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 e ff ective 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 N X X 1 x i 2 R 3 H = � ∆ x i + � ij � ( x i � x j ) , 2 m i i =1 1 ≤ i<j ≤ N 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- 0 ( R 3 \ { 0 } ) . adjoint extensions of � ∆ on C ∞ There exists a one-parameter family of such extensions, denoted by h α for ↵ 2 R , with ⇢ � Z � � ↵ + 2 ⇡ 2 p µ ⇠ 2 L 2 ( R 3 ) | ˆ ( p ) = ˆ ˆ p 2 + µ, � 2 H 2 ( R 3 ) , D ( h α ) = � ( p ) + � = ⇠ for µ > 0 and ( h α + µ ) = ( � ∆ + µ ) � Functions in D ( h α ) satisfy ✓ 2 ⇡ 2 ◆ ⇠ ( x ) ⇡ | 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 ⇢ 0 for ↵ � 0 � α � 2 inf spec h α = � for ↵ < 0 2 π 2 Moreover, the quadratic form for the energy reads h | h α i = E α ( ) = h � | ( � ∆ + µ ) � i � µ k k 2 + | ⇠ | 2 � � ↵ + 2 ⇡ 2 p µ with ⇢ � ⇠ 2 L 2 ( R 3 ) | ˆ ( p ) = ˆ p 2 + µ, � 2 H 1 ( R 3 ) , ⇠ 2 C D ( E α ) = � ( p ) + The Hamiltonians h α can be obtained by a suitable limiting procedure , e.g., taking R ! 0 for � ∆ + V R ( x ) with ✓ ⇡ 2 ◆ ⇢ R − 2 4 + 2 R for | x | R V R ( x ) = � 0 for | x | � R a 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 e ff ect . It is closely related to the Efimov e ff ect. For n -component fermions, only particles in di ff erent “spin” states interact. Instability problem persists for n � 3 . For two-component fermions , stability fails if the mass ratio m 1 /m 2 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 N N X X H = � 1 2 m ∆ x 0 � 1 ∆ x i + � � ( x 0 � x i ) 2 i =1 i =1 acting on wave functions ( x 0 , x 1 , . . . , x N ) antisymmetric in ( x 1 , . . . , x N ) . 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 F α with domain n o = � + G ⇠ | � 2 H 1 ( R 3 ) ⌦ H 1 as ( R 3 N ) , ⇠ 2 H 1 / 2 ( R 3 ) ⌦ H 1 / 2 as ( R 3( N − 1) ) D ( F α ) = ⇣ ⌘ − 1 P N 1 0 + 1 2 m k 2 i =1 k 2 where G ( k 0 , k 1 , . . . , k N ) = i + µ and G ⇠ is short for the function 2 with Fourier transform N X ( � 1) i +1 ˆ c G ⇠ ( k 0 , k 1 , . . . , k N ) = G ( k 0 , k 1 , . . . , k N ) ⇠ ( k 0 + k i , k 1 , . . . , k i − 1 , k i +1 , . . . , k N ) i =1 For 2 D ( F α ) , we have � � * + � � N X � � 1 2 m ∆ x 0 � 1 � � � µ k k 2 F α ( ) = � ∆ x i + µ � � � � 2 i =1 ✓ 2 m ◆ m + 1 ↵ k ⇠ k 2 + N L 2 ( R 3 N ) + T diag ( ⇠ ) + T o ff ( ⇠ ) R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017 # 6
The Model, Part 2 where Z R 3( N − 1) | ˆ ⇠ ( k 0 , s, ~ k ) | 2 L ( k 0 , s, ~ k ) d k 0 d s d ~ T diag ( ⇠ ) = k Z ˆ k )ˆ ⇠ ∗ ( k 0 + s, t, ~ ⇠ ( k 0 + t, s, ~ k ) G ( k 0 , s, t, ~ k ) d k 0 d s d t d ~ T o ff ( ⇠ ) = ( N � 1) k R 3( N +1) with ~ k = ( k 1 , . . . , k N − 2 ) and ◆ 3 / 2 ! 