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
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