Bound states in N-body quantum systems

Physica 124A (1984) 311-316 North-Holland, Amsterdam

31 l

BOUND STATES IN N-BODY QUANTUM SYSTEMS

R. FROESE Department of Mathematics, University of Virginia, Charlottesville, VA 22903 I would like to talk about some results on the large Ixl solutions,

behavior of L2

~(x) , to the Schrodinger equation

H~ = E~

where H is an N-body quantum Hamiltonian. bounds for ~.

I w i l l concentrate on finding lower

The connection between the permissible rates of decay for

¢ and

the spectrum of H w i l l show that for a large class of H positive eigenvalues do not occur. These results were obtained j o i n t l y with Ira Herbst4'5 and are closely r e l a t ed to results which we obtained earlier in collaboration with Maria and Thomas Hoffmann-Ostenhof.6 The class of Schr6dinger operators under consideration is actually somewhat }N larger than the class of N-body Hamiltonians arising in physics. Let {~i i=l be a set of orthogonal projections onto the subspaces {xi}Ni=l of ~Rn . Suppose vi(Y) is a real valued function of y ~ Xi which "goes to zero" as [yl ÷ ~ in a sense to be made precise later. in L2(• n) given by

Then H is the operator acting

N

H : -L + ~ v i ( ~ i ( x ) ) i=l with domain D(H) = D(z~). Two sets of conditions on V w i l l be used in what follows: Let Ai and y denote the Laplacian and variable corresponding to Xi .

Then the f i r s t set

o f conditions is

(a)

v i ( - ~ i + I ) -I (-mi + l ) - l y

is compact on L2(Xi ) • v Vi(-A i + l ) -I

is compact on L2(Xi ) .

Presented at the V l l t h INTERNATIONAL CONGRESS ON MATHEMATICAL PHYSICS, Boulder, Colorado, 1983.

312

R. FROESE

Let ~. = dim (Xi) and Pi = max {2, ~i - I } . 1 is

(b)

Then the second set o f c o n d i t i o n s

v i e LPi(xi ) + L~(Xi ) vi =

v1(l . ) + vi(2)

and f o r each

with

c > 0

(I + I Y l ) v i ( 1 )

y • vvi(2)

e k

Pi

<__c(-zl) + b E

(X i ) + L (X i ) f o r some

b

In the above conditions the gradients on v i are taken in the distribution sense; i t is not assumed that classical derivatives exist.

A bounded measurable func-

tion with o ( I x l - l ) decay at i n f i n i t y satisfies both (a) and (b).

On the other

hand i f classical derivatives exist and have suitable decay, these conditions are also satisfied by long range potentials, for example vi(Y) = lyl -~ holds are defined as follows.

Thres-

For any proper subspace Y C R n of the form

Y = span {Xi} i cJ consider the operator Hy acting in L2(y) given by

Hy = -Ay +

~ vi ( ~ i (y)) l xi~Y

where ~y is the Laplacian for Y.

The set of thresholds is defined to be all

eigenvalues of such operators and is denoted %. The spectrum of such an N-body operator looks like:

v

;

v .. 0

X

ei genval ues

@

thresholds

.,,

continuous spectrum

I w i l l describe in a moment how to prove that there are no positive thresholds or eigenvalues. Let H be as above and suppose ~ is an L2-solution to the Schr~dinger equation H~ = E~. We ask the question: when is exp(~Ixl) ~ ~ L2(~n ) ? Consider increasing ~ in exp(~Ixl)~ starting from ~ = O.

For ~ = 0 we have

BOUND STATES IN N-BODY QUANTUM SYSTEMS

exp(mlxl)~ = ~ ~ L2(~ n ) by hypothesis. some p o i n t mO~exp(mlxl)~

313

I t might happen t h a t as we move past

changes from being in L 2 ( ~ n )

to not being in L2(~n).

Theorem 1 says t h a t i f V s a t i s f i e s (a) t h i s can only happen i f m02 + E is a threshold.

