dolbeaul@ceremade.dauphine.fr

http://www.ceremade.dauphine.fr/

edolbeaul/

A joint work in collaboration with

G. Rein, C. Cid, P. Felmer, E. Paturel and M. Loss Toulouse – 20 d´ecembre 2004

Some issues on stability in quantum mechanics

I. Defocusing Nonlinear Schr¨odinger equation: confinement, sta- bility and asymptotic stability - The “boson” case

[J.D., Rein], [Cid,J.D.]

II. Stability of mixed states and applications to molecular dynam- ics - The “fermion” case

[J.D., Felmer, Paturel, Rein]

III. Lieb-Thirring type inequalities

[J.D., Felmer, Loss, Paturel]

I. Defocusing Nonlinear Schr¨ odinger equation: confinement, stability and asymptotic stability - The “boson” case

[Cid, J.D., Rein]

Nonlinear Schr¨odinger equation with confinement:

• Minimizers, Convexity

• Csisz´ar-Kullback type inequalities

• Nonlinear stability of NLS

Asymptotic stability and decay estimates (no confinement)

• Time-dependent rescalings

• Decay estimates

• Asymptotic nonlinear stability

Minimizers, Convexity

[Cid,J.D.] Let V be a nonnegative potential such that

r→+∞lim infess_|x|>r V (x) = +∞ .

Assume that p ∈ [1, ^d+2_d−2) if d ≥ 3 and p ∈ [1,+∞) if d = 1 or 2, and consider a minimizer φ_∞ of the functional

E[φ] := A 2

Z

IR^d |∇φ|² dx + B p + 1

Z

IR^d |φ|^p+1 dx + 1 2

Z

IR^d V (x)|φ|² dx with A, B > 0, under the constraint kφk

L²(IR^d) = 1.

Euler-Lagrange equations:

−A∆φ_∞ + B |φ_∞|^p−1φ_∞ + V (x)φ_∞ − λ φ_∞ = 0 (1) If V is radial and increasing, then the real positive solution φ_∞ has to be radial and strictly decreasing by [Gidas-Ni-Nirenberg79].

φ_∞ realizes the minimum of the functional G[φ] := F[φ] − F[φ_∞] ,

where F[φ] := E[φ] − ^λ₂ ^R_IRd |φ|² dx. The functional G can be rewritten as

G[φ] := F[φ] − F[φ_∞] − DF[φ_∞] · (φ − φ_∞) using the fact that DF[φ_∞] = 0.

G[φ] = ^A₂ ^R_IRd |∇φ − ∇φ_∞|² dx

+ ^B

p+1 R

IR^d

|φ|^p+1−|φ_∞|^p+1−(p+1)|φ_∞|^p−1φ_∞ · (φ − φ_∞)dx +¹₂ ^R

IR^d V (x)|φ − φ_∞|² dx − ^λ₂ ^R_IRd |φ − φ_∞|² dx .

λ < 0: G is not a convex functional =⇒ introduce ρ = |φ|², ρ_∞ = |φ_∞|², and F^{[ρ] = F[√ρ]}

F^{[ρ] = A} 2

Z

IR^d|∇√

ρ|² dx + B p+1

Z

IR^dρ^p+12 dx + 1 2

Z

IR^d V ρ dx−λ 2

Z

IR^d ρ dx

[Benguria79,Benguria-Brezis-Lieb] With ρ_t = tρ₁ + (1 − t)ρ₂, d²

dt²

Z

IR^d

|∇ρ_t|²

ρ_t dx =

Z

IR^d

2

ρ³_t |ρ_t∇(ρ₂ − ρ₁) − (ρ₂ − ρ₁)∇ρ_t|² dx > 0 . F is strictly convex with a unique minimizer φ_∞ = √

ρ_∞

−A ∆(√

ρ_∞)

√ρ_∞ + B ρ

p−1

∞2 + V (x) − λ = 0 . (2)

F^{[ρ] −} F^{[ρ∞] = A} 2

Z

IR^d

|∇√

ρ|² − |∇√

ρ_∞|² dx + B p + 1

Z

IR^d ρ^p+12 − ρ

p+1

∞2

!

dx

+1 2

Z

IR^d(V − λ)(ρ − ρ_∞) dx

= A 2

Z

IR^d

|∇√

ρ|² − |∇√

ρ_∞|² dx + B p + 1

Z

IR^d

ρ^p+12 − ρ

p+1

∞2

!

dx

+1 2

Z

IR^d A ∆√ ρ_∞

√ρ_∞ − B ρ_∞

p−1 2

!

