## Linear Operators: Spectral theory |

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Page 1559

As usual , let N ( t ) be the number of zeros in the interval [ 0 , t ) of a solution of

the equation of and

2 . ( a ) Show that SiQ ( s ) -- dN ( 8 ) $ SiQ ( 8 ) -1ds + K S ; Q = 1-4dQ ( 8 ) .

As usual , let N ( t ) be the number of zeros in the interval [ 0 , t ) of a solution of

the equation of and

**suppose**that N ( t ) = f ' ( ( s ) ds + KQ ( € ) 2– € where 0 < e <2 . ( a ) Show that SiQ ( s ) -- dN ( 8 ) $ SiQ ( 8 ) -1ds + K S ; Q = 1-4dQ ( 8 ) .

Page 1563

G41

belongs to the essential spectrum of t . ( a ) Let { { n } be a sequence in D ( To ( t )

) such that inl = 1 , Tin → 0 , and such that in vanishes in the interval [ 0.n ) . Set

gn ( t ) ...

G41

**Suppose**that the function q is bounded below .**Suppose**that the originbelongs to the essential spectrum of t . ( a ) Let { { n } be a sequence in D ( To ( t )

) such that inl = 1 , Tin → 0 , and such that in vanishes in the interval [ 0.n ) . Set

gn ( t ) ...

Page 1597

( 18 ) In the interval ( 0 , 0 ) ,

< 0 , 19 ( t ) | 3 ( q ' ( t ) ) 2 ( c ) | g for large M. Then the essential spectrum of t is

empty ( Wintner [ 8 ] ) . ( 19 ) In the interval ( a , ) ,

...

( 18 ) In the interval ( 0 , 0 ) ,

**suppose**that ( a ) lim q ( t ) -00 , ( q ' ( t ) ) ( b ) lim sup< 0 , 19 ( t ) | 3 ( q ' ( t ) ) 2 ( c ) | g for large M. Then the essential spectrum of t is

empty ( Wintner [ 8 ] ) . ( 19 ) In the interval ( a , ) ,

**suppose**that Sie dt – đó , lim q...

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