Stability of quantum many-body systems with point interactions - - PowerPoint PPT Presentation

Stability of quantum many-body systems with point interactions

Robert Seiringer IST Austria Joint work with Thomas Moser arXiv:1609.08342, Commun. Math. Phys. (in press) Quantissima in the Serenissima II Venice, August 21–25, 2017

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 1

Preface: Point Interactions

Point interactions are ubiquitously used in physics, as effective models whenever the range of the interparticle interactions is much shorter than other relevant length scales. Examples: Nuclear physics, polaron models, cold atomic gases, . . . Roughly speaking, one tries to make sense of a formal Hamiltonian of the form H =

N

X

i=1

1 2mi ∆xi + X

1≤i<j≤N

ij(xi xj) , xi 2 R3 The problem is completely understood for N = 2, but there are many open questions for N 3:

• Does there exist a suitable self-adjoint Hamiltonian modeling point interactions

between pairs of particles?

• If yes, is it stable, i.e., bounded from below?
• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 2

The N = 2 Problem

Separating the center-of-mass motion, one can rigorously define ∆ + (x) via self- adjoint extensions of ∆ on C∞

0 (R3 \ {0}).

There exists a one-parameter family of such extensions, denoted by hα for ↵ 2 R, with D(hα) = ⇢ 2 L2(R3) | ˆ (p) = ˆ (p) + ⇠ p2 + µ, 2 H2(R3), Z ˆ =

• ↵ + 2⇡2pµ
• ⇠
• for µ > 0 and

(hα + µ) = (∆ + µ) Functions in D(hα) satisfy (x) ⇡ ✓2⇡2 |x| + ↵ ◆ ⇠ (2⇡)3/2 + o(1) as |x| ! 0 hence ↵ = 2⇡2/a with a the scattering length of the pair interaction.

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 3

The N = 2 Problem, Continued

One checks that inf spec hα = ⇢ 0 for ↵ 0

• α

2π2

2 for ↵ < 0 Moreover, the quadratic form for the energy reads h |hα i = Eα( ) = h| (∆ + µ) i µk k2 + |⇠|2 ↵ + 2⇡2pµ

• with

D(Eα) = ⇢ 2 L2(R3) | ˆ (p) = ˆ (p) + ⇠ p2 + µ, 2 H1(R3), ⇠ 2 C

• The Hamiltonians hα can be obtained by a suitable limiting procedure, e.g., taking

R ! 0 for ∆ + VR(x) with VR(x) = ✓⇡2 4 + 2R a ◆ ⇢ R−2 for |x|  R for |x| R

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 4

Stability for N > 2

It is known that stability fails, in general, for N 3, unless the particles are fermions. This is known as the Thomas effect. It is closely related to the Efimov effect. For n-component fermions, only particles in different “spin” states interact. Instability problem persists for n 3. For two-component fermions, stability fails if the mass ratio m1/m2 for the two components is too large (& 13.6) or too small (. 1/13.6). For the 2 + 1 problem, stability is known in the opposite mass ratio regime. The general N + M problem is open, however! We consider here the simplest many-body problem, namely the N + 1 problem, formally defined by H = 1 2m∆x0 1 2

N

X

i=1

∆xi +

N

X

i=1

(x0 xi) acting on wave functions (x0, x1, . . . , xN) antisymmetric in (x1, . . . , xN).

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 5

The Model, Part 1

Our model is defined via a quadratic form Fα with domain D(Fα) = n = + G⇠ | 2 H1(R3) ⌦ H1

as(R3N), ⇠ 2 H1/2(R3) ⌦ H1/2 as (R3(N−1))

• where G(k0, k1, . . . , kN) =

⇣

1 2mk2 0 + 1 2

PN

i=1 k2 i + µ

⌘−1 and G⇠ is short for the function with Fourier transform c G⇠(k0, k1, . . . , kN) = G(k0, k1, . . . , kN)

N

X

i=1

(1)i+1 ˆ ⇠(k0 + ki, k1, . . . , ki−1, ki+1, . . . , kN) For 2 D(Fα), we have Fα( ) = *

• 1

2m∆x0 1 2

N

X

i=1

∆xi + µ

• +

µ k k2 + N ✓ 2m m + 1↵ k⇠k2

L2(R3N) + Tdiag(⇠) + Toff(⇠)

◆

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 6

The Model, Part 2

where Tdiag(⇠) = Z

R3(N−1) |ˆ

⇠(k0, s,~ k)|2L(k0, s,~ k) dk0 ds d~ k Toff(⇠) = (N 1) Z

R3(N+1)

ˆ ⇠∗(k0 + s, t,~ k)ˆ ⇠(k0 + t, s,~ k)G(k0, s, t,~ k) dk0 ds dt d~ k with ~ k = (k1, . . . , kN−2) and L(k0, k1, . . . , kN−1) = 2⇡2 ✓ 2m m + 1 ◆3/2 k2 2(m + 1) + 1 2

N−1

X

i=1

k2

i + µ

!1/2 The dangerous term is Toff(⇠), which is unbounded from below and multiplied by (N1). It has to be controlled by Tdiag(⇠). Note that even though all terms above depend on the choice of µ, Fα( ) is actually independent of µ!