(ρ − ρ_∞) dx

= A 2

Z

IR^d |∇√ ρ −

s ρ

ρ_∞ ∇√

ρ_∞|² dx

B ^{Z p+1 p+1}₂ p + 1 ^p−1₂ ^!

Theorem 1 There exists a unique up to a constant phase factor (resp. unique) minimizer of E (resp. E) under the constraint kφkL²(IR^d) = 1 (resp. ρ ≥ 0, kρk

L¹(IR^d) = 1) that we shall denote by φ_∞ (resp. ρ_∞).

Moreover, for any φ such that kφk

L²(IR^d) = 1 (resp. for any nonnegative ρ such that kρk

L¹(IR^d) = 1), provided ρ = |φ|², E[φ] − E[φ_∞]

≥ A 2

Z

IR^d ρ_∞

∇^q_ρ^ρ_∞

2

dx + B

p+1 Z

IR^d ρ^p+12 − ρ

p+1

∞2 − ^p+1₂ ρ

p−1

∞2 (ρ − ρ_∞)

!

dx

where equality holds for φ = √ ρ.

Limiting cases of Theorem 1

Case A = 0, B > 0: let φ_∞ and ρ_∞ be defined by φ_∞(x) =

q

ρ_∞(x) =

1

B(λ − V (x))₊

_p−1²

,

Case A > 0, B = 0: let φ_∞ = √

ρ_∞ be the first eigenfunction of the operator (−A∆ + V ).

Csisz´ ar-Kullback type inequalities

Consider a strictly convex function σ on IR⁺, taking finite values on (0,+∞) and define on L¹(IR^d) the functional

Σ[ρ] =

Z

IR^d

[ σ(ρ) + V (ξ) ρ] dx − C with C such that Σ[ρ_∞] = 0.

Lemma 2 [Caceres-Carrillo-J.D.] Assume that d ≥ 1, 1 ≤ p ≤ 3, max(1,2/(4 − p)) ≤ s < 2, and let q = s(3 − p)/(2 − s). If κ= inf_s>0 s^−(p−3)/2σ⁰⁰(s) > 0, then for any nonnegative function ρ in L¹ ∩ L^(p+1)/2(IR^d),

kρ − ρ_∞k²

L^s(IR^d) ≤ 1 κ

2^2(1+s)/s

p − 1 K_q^(3−p)/2 Σ[ρ]

where K_q = max{kρk

L^q/2(IR^d),kρ_∞k

L^q/2(IR^d)}.

Special case: 2s = q = p + 1.

Corollary 1 With |φ|² = ρ, |φ_∞|² = ρ_∞, A

2

Z

IR^d

∇(^q_ρ^ρ_∞)

2ρ_∞ dx+ C

p + 1kρ−ρ_∞k²

L

p+1 2

≤E^[ρ]−E^{[ρ∞] ≤ E[φ]−E[φ∞]}

Nonlinear stability of NLS

i∂φ

∂t = −A∆φ + B |φ|^p−1φ + V (x) φ (3) Existence: [Cazenave].

Corollary 2 Consider a global in time solution of (3) with initial condition φ₀ ∈ H¹(IR^d) such that √

V φ₀ ∈ L²(IR^d). Then for any > 0, there exists δ > 0 such that

E[φ₀]−E[φ_∞] < δ =⇒ k |φ(·, t)|²−|φ_∞|²k

L

p+1

2 (IR^d)

< ∀ t > 0 .

Asymptotic stability and decay estimates

Defocusing nonlinear Schr¨odinger equation (NLS)

i ψ_t = −∆ψ + |ψ|^p−1ψ (4) p ∈ (1, p_∗) with p_∗ := 1 + 4/(d−2) if d > 2, p_∗ := +∞ if d = 1,2.

Define p_c := 1 + 4/d < p_∗ as the critical exponent:

(i) subcritical case if p ∈ (1, p_c),

(ii) critical case (or pseudo-conformal invariant case) if p = p_c, (iii) supercritical case if p ∈ (p_c, p_∗).

RR¨ = R^−cp−1 with c_p = min (d

2 (p−1),2) , R(0) = 1 , R(0) = 0˙ .