So i f we define

m(~) : sup{m 2 + E :m ~ 0 and exp(mIx I) @ ~ L 2 ( R n ) } then we have Theorem I :

If V satisfies

(a) then

T(~) c Z U { + ~} This theorem gives an L2 upper bound, namely i f T O is the f i r s t above E then e x p ( ~ I x l ) ~E L 2 ( ~ n )

for all a < ~

the possibility that T(@) = +~ , i.e. that exp(~Ixl) ~ Theorem 2: I f V s a t i s f i e s

.

L2(~ n) for all

~.

(b) then

Thus i f V s a t i s f i e s both (a) and (b) we know t h a t T(~) c I . Corollary I.

threshold at or

Theorem 2 e l i m i n a t e s

If V satisfies

NOW we can prove

(a) and (b) then H has no p o s i t i v e eigenvalues

or thresholds. The proof o f t h i s c o r o l l a r y is by i n d u c t i o n on the set of subspaces Y o f the form Y = span { X i } i ~ j

.

The induction step goes as f o l l o w s :

each W of t h i s form s t r i c t l y

thresholds. Then, by d e f i n i t i o n , and 2 ~(~) f o r any e i g e n f u n c t i o n not p o s i t i v e .

Suppose f o r

contained in YoHw has no p o s i t i v e eigenvalues o f Hy has no p o s i t i v e thresholds.

By Theorems 1

@ with eigenvalue E is a threshold and thus

But from the d e f i n i t i o n o f T(~) i t is c l e a r t h a t E ~ T ( ~ )

.

Thus E ~ 0 . Now we can also prove C o r o l l a r y 2: I f V s a t i s f i e s (a) and (b) then

exp(~Ixl) ~ # L2( ~Rn) for ~ > ~ E To prove t h i s c o r o l l a r y note t h a t i f f o r some ~

with 2

+ E > O.

i t is f a l s e we have e x p ( ~ I x l ) @ ~ L2 (IRn)

Thus T(@) >__2

+ E > 0 , which is a c o n t r a d i c t i o n

since T(~) is a threshold. C o r o l l a r y 2 can be viewed as a crude s o r t o f lower bound and the question

314

R. FROESE

n a t u r a l l y a r i s e s : can b e t t e r lower bounds be obtained?

If

~ is the ground

s t a t e then d e t a i l e d p o i n t w i s e l o w e r bounds have been obtained by Carmona and Simon. 2

If

9

is not the ground s t a t e , then the p o t e n t i a l l y

complicated na-

t u r e o f the set where ~(x) = 0 i n d i c a t e s t h a t p o i n t w i s e lower bounds f o r I~(x)l Merigot,

would be very d i f f i c u l t to o b t a i n . I n s t e a d , f o l l o w i n g Bardos and 1 we consider the average o f ~ over spheres. Let S(R) be the sphere

o f radius R and d e f i n e I~l R by

I~IR 2 = ~

I-~12 ds

S(R) Then we have the f o l l o w i n g theorem, which gives exponential upper and lower bounds f o r

I~IR when E l i e s below the e s s e n t i a l spectrum o f H.

Theorem 3. lira R-I

R-~o:

If V satisfies

(a) and (b) and E < z = i n f ~ess(H) then

log(IC, IR) = -%

where a 0 c Z . When E does not n e c e s s a r i l y l i e

below the infimum o f the e s s e n t i a l spectrum

then the theorem remains t r u e w i t h I~IR replaced by the L2 norm o f

~ over a

s p h e r i c a l shell w i t h r a d i u s R and thickness 6(R) where ~(R) does not decrease

too q u i c k l y .

For a precise statement and p r o o f of these r e s u l t s ,

see r e f s . 5

and 7. To conclude t h i s t a l k I w i l l o f Theorem l .

very roughly sketch the main step o f the p r o o f

The many o m i t t e d d e t a i l s can be found in r e f . 3.

generator of dilations,

Let A be the

i.e.

