On stability of systems with singular potentials

Volume40B, number 1

ON

PHYSICS LETTERS

STABILITY

OF

SYSTEMS

WITH

SINGULAR

12 June 1972

POTENTIALS

L V. SIME NOG

Institute for Theoretical Physzcs, Academy of Sciences of the Ukraunan SSR, K~ev , USSR Received 12 April 1972

It is shown for the s y s t e m s with singular attractive potentials, that the concept of independent particles m the s e n s e of H a r t r e e is exactly r e a l i z e d m the asymptotic limit N ~

A rigorous investigation of the stability of the systems with Coulomb interactlons has bee made by Dyson [I] using the basic results of Fisher and Ruelle. Both for the Bose and Fermi particles his r e sults taking on the form of inequalities for the ground-state energies lead to unambiguous concluslons about the energies in the llmlt of a great number of particles. In some particular cases he has suggested that an absolute minimum for the energy of Fermi particles is realized on the states of the type of a Thomas- Fermi self-consistent field. In the present work we show for what class of potentials a self-conslstent Hartree field serves as a limiting problem for the many-particle SchrSdinger equation in the asymptotic limit of a great number of particles N ~ co Now we give a statement of the problem in more detail. Give a Hamiltoman

H= ~

z~=lpi - i
(1)

where 0 --< ~ < 2, and

s(Iri-

Ilr-l r s]--O, -rjl- ,

g+airz-rjI ~,

(g>O, fi>O)

(2)

C, (c>o),

-f(r)>~ i s the m o n o t o n i c p o t e n t i a l . We now i n v e s t i g a t e the l o w e s t s y m m e t r i c a l b o u n d s t a t e of the H a m i l t o n i a n (1) i n the l i m i t N ~ . ~ . T h e f o l l o w i n g t h e o r e m h o l d s t h e n : T h e o r e m 1. T h e l o w e s t e l g e n v a l u e of H (1) in the c l a s s of s q u a r e i n t e g r a b l e f u n c t i o n s i s g i v e n a s y m p t o t i c a l l y by N ~ co

,,m o = -

g

N--~o

t1 + O

,

(3)

m i n ( 1 , f l / ( 2 - a))

where -s

= i n f t f d x ~ ( x ) p 2 ~ ( x ) - ½f dx f d )

t

~_~(x) ~U'(y)

21,<-yl 2~_ 1 and ~(x) is the square integrable function (fdx ~2(x)= i).

(4)

The eigenfunction I for H, corresponding to

energy Eo, has a single-particle form N

= ~ ¢((2mNg)l/(2-a)ri).

(5)

,=i Here we shall explain in brief the proof of this theorem while its details will be given elsewhere. Only the behaviour of the potential V(ri?) =-f(ri~)/~a a t r / . ~ 0 is important for the HamIltonian H (i) in the limit N ~ co That is, the lowest'eigenvalue a{ N ~ is defined by the character of the singularity of the potential at zero:

53

PHYSICS

Volume 40B, number 1

12 June 1972

LETTERS

E o ~ Nl+2/(2-°t) • If we perform the scaling transformation in H (1)

(6)

r z = (2mNg)-l/(2-°t) x i , then we have for eq. (1) in l i m i t N ~ H = (2m)

cl/(2-a)(Ng)2/(2-aff

= (2m)a/(2-a)(g~2/(a-a)

p2 _ -ff ~

1

a

i
1

g

(71

Now the p r o b l e m r e d u c e s to the p r o o f that a s N ~ 0% the l i m i t i n g H a m i l t o n i a n

i=I

"

Ixi-

x3]a

h a s the l o w e s t s t a t e of a s i n g l e - p a r t i c l e c h a r a c t e r H

(lim)

q' = Eo~I',

(8)

N

gz = ~

t~(x~),

E o =-sN.

i=1 Now we i n t r o d u e e in H(lim), a c o m p e n s a t i n g field ~o(xi) N

H ( h m ) = i=1 ~ (pz+ ~°(xi)) -

"= P(xz) + ' ~ z < j

Ix i - x l } a

o

and u s e the p e r t u r b a t i o n t h e o r y in W. If we choose a c o m p e n s a t i n g field p r o v i d e d that a s i n g l e - p a r t i c l e t r a n s i t i o n is (0 IWII> ~ 1/~fN,

(9)

then the field ~o(x) takes on the form ~o(x) = - f d y

~ 2 ( y ) / ] x - y ]a,

(10)

w h e r e ~(x) g i v e s the l o w e r bound (4). With s u c h a c h o i c e of the c o m p e n s a t i n g field ~o(xi) we get for the lowest eigenvalue Z ° = ( 0 1 H o l 0 ) + ( 0 IW]0> + O(1) = N i n f {

fdx q/(x)p 2 ~/(x) + ½fdx ~92(x)q~(x)}.

(11)

It is important to mention in this connection, that in (II), the contributions of the order of unity to the second and higher orders of perturbation theory in W are given by the virtual transitions with two-particle excitations. These contributions can be calculated. Thus, the system with a limiting H(lim) (I') satisfies always the stability condition (Eo ~N) i r r e spective of the form of a pair potential and such a system is localized in space There exists also a generalization of Theorem 1 both to the cases of antisymmetric states and monotonic potentials for which the lower bound is realized when r. ~ 0. Theorem 1 is also extented to the case when there is an e~ternal field singular at zero such as~3the atomic system with a fixed nucleus, etc. A detailed account of these results will be given elsewhere. Here we can also make an assumption that for the case of a certain class of the short-range repulsive potentials, the limit of independent pairs, but not of independent particles, is realized asymptotically. To conclude, I extend my gratitude to I. M. Burban, V. P. Gachok and D. Ya Petrina for discussions of the points given above.

References [1] F . J . Dyson, J Math. Phys., 8 (1967) 1538. 54