ψ(t, x) = R(t)^−2de^{2i S(t)|x|2}φ τ(t), x R(t)

!

Time-dependent rescalings [J.D.,Rein]. Rescaled equation:

iτ φ˙ _τ =−¹

R2∆φ+R^−2d(p−1)|φ|^p−1φ+^R₂²( ˙S+2S²)|ξ|²φ+i^R_R^˙−2S

d

2 φ+ξ·∇φ, Choice: S = ˙R/2R,

τ˙ = 1 2

RR¨ = R^−cp where c_p = min

d

2 (p − 1),2

.

Thus c_p = d(p − 1)/2 if p is subcritical and c_p = 2 if p is critical or supercritical, and φ solves the equation

iφ_τ = −R^cp−2∆φ + R^{cp−2d(p−1)}|φ|^p−1φ + 1

2|ξ|² φ . (5)

t→+∞lim τ(t) = τ_∞ > 0 ,

where τ_∞ = +∞ if p ≤ 1 + 2/d and τ_∞ < +∞ if p > 1 + 2/d.

Energy functional:

E(τ) = 1

2R^cp−2

Z

IR^d|∇φ|²dξ+1 4

Z

IR^d|ξ|²|φ|²dξ+R^{cp−2d(p−1)} p + 1

Z

IR^d|φ|^p+1dξ

E⁰ = −RR˙ ^2cp−1

2−cp 2R2

Z

IR^d|∇φ|²dξ + ^{d(p−1)−2cp}

2 (p+1)Rd(p−1)/2

Z

IR^d|φ|^p+1dξ

Case p ≥ p_c:

R²

Z

IR^d

∇ψ − iR˙

2R ψx

2

dx = 1 2

Z

IR^d |∇φ|² dξ ≤ E(τ) ≤ E(0) .

Z

IR^d |∇|ψ| |² dx =

Z

IR^d

∇|e⁻ⁱ

R˙

2R|x|²ψ|

2

dx ≤

Z

IR^d

∇ψ − iR˙

2R x ψ

2

dx

Case p ≤ p_c: 1

p + 1 R^d(p+1)

1

2−_p+1¹ Z

IR^d |ψ|^p+1dx = 1 p + 1

Z

IR^d |φ|^p+1dξ ≤ E(0) , 1

2R^2d(p−1)

Z

IR^d |∇ψ − iR˙

2R x ψ|² dx = 1

2 R^cp−2

Z

IR^d |∇φ|² dξ ≤ E(0) .

Decay estimates

Theorem 3 Assume that p ∈ (1, p_∗), r ∈ [2, p_∗ + 1). Let ψ be a solution of (4) with an initial data ψ₀ ∈ H¹(IR^d) such that (1 + | · |²)^1/2 ψ₀ ∈ L²(IR^d). Then there exists a constant C > 0 such that

kψ(t, ·)k

L^r(IR^d) ≤ CR(t)^−d

1

2−¹_r

(1−) ∀ t ≥ 0 ,

where = 0 if p ∈ [p_c, p_∗) or r ∈ [2, p+1], and = (r−(p+1))(4−d(p−1)) (r−2)(4−d(p−1)+2(p−1))

otherwise. Moreover C depends only on d, p, r and

E₀ := 1

2k∇ψ₀k²_L2 + 1

4k |x|ψ₀k²_L2 + 1

p + 1kψ₀k^p+1

L^p+1 .

Asymptotic nonlinear stability

Theorem 4

k |ψ|² − |ψ_∞|²k²

L^(p+1)/2(IR^d) ≤ C R(t + t₀)−2d(p−1)/(p+1)

where ψ_∞(t, x) = ¹

R(t)^d/2φ_∞( ^x

R(t)).

Other example: the logarithmic NLS [Cid,J.D.]

iψ_t = −∆ψ + log(|ψ|²) ψ , ψ_|t=0 = ψ₀ .

