AMREIN BOUTET DE MONVEL"BERTHIER GEORGESCU

(d) for each e 0 there is r « such that for all u £ C (f t )

o r

| x |

b j

u | |

L

2

( n )

i

e

| | u | |

H

l

( n )

,

| x | c u | |

L

2

( f J )

S

e

| | u | |

H

2

( n )

.

Le t cp : ( r ,°°) - JR , f o r some r °° , be of c l a s s C , n o n - d e c r e a s i n g

o o

and such t h a t f o r some a € ( 0 , 2 ) and 3 0 :

(a ) cp'(t ) ^ c o n s t - t , | t p " ( t ) | S c o n s t - ( l + c p f ( t ) ) , tip" T ( t ) (l+tp ? ( t ) ) "1- Q

a s t - °° ,

(3) li m i n f ( a c p '

2

+. 2tcp'(p") - 3 a s t + »

9

(y) t c p

f

( t ) ^ c o n s t , o r t c p

f

( t ) • °° a s t - °° .

Let X 0 and m € {0,1,2,...}. Then there are finite constants K and r

such that the inequality (1.8) holds for each s £ [0,2] and all

u € H2(ft ) (the closure in tf2(]Rn) of the set of all functions in H2(]Rn)

having compact support in 0. ) .

We add a few remarks in connection with this theorem . Our results

apply to a more general class of operators A which are such that the

coefficient c may be decomposed into a sum c = c + c , where c. satis-

fies the conditions (c) and (d) of Theorem 1.2 , while cp may contain

so-called long range potentials and a sum of two-body interactions of

homogeneous type ( in the language used for Schrodinger operators in

quantum mechanics ) . In this case a precise relation must be fulfilled

between the quantities a, 3 , c and X , and a $ 0 or X i 0 are allowed .

The coefficient cp may be such that -A + o, is unbounded below ; for

example we can treat the case where cJx) = ax- for any a € M and

I £ {0,1,2,...} . In the last section of this paper we give several exam-

ples of applications of our abstract theorems to differential operators

of the form (1.1); however we shall only treat operators for which