,~ = ½ (X o D + D o x) where D is the g r a d i e n t o p e r a t o r . [H,A] = - 2 & The f i r s t

Then the commutator o f H w i t h A is given by

- x • v V

i n g r e d i e n t in the p r o o f is the Mourre estimate 4'8 f i r s t

body H by Perry, S i g a l , and Simon. then there e x i s t s an open i n t e r v a l

E(1) [H,A] E(1) ~ c

I t says t h a t i f

•

proved f o r N-

is a non-threshold p o i n t

I c o n t a i n i n g ~ such t h a t

E(1) + K

where E(1) i s the s p e c t r a l p r o j e c t i o n f o r H, c is p o s i t i v e and K is compact.

BOUND STATES IN N-BODY QUANTUM SYSTEMS

The second ingredient is some equations satisfied by @ ~ exp(F) .~ F A computation shows that

315

where

F is an increasing function of Ixl . H~ F : (E + (VF)2) ~F - B~F

where B is an antisymmetric f i r s t order operator.

Now using the formal relation

: 0 with A = exp(F)A exp(F) we obtain <~F,EH,A] ~F> : -4

II

g½A~FII2 + <~F' G ~F)

where g is defined by vF = xg and G is a function which w i l l be small for large [xl

for the choice of F that we make. What we want to show in this step of the proof is that T(~) cannot be a non-

threshold point greater than E.

Assumethe contrary.

with 2 + E < T(O) < (~ + y)2 + E where ~ and y

Then we can find ~ > 0

are such that the Mourre est-

imate holds for some I containing [ 2 + E, (~ + ¥)2 + E]. exp(~Ixl) 0 ~ L2(Rn) while exp((~ + y ) I x l ) ~ L 2 ( R n ) .

Furthermore To interpolate we de-

fine F~ by exp(Fx) = exp(~[xl) (l + y l x l / x )~ Then exp (Fx) ~ ~ L2(~ n) for each

but

l im llexp(Fh)VII= ~ , Define

Then i t is easily seen that for a bounded set f i l l 2 dnx ~ 0 as

~~

so that ~X ~ O . The next step is to show that IIB~II ÷0. We omit the proof of t h i s . Now using this and the fact that (VF~)2 is approximately 2 we have for large 1 HI~ = (E + c=2) I~

316

R. FROESE

From t h i s i t f o l l o w s t h a t , roughly speaking, E(1).

Here we are ignoring an e r r o r term.

~X converges i n t o the range of From the Mourre estimate we have

<~E), E(1)[H,A] E(1) I)> _> c liE(1) ~)~II2 + ,,K !i)> In the l i m i t

~ ÷ = we can more or less ignore the E(1), and the l a s t term van-

ishes since K is compact and ~x ~ O.

So up to a small e r r o r which goes to zero

with Y, lim inf <~,

[H,A] I X )

>

c

On the other hand using the f a c t t h a t -411 g½A~ll 2 < 0 we obtain

(~),[H,A] I~) <__( I x , G IX) Now f o r l a r g e Ixl G is small on the order of y . I x l regions as I x l + =

Since T~

is moving to large

we have, up to an O(y) e r r o r term

lim sup ~I), [H,A] ~>)

<

0

This c o n t r a d i c t s the previous equation ( f o r small enough y ) and completes the proof. REFERENCES I ) C Bardos and M Merigot, Proc. Roy. Soc. Edin. 76A, 323-344 (1977). 2) R Carmona, B Simon, Comm. Math. Phys. 80 (1981) 59. 3) R Froese and I Herbst, Comm. Math. Phys. 87 (1982) 429-447. 4) R Froese and I Herbst, Duke Math. J. 49 (1982) 1075-1085. 5) R Froese and I Herbst, Exponential Lower Bounds f o r Solutions of the Schr~dinger Equation: Lower Bounds on the Shperical Average, p r e p r i n t . 6) R Froese and I Herbst, M Hoffman-Ostenhof and T Hoffman-Ostenhof, Comm. Math Phys. 87 (1982) 265-286. 7) R Froese, Lower Bounds for Solutions of the SchrSdinger Equation, U. o f V i r g i n i a D i s s e r t a t i o n , August 1983. 8) P Perry, I M S i g a l , and B Simon, Ann. Math. 114 (1981) 519-567.