II. Stability of mixed states an applications to molecular dynamics - The “fermion” case

[J.D., Felmer, Paturel, Rein]

• One-particle linear Schr¨odinger equation: free energy and sta- bility

• N-particles Schr¨odinger equation

• Stability for the Hartree-Fock model with temperature An extension of the results of [Markowich-Rein-Wolansky]

One-particle linear Schr¨ odinger equation

Consider the linear Schr¨odinger equation

i∂_tψ = −∆ψ + V ψ , x ∈ IR^d , t > 0 (1) Let

E(ψ) :=

Z

IR^d (|∇ψ|² + V |ψ|²) dx

Assume that V is a potential such that for any i ∈ IN,

λ_i := inf

(ψ_j)ⁱ_j=1∈(L²(IR^d))ⁱ (ψ_j, ψ_k)_L2 = δ_jk

sup

ψ∈Vect(ψ_j)ⁱ_j=1 kψk_L2

E(ψ)

is a nondecreasing sequence such that for any i ∈ IN, there exists a j ∈ IN with j > i such that λ_j > λ_i: Assumption (H1)

(λ_i)_i∈IN is a sequence of eigenvalues counted with multiplicity.

Let (ψ_i)_i∈IN be the associated eigenstates:

(ψ_i, ψ_j)

L²(IR^d) = δ_ij ∀ i , j ∈ IN .

Free energy functional F ^{: [0,+∞)IN × (L2(IRd))IN 3→ IR} F^{(ν, ψ) := X}

i∈IN

(β(ν_i) + ν_i E(ψ_i))

Here β is a convex nonnegative and nondecreasing function such that lim_ν→0 β(ν) = 0 (at least when there is a confining potential): Assumption (H2). Then λ 7→ (β⁰)⁻¹(λ − λ_i) is an increasing function.

Assumption (H3) (on V and β): ∃λ ∈ IR such that ν_i = (β⁰)⁻¹(λ − λ_i) , ^X

i∈IN

ν_i = 1 , ^X

i∈IN

β(ν_i) < ∞ , ^X

i∈IN

ν_i λ_i < ∞

F^{(ν, ψ) := XN}

i=1

(β(ν_i) + ν_i E(ψ_i))

E(ψ_i) :=

Z

IR^d (|∇ψ_i|² + V |ψ_i|²) dx

Lemma 5 There exists a minimizer (¯ν,ψ)¯ ∈ [0,+∞)^IN×(L²(IR^d))^IN of F under the constraints

X

i∈IN

ν_i = 1 and (ψ_i, ψ_j)

L²(IR^d) = δ_ij ∀ i , j ∈ IN .

¯ν_i = (β⁰)⁻¹(λ − λ_i)

and the sequence ψ¯ = ( ¯ψ_i)_i∈IN is unique up to any unitary trans- formation which leaves all eigenspaces of −∆ + V invariant.

Lemma 6 For any (ν, ψ) ∈ [0,+∞)^IN × (L²(IR^d))^IN F^{(ν, ψ) = X}

i∈IN

(β(ν_i)−β(¯ν_i)−β⁰(¯ν_i)(ν_i−¯ν_i))+ ^X

i∈IN

ν_i (E(ψ_i)−E( ¯ψ_i)). If ψ = (ψ_i)_i∈IN ∈ (L²(IR^d))^IN satisfies the orthogonality conditions

(ψ_i, ψ_j)

L²(IR^d) = δ_ij ∀ i , j ∈ IN .

and is ordered such that (E(ψ_i))_i∈IN is a nondecreasing sequence, then for any i ∈ IN

E(ψ_i)−E( ¯ψ_i) =

Z

IR^d (|∇ψ_i−∇ψ¯_i|²+V |ψ_i−ψ¯_i|²−λ_i |ψ_i−ψ¯_i|²)dx (2) is nonnegative.

For an orthogonal sequence ψ = (ψ_i)_i∈IN ∈ (L²(IR^d))^IN, define d_i(ψ_i, ψ¯_i) := E(ψ_i) − E( ¯ψ_i) .

d((ν, ψ),(¯ν, ψ)) =¯ F^{(ν, ψ) −} F^(¯ν,ψ)¯

= ^X

i∈IN

(β(ν_i) − β(¯ν_i) − β⁰(¯ν_i)(ν_i − ¯ν_i)) + ^X

i∈IN

ν_i d_i(ψ_i,ψ¯_i) .

defines a ”distance” on (0,+∞)^IN ×(L²(IR^d))^IN/UV where UV is the the group of unitary transformations that leave invariant all eigenspaces of −∆ + V .

Theorem 7 Let (ν, ψ⁰) ∈ [0,+∞)^IN × (L²(IR^d))^IN be such that

X

i∈IN

ν_i = 1 and kψ_ik_L2 = 1 ∀ i ∈ IN

If ψ_i(x, t) is the solution to Equation (1) with initial data ψ_i⁰, then

d((ν, ψ(·, t)),(¯ν, ψ)) =¯ d((ν, ψ⁰),(¯ν, ψ))¯ .

