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Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory Hongshuo Chen joint work with Heinz Siedentop Chongqing University, China October 22 2019, CIRM Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems The Thomas-Fermi-Weizsäcker theory The functional for molecules in non-relativistic Thomas-Fermi-Weizsäcker theory with atomic nuclei of atomic number Z k situated at R k reads ⎛ √ E nTFW ( ρ ) = ∫ R 3 ρ ( x )∣ 2 − ⎞ Z k ρ ( x ) 3 ( x ) + 1 K 3 2 ∣ ∇ ∑ 5 ⎝ 10 γρ ∣ x − R k ∣ ⎠ d x k = 1 + D [ ρ ] + Z k Z l ∑ ∣ R k − R l ∣ 0 ≤ k < l ≤ K where ρ ( x ) ρ ( y ) D [ ρ ] ∶= ∬ R 3 × R 3 ∣ x − y ∣ d x d y . The functional is naturally defined on all non-negative ρ ∈ L 5 / 3 ( R 3 ) with finite Weizsäcker energy and finite Coulomb energy D [ ρ ] . The physically γ = ( 3 π 2 ) 2 3 . The other constants in front of the Weizsäcker term than 1 / 2 have been also discussed. Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems The Thomas-Fermi-Weizsäcker theory The relativistic Thomas-Fermi-Dirac-Weizsäcker model is given by Engel and Dreizler 1 . Following them, we write it in terms of the Fermi momentum p given by p ( x ) ∶= ( 3 π 2 ρ ( x )) 1 / 3 , instead of density ρ . The energy is 3 π 2 ∫ R 3 d x p 3 ( x ) E TFW ( p ) ∶=T W ( p ) + T TF ( p ) − K α Z k + 18 π 4 D [ p 3 ] + α α Z k Z l ∑ ∑ ∣ x − R k ∣ ∣ R k − R l ∣ k = 1 0 ≤ k < l ≤ K where α is the Sommerfeld fine structure constant, which is 1 / c in Hartree units and has the physical value of about 1 / 137. The Thomas-Fermi part of the kinetic energy is T TF ( p ) ∶= 8 π 2 ∫ R 3 t TF ( p ( x )) d x 1 with t TF ( s ) ∶= s ( s 2 + 1 ) 3 / 2 + s 3 ( s 2 + 1 ) 1 / 2 − arsinh ( s ) − 8 3 s 3 . The Weizsäcker part of the kinetic energy is T W ( p ) ∶= 3 λ 8 π 2 ∫ R 3 d x ∣ ∇ p ( x )∣ 2 f W ( p ( x )) 2 √ with f W ( t ) ∶= t 2 + 1 + 2 t 2 + 1 arsinh ( t ) . The constant λ is positive. The space √ t 2 t of allowed p is P ∶= { p ∈ L 4 ( R 3 )∣ p ≥ 0 , D [ p 3 ] < ∞ , F ○ p ∈ D 1 ( R 3 )} 0 f W ( s ) d s . where F is the antiderivative F ( t ) ∶= ∫ t 1 E. Engel and R. M. Dreizler. “ Field-theoretical approach to a relativistic Thomas-Fermi-Dirac-Weizsäcker model”. In: Physical Review A 35.9 (May 1987), pp. 3607–3618; E. Engel and R. M. Dreizler. “Solution of the relativistic Thomas-Fermi-Dirac-Weizsäcker model for the case of neutral atoms and positive ions”. In: Physical Review A 38.8 (Oct. 1988), pp. 3909–3917. Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems The Thomas-Fermi-Weizsäcker theory The following table shows the asymptotic behaviors of the integrands of the Thomas-Fermi and Weizsäcker term. The relativistic TFW model s → 0 s → ∞ t TF ( s ) 5 s 5 − 1 7 s 7 + O ( s 9 ) 2 s 4 − 8 3 s 3 + 2 s 2 + 1 4 4 − ln s − ln 2 − 4 s 2 + O ( 1 1 s 4 ) 2 ln s + 2 ln 2 + 1 − 2 ln s + 2 ln 2 ( f W ( s )) 2 2 s 3 + O ( s 5 ) s + 3 + O ( 1 s 4 ) s 2 Comparing them with the non-relativistic model as following. Rel Model Non-rel Model 8 π 2 ∫ R 3 t TF (( 3 π 2 ρ ( x )) 3 ( 3 π 2 ) 2 1 3 ) d x 1 Thomas-fermi 3 5 3 ( x ) d x ∫ R 3 ρ 10 ∫ R 3 d x ∣∇ ρ ( x )∣ 2 f W (( 3 π 2 ρ ( x )) 2 3 ) 1 ∣ ∇ ρ ( x )∣ 2 ( 3 π 2 ) 2 1 Weizsäcker 3 λ 8 ∫ R 3 ρ ( x ) d x 3 ( x ) 24 π 2 4 ρ Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems Results in the Thomas-Fermi-Weizsäcker theory Results in the relativistic Thomas-Fermi-Weizsäcker theory The relativistic energy E TFW ( p ) for molecules is bounded from below. The energy E TFW ( p ) for atoms has a global minimizer. For p ∈ P , there is a p 0 , such Z that E TFW ( p 0 ) = inf P E TFW ( p ) . Z Z The ground state energy E TFW ( N ) for ∫ ρ = N fixed is monotone decreasing in N . Z The particle number of the minimizer is N C ∶= cTF ∫ p 3 0 ( x ) d x . It satisfies 1 Z ≤ N C < 2 . 56 Z . An atom of atomic number Z can not bind more than 2 . 56 Z electrons. Strategy to prove the excess charge problem We follow the strategy given by Benguria. We multiply the Euler equation by ∣ x ∣ F ( p )( x ) and integrate, √ d x ( 8 ∣ x ∣ H ( p )( x ) p 3 ( x )( p 2 ( x ) + 1 − 1 ) − 6 λ F ( p )( x )∣ x ∣ ∆ F ( p )( x ) ∫ ∣ x ∣ − 8 α ZH ( p )( x ) p 3 ( x ) + 4 q α H ( p )( x ) p 3 ( x ) p 3 ( y )) = 0 3 π 2 ∫ d y ∣ x − y ∣ F ( p ) where H ( p ) ∶ = pF ′ ( p ) and c F ∶ = 0 . 612 < H < 1. Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems Some unsolved problems The relativistic Weizsäcker term breaks the convexity. The uniqueness of the minimizer is unknown. Unremovable case The existence of the minimizer for ∫ ρ = N < N C , especially for N ≤ Z . We are trying to multiply the Euler equation by some other functions than ∣ x ∣ F ( p )( x ) , to get a bound better than N < 2 . 56 Z . The ideal result in our strategy is N < 2 Z . Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory

Introduction Results in the Thomas-Fermi-Weizsäcker theory Some unsolved problems Thanks for your attention! Hongshuo Chen The excess charge problem in the relativistic Thomas-Fermi-Weizsäcker theory