1 / 2 ✓ 2 m N − 1 X k 2 2( m + 1) + 1 0 L ( k 0 , k 1 , . . . , k N − 1 ) = 2 ⇡ 2 k 2 i + µ m + 1 2 i =1 The dangerous term is T o ff ( ⇠ ) , which is unbounded from below and multiplied by ( N � 1) . It has to be controlled by T diag ( ⇠ ) . Note that even though all terms above depend on the choice of µ , F α ( ) 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 lim m →∞ Λ ( m ) = 0 , such that T o ff ( ⇠ ) � � Λ ( m ) T diag ( ⇠ ) A numerical evaluation of the explicit expression for 4 Λ ( m ) shows that Λ ( m ) < 1 for m � 0 . 36 . 3 In particular, if m is such that Λ ( m ) < 1 , then � ( m ) 2 ( 0 1 for ↵ � 0 ⇣ ⌘ 2 F α ( ) � 0 α k k 2 � for ↵ < 0 0.0 0.5 1.0 1.5 2.0 2 π 2 (1 − Λ ( m )) m This lower bound is sharp as m ! 1 ! Recall that F α 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 , F α is closed and bounded from below, and thus gives rise to a self- adjoint Hamiltonian H α . To define it, we need the positive operator Γ on L 2 ( R 3 ) ⌦ L 2 as ( R 3( N − 1) ) defined by the quadratic form T diag ( ⇠ ) + T o ff ( ⇠ ) = h ⇠ | Γ ⇠ i We have ⇢ = � + G ⇠ | � 2 H 2 ( R 3 ) ⌦ H 2 as ( R 3 N ) , ⇠ 2 D ( Γ ) , D ( H α ) = ✓ 2 m ↵ ◆ � � � x N = x 0 = ( � 1) N +1 m + 1 + Γ ⇠ (2 ⇡ ) 3 / 2 and ! N X � 1 2 m ∆ x 0 � 1 ( H α + µ ) = ∆ x i + µ � 2 i =1 The Hamiltonian H α commutes with translations and rotations, and transforms under λ = � 2 H λ − 1 α . scaling as U λ H α U ∗ R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017 # 9
Boundary Condition For = � + G ⇠ , the boundary condition ✓ 2 m ↵ ◆ � � x N = x 0 = ( � 1) N +1 m + 1 + Γ ⇠ (2 ⇡ ) 3 / 2 means that ✓ ◆ | x 0 � x N | � 1 1 ( x 0 , x 1 , . . . , x N ) ⇠ a + o (1) as | x 0 � x N | ! 0 . More precisely: For any 2 D ( H α ) , ✓ ◆ r mr R + 1 + m, x 1 , . . . , x N − 1 , R � 1 + m ✓ 2 ⇡ 2 ◆ ( � 1) N +1 2 m = | r | + ↵ (2 ⇡ ) 3 / 2 ⇠ ( R, x 1 , . . . , x N − 1 ) + � ( R, x 1 , . . . , x N − 1 , r ) m + 1 with lim r → 0 k � ( · , r ) k L 2 ( R 3 N ) = 0 . R. Seiringer — Stability of many-body systems with point interactions — August 25, 2017 # 10
Tan Relations For 2 D ( H α ) , define the contact ✓ 2 m ◆ 2 N k ⇠ k 2 C = m + 1 It shows up in a number of physically relevant quantities: • The two-particle density ✓ 1 ◆ Z 2 1+ m ) d R ⇡ ⇡ r mr % ( R + 1+ m , R � | r | 2 � C as | r | ! 0 2 | r | a • The momentum distributions , n ↑ ( k ) ⇡ n ↓ ( k ) ⇡ C| k | − 4 as | k | ! 1 ∂α F α ( ) = m +1 ∂ • 2 m C at fixed (“adiabatic sweep theorem”) • The energy k 2 ✓ ◆ ✓ ◆� Z + k 2 C C d k � m + 1 h |H α i = n ↑ ( k ) � n ↓ ( k ) � 2 m C ↵ | k | 4 | k | 4 2 m 2 R 3 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 T o ff ( ⇠ ) ; this part is crucial, it is known that the inequality T o ff ( ⇠ ) T diag ( ⇠ ) 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: Z 1 k K k sup | K ( x, y ) | h ( y ) d y h ( x ) x 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
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