N

-particles Schr¨ odinger equation

Consider now an N-particles wave function ψ(x₁, x₂, . . . , x_N; t) defined on (IR^d)^N × IR⁺ and evolving under the action of a Schr¨odinger operator:

i∂_tψ = −

N X

i=1

∆_xiψ + V ψ , x ∈ IR^d, t > 0 (3) Case of interest in molecular dynamics:

V (x₁, x₂, . . . , x_N) =

M X

k=1

Z_k

|x − x¯_k| + ^X

i,j=1,...,N

1

|x_i − x_j| . (4) By Pauli’s exclusion principle, the N-particles wave function is antisymmetric:the natural space is Λ^N(IR³), which is invariant under the evolution according to (3).

Stability for the Hartree-Fock model with temperature

Ansatz (zero-temperature Hartree-Fock model): replace Λ^N(IR³) by

X := {ψ = 1

√N!

X

σ∈SN

ε(σ) ^Y

1≤i,j≤N

ψ_i(x_σ(j)) : (ψ_i, ψ_j)

L²(IR^d) = δ_ij}

E^HF(ψ₁, ψ₂, . . . , ψ_N) = ^X

1≤i≤N Z

IR³ |∇ψ_i|² dx

+ ^X

1≤i,j≤N Z

IR³

(1

2|ψ_j|² ∗ 1

|x| −

M X

k=1

Z_k

|x − ¯x_k|) |ψ_i|² dx

−1 2

X

1≤i,j≤N Z

IR³ Z

IR³

ψ_i(x)ψ_i(y)ψ_j(x)ψ_j(y)

|x − y| dx dy

Evolution equation:

i∂_tψ_i = −∆ψ_i −

M X

k=1

Z_k

|x − ¯x_k| ψ_i + 1

2( ^X

1≤j≤N

|ψ_j|²) ∗ 1

|x| ψ_i (5)

− 1 2

X

1≤j≤N Z

IR³

ψ_i(y)ψ_j(y)

|x − y| dy ψ_j(x)

Hartree-Fock model with temperature:

i∂_tψ_i = −∆ψ_i −

M X

k=1

Z_k

|x − x¯_k| ψ_i + 1

2 ρ(x) ∗ 1

|x| ψ_i (6)

− 1 2

X

1≤j≤N Z

IR³

ρ(x, y)ψ_i(y)

|x − y| dy ρ(x, y) = ^X

j∈IN

ν_j ψ_j(x) ψ_j(y) , ρ(x) = ρ(x, x)

Free energy:

F^{(ν, ψ) := X}

i∈IN

β(ν_i) + E_ν^HF(ψ) .

Lemma 8 Under Assumption (H2), F^{(ν, ψ)} has a minimizer un- der the constraints

X

i∈IN

ν_i = 1 and (ψ_i, ψ_j)

L²(IR^d) = δ_ij ∀ i , j ∈ IN ,

which can be written as ν_i = β(λ − λ_i) ,

−∆ ψ_i + (ρ ∗ _|x|¹ − ^PM_{k=1 |x−¯}^Zk

x_k|) ψ_i − ^R_IR3 ρ(x,y)

|x−y| ψ_i(y) dy = λ_i ψ_i (7)

e_ρ(ψ) :=

Z

IR³ |∇ψ|² dx +

Z

IR³ (ρ ∗ 1

|x| −

M X

k=1

Z_k

|x − ¯x_k|) |ψ|² dx

−

Z

IR³ Z

IR³ ψ(x)ψ(y) ρ(x, y)

|x − y| dx dy .

λ_i = e_ρ(ψ_i) , E_ν^HF = ^X

i∈IN

ν_i e_ρ(ψ_i) and d_i(ψ_i,ψ¯_i) := e_ρ(ψ_i) − e_ρ¯( ¯ψ_i)

Lemma 9 Let (ν, ψ⁰) ∈ [0,+∞)^IN ×(L²(IR^d))^IN and assume that

X

i∈IN

ν_i = 1 and kψ_ik_L2 = 1 ∀ i ∈ IN .

If (e_ρ(ψ_i)_i)_i∈IN is nondecreasing, then d_i(ψ_i,ψ¯_i) is nonnegative.