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 7

Main Result

THEOREM 1. There exists Λ(m) > 0, independent of N, with limm→∞ Λ(m) = 0, such that Toff(⇠) Λ(m)Tdiag(⇠) A numerical evaluation of the explicit expression for Λ(m) shows that Λ(m) < 1 for m 0.36. In particular, if m is such that Λ(m) < 1, then Fα( ) ( 0 for ↵ 0

• ⇣

α 2π2(1−Λ(m))

⌘2 k k2 for ↵ < 0

0.0 0.5 1.0 1.5 2.0 1 2 3 4 m (m)

This lower bound is sharp as m ! 1! Recall that Fα is known to be unbounded from below for any N 2 for m  0.0735. In particular, the critical mass for stability satisfies 0.0735 < m∗ < 0.36.

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 8

The Hamiltonian

For Λ(m) < 1, Fα is closed and bounded from below, and thus gives rise to a self- adjoint Hamiltonian Hα. To define it, we need the positive operator Γ on L2(R3) ⌦ L2

as(R3(N−1)) defined by the quadratic form

Tdiag(⇠) + Toff(⇠) = h⇠|Γ⇠i We have D(Hα) = ⇢ = + G⇠ | 2 H2(R3) ⌦ H2

as(R3N), ⇠ 2 D(Γ),

xN=x0= (1)N+1 (2⇡)3/2 ✓ 2m↵ m + 1 + Γ ◆ ⇠

• and

(Hα + µ) = 1 2m∆x0 1 2

N

X

i=1

∆xi + µ !

• The Hamiltonian Hα commutes with translations and rotations, and transforms under

scaling as UλHαU ∗

λ = 2Hλ−1α.

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 9

Boundary Condition

For = + G⇠, the boundary condition xN=x0= (1)N+1 (2⇡)3/2 ✓ 2m↵ m + 1 + Γ ◆ ⇠ means that (x0, x1, . . . , xN) ⇠ ✓ 1 |x0 xN| 1 a + o(1) ◆ as |x0 xN| ! 0. More precisely: For any 2 D(Hα), ✓ R + r 1 + m, x1, . . . , xN−1, R mr 1 + m ◆ = ✓2⇡2 |r| + ↵ ◆ 2m m + 1 (1)N+1 (2⇡)3/2 ⇠(R, x1, . . . , xN−1) + (R, x1, . . . , xN−1, r) with limr→0 k( · , r)kL2(R3N) = 0.

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 10

Tan Relations

For 2 D(Hα), define the contact C = ✓ 2m m + 1 ◆2 Nk⇠k2 It shows up in a number of physically relevant quantities:

• The two-particle density

Z %(R +

r 1+m, R mr 1+m) dR ⇡ ⇡

2 ✓ 1 |r|2 2 |r|a ◆ C as |r| ! 0

• The momentum distributions, n↑(k) ⇡ n↓(k) ⇡ C|k|−4 as |k| ! 1
• ∂

∂αFα( ) = m+1 2m C at fixed (“adiabatic sweep theorem”)

• The energy

h |Hα i = Z

R3

 k2 2m ✓ n↑(k) C |k|4 ◆ + k2 2 ✓ n↓(k) C |k|4 ◆ dk m + 1 2m C↵

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 11

Sketch of the Proof of the Main Theorem

• Separate center-of-mass motion to eliminate one degree of freedom; this leaves us

with a problem of N fermions only.

• Identify the negative part of the operator corresponding to Toff(⇠); this part is

crucial, it is known that the inequality Toff(⇠)  Tdiag(⇠) fails for all m > 0 (and suitable ⇠)

• Replace the factor N 1 by a sum over particles, using the anti-symmetry.
• Use a suitable version of the Schur test to bound the corresponding operator:

kKk  sup

x

1 h(x) Z |K(x, y)|h(y) dy for any positive function h.

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 12

Conclusions

• We proved stability of the N +1 system of fermions with point interactions, for mass

ratio m 0.36 independent of N.

• We constructed the corresponding self-adjoint Hamiltonian.
• We showed the validity of the Tan relations for all functions in the domain of this

Hamiltonian.

• Main open problem: Investigate the stability for the general N + M system. For

N = M = 2, numerical studies suggest stability in the whole parameter regime where the 2 + 1 problem is stable.

• R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017

# 13