Theorem 10 Consider (ν, ψ⁰) ∈ [0,+∞)^IN × (L²(IR^d))^IN and as- sume that

X

i∈IN

ν_i = 1 and kψ_ik_L2 = 1 ∀ i ∈ IN .

Let ψ be the solution of (6) with initial data ψ_i⁰. Let σ(t, i) be a reordering such that (e_ρ(·,t)(ψ_i(·, t))

i)_i∈IN is nondecreasing for any t ≥ 0. Then

F(ν, ψ(·, t)) − F^{(¯ν,ψ) =¯ X}

i∈IN

(β(ν_σ(t,i)) − β(¯ν_i) − β⁰(¯ν_i)(ν_σ(t,i) − ν¯_i)) + ^X

i∈IN

ν_σ(t,i) d_i(ψ_σ(t,i),ψ¯_i) does not depend on t

and measures the distance between ((ν(t), ψ(t))_i)_i∈IN and ((¯ν,ψ)¯ _i)_i∈IN

III. Lieb-Thirring type inequalities

[J.D., Felmer, Paturel, Loss]

Lieb-Thirring type inequalities and “the stability of matter” in quantum mechanics.

Let V be a smooth bounded nonpositive potential on IR^d, H_V = −_2m^¯h2 ∆ + V with eigenvalues

λ₁(V ) < λ₂(V ) ≤ λ₃(V ) ≤ . . . λ_N(V ) < 0 The Lieb-Thirring inequality.

N X

i=1

|λ_i(V )|^γ ≤ C_LT(γ)

Z

IR^d |V |^γ+2d dx (1) γ = 1: the sum ^PN_i=1|λ_i(V )| is the complete ionization energy,

[Hundertmark-Laptev-Weidl00,Laptev-Weidl00]

The Lieb-Thirring conjecture

C_LT(γ) = C_LT⁽¹⁾(γ) := inf

V ∈ D^(IRd) V ≤ 0

|λ₁(V )|^γ

R

IR^d |V |^γ+2d dx

. (2)

Plan

• Connection of the best constant C_LT⁽¹⁾(γ) with the best constant in Gagliardo-Nirenberg inequalities

• A new inequality of Lieb-Thirring type

• The “dual” case in Gagliardo-Nirenberg inequalities

Connection with Gagliardo-Nirenberg inequalities

(Usual case of the Lieb-Thirring inequalities) X_γ :=

V ∈ L^γ+2d(IR^d) : V ≥ 0, V 6≡ 0 a.e.

Best constant in the Lieb-Thirring inequality:

C_LT⁽¹⁾(γ) = sup V ∈ X_γ

V ≥ 0, V 6≡ 0 a.e.

|λ₁(−V )|^γ

R

IR^d V ^γ+2d dx .

where

λ₁(−V ) = inf

u ∈ H¹(IR^d) u 6≡ 0 a.e.

R

IR^d |∇u|² dx − ^R_IRd V |u|² dx

R

IR^d |u|² dx .

Gagliardo-Nirenberg inequality:

C_GN(γ) = inf

u ∈ H¹(IR^d) u 6≡ 0 a.e.

k∇uk

d 2γ+d

L²(IR^d)kuk

2γ 2γ+d

L²(IR^d)

kuk

L²

2γ+d

2γ+d−2(IR^d)

. (3)

Theorem 11 Let d ∈ IN, d ≥ 1. For any γ > 1 − ₂^d,

C_LT⁽¹⁾(γ) = κ₁(γ) [C_GN(γ)]^−κ2(γ) , (4) where κ₁(γ) = ²

d

d 2γ+d

₁₊ ^d

2γ and κ₂(γ) = 2 + ^d

γ. Moreover, the constant C_LT⁽¹⁾(γ) is optimal and achieved by a unique pair of functions (u, V ), up to multiplications by a constant, scalings and translations.

The scaling invariance can be made clear by redefining [C_LT⁽¹⁾(γ)]

1

γ = sup

V ∈ X_γ

V ≥ 0, V 6≡ 0 a.e.

sup

u ∈ H¹(IR^d) u 6≡ 0 a.e.

R(u, V )

where

R(u, V ) =

R

IR^d V |u|² dx − ^R_IRd |∇u|² dx

R

IR^d |u|² dx kV k¹⁺

d 2γ

L^γ+d2(IR^d)

.

Note indeed that λ₁(V ) ≤ 0, and R(u, V ) is invariant under the transformation

(u, V ) 7→ (u_λ = u(λ ·), V_λ = λ²V (λ ·)) , i.e.,

R(u_λ, V_λ) = R(u, V ) ∀ λ > 0 .

Proof of Theorem 11. By H¨older’s inequality,

Z

IR^d

V |u|² dx ≤ Akuk²

L^2q(IR^d), A = kV k

L^γ+2d(IR^d)

, q = 2γ + d 2γ + d − 2 . With x = ^kukL2q(IRd)

kuk_L2(IRd

)

and θ = ^d

2γ+d, the Gagliardo-Nirenberg in- equality (3) can be rewritten as

k∇ukL²(IR^d)

kukL²(IR^d)

≥ [C_GN(γ) x]^1θ so that

R(u, V ) ≤ A x² − [C_GN(γ)]^2θ x^2θ A¹⁺

d 2γ

Optimize on x.

The estimate is achieved:

V ^γ+2d−1 = |u|² ⇐⇒ V = V_u = |u|

4

2γ+d−2 = |u|^2(q−1) , (5) where u is a solution of

∆u + |u|^2(q−1)u − u = 0 in IR^d .

Up to a scaling, these two equations are the Euler-Lagrange equations corresponding to the maximization in V and u.

Gagliardo-Nirenberg inequality:

R(u, V ) ≤ R(u, V_u) =

R

IR^d u^2q dx − ^R_IRd |∇u|² dx

R

IR^d u² dx ^R_IRd u^2q dx

1 γ

t u

A new inequality of Lieb-Thirring type

Let V be a nonnegative unbounded smooth potential on IR^d: the eigenvalues of H_V are

0 < λ₁(V ) < λ₂(V ) ≤ λ₃(V ) ≤ . . . λ_N(V ) . . .

Main Theorem

For any γ > d/2, for any nonnegative V ∈ C^∞(IR^d) such that V ^d/2−γ ∈ L¹(IR^d),

N X

i=1

λ_i(V )^−γ ≤ C_LT,d(γ)

Z

IR^d V ^2d−γ dx.

Theorem 12 An inequality by Golden, Thompson and Symanzik

[Symanzik, B. Simon] Let V be in L¹_loc(IR^d) and bounded from below.

Assume moreover that e^{−t V} is in L¹(IR^d) for any t > 0. Then

Tre^{−t(−∆+V )} ≤ (4πt)^−2d

Z

IR^d e^{−t V (x)} dx . (6)

Proof. The usual proof is based on the Feynman-Kac formula.

Here: an elementary approach.

G(x, t) = (4πt)^−2d e⁻

|x|2

4t , e^t∆ f = G(·, t) ∗ f . Trotter’s formula:

e^{−t(−∆+V )} = lim

n→∞

e^{nt ∆} e^{−nt V}

_n

Compute the trace...

. . . =

Z

(IR^d)ⁿ dx dx₁ dx₂ . . . dx_n G

t

n, x − x₁

e^{−nt V (x1)} G

t

n, x₁ − x₂

·e^{−nt V (x2)} . . . G

t

n, x_n − x

e^{−nt V (x)} . Notation x = x₀ = x_n+1:

Z

(IR^d)ⁿ dx₀ dx₁ dx₂ . . . dx_n

n Y

j=0

G

t

n, x_j − x_j+1

e^−nt

Pn−1

k=0V (x_k)

.

Convexity of x 7→ e^−x: e^−nt

P_n−1

k=0V (x_k)

≤ 1 n

n−1 X

k=0

e^{−t V (xk)} .

Main ingredient:

Z

(IR^d)ⁿ⁻¹dx₀ dx₁ dx₂ . . . dx_k−1 dx_k+1 . . . dx_n

n Y

j=0

G

t

n, x_j − x_j+1

= G(t, x_k − x_k) = (4πt)^−2d . Tr

e^{nt ∆} e^{−nt V}

_n

≤ 1 n

n−1 X

k=0 Z

(IR^d)ⁿ

dx₀ dx₁ dx₂ . . . dx_n

·

n Y

j=0

G

t

n, x_j − x_j+1

e^{−t V (xk)}

= (4πt)^−2d

Z

(IR^d)² e^{−t V (x)} dx

t u

Proof of the Main Theorem. By definition of the Γ function, for any γ > 0 and λ > 0,

λ^−γ = 1 Γ(γ)

Z _+∞

0

e^−tλ t^γ−1dt .

The operator −∆ + V is essentially self-adjoint on L²(IR^d), and positive:

Tr(−∆ + V )^−γ = 1 Γ(γ)

Z _+∞

0

Tr e^{−t(−∆+V )} t^γ−1dt . Using (6), since V ^d2−γ ∈ L¹(IR^d), we get

Tr (−∆ + V )^−γ ≤ 1 Γ(γ)

Z _+∞

0

Z

IR^d(4πt)^−d2e^{−tV (x)} t^γ−1dx dt

≤ Γ(γ − ₂^d) (4π)^2dΓ(γ)

Z

IR^d V (x)^2d−γdx .

t u

The “dual” case in Gagliardo-Nirenberg inequalities

Let V ∈ C^∞(IRd) such that lim_|x|→+∞ V (x) = +∞ and denote by λ₁(V ), λ₂(V ), . . . λ_N(V ) the positive eigenvalues of −∆ + V .

Y_γ =

V ^2d−γ ∈ L¹(IR^d) : V ≥ 0, V 6≡ +∞ a.e.

Second type Gagliardo-Nirenberg inequalities:

C_GN,d(γ) = inf

u ∈ H¹(IR^d) u 6≡ 0 a.e.

R

IR^d |u|^2q dx < ∞

k∇uk

d(1−q) d(1−q)+2q

L²(IR^d) kuk

2q d(1−q)+2q

L^2q(IR^d)

kukL²(IR^d)

(7) with q := ^2γ−d

2(γ+1)−d ∈ (0,1).

Theorem 13 Let d ∈ IN, d ≥ 1. For any γ > 1 − ₂^d, there exists a positive constant C_LT,d⁽¹⁾ (γ) such that, for any V ∈ Y_γ,

λ₁(V )^−γ ≤ C_LT,d⁽¹⁾ (γ)

Z

IR^d V ^2d−γ dx . (8) The optimal value of C_LT,d⁽¹⁾ (γ) is such that

C_LT,d⁽¹⁾ (γ) = κ₁(γ) [C_GN,d(γ)]^−κ2(γ) , (9) where, with q := _2γ−d+2^2γ−d ,

κ₁(γ) = (2q)^γ−2d(d(1 − q))^d2

(d(1 − q) + 2q)^γ and κ₂(γ) = 2γ .

Moreover, the constant C_LT,d⁽¹⁾ (γ) is achieved by a unique pair of functions (u, V ), up to multiplications by a constant, scalings and translations.

Notice that if γ > 1 − ₂^d, then 2q > 1.

C_LT⁽¹⁾(γ) := sup V ∈ Y_γ

V ≥ 0, V 6≡ 0 a.e.

[λ₁(V )]^−γ

R

IR^d V ^2d−γ dx .

Scaling invariance:

[C_LT,d⁽¹⁾ (γ)]

1

γ = sup

V ∈ X_γ

V ≥ 0, V 6≡ 0 a.e.

sup

u ∈ H¹(IR^d) u 6≡ 0 a.e.

R(u, V )

R(u, V ) :=

R

IRd |u|² dx kV ⁻¹k

1− d 2γ

Lγ−d

2 (IRd)

R

IRd |∇u|² dx+R

IRd V |u|² dx is invariant under the trans- formation

(u, V ) 7→ (u_λ = u(λ ·), V_λ = λ²V (λ ·)) .

Proof of Theorem 13. By H¨older’s inequality,

Z

IR^d u^2q dx =

Z

IR^d u^2q V ^q·V ^−q dx ≤

Z

IR^d V |u|² dx

_q Z

IR^d V ⁻

q

1−q dx

_1−q

.

With C :=

R

IR^d V ⁻

q

1−q dx

₋^1−q

q = kV ⁻¹k

L^γ−2d(IR^d)

, this means that

R(u, V ) ≤

kuk²

L²(IR^d) C¹⁻

d 2γ

k∇uk²

L²(IR^d) + C kuk²

L²(IR^d)

.

Optimize R(u, V ) under the scaling λ 7→ λ^−d/2u(·/λ). tu Remark 1 Optimal functions are explicitly known only for d = 1.

Solutions have compact support and minimal ones are radially symmetric and unique up to translations.