Monashova
A.Z.
L.N.Gumilyov
Eurasian National University, Astana
ABOUT ONE MODIFICATION OF OTELBAEV FUNCTION 
In paper [1] it is obtained
strong asymptotic form of distribution function of discrete spectrum for Sturm-Liouville
operator
, 
,
where 
, 
. Asymptotic formula was obtained in terms of Otelbaev
function
.
Our aim is studying of properties of the
following modification of function 
:
,
where 
is locally summable
function on half-axle 
.
From properties of absolutely
continuity of measure 
it follows that there exists sufficiently small number
, such that
, if 
.
Therefore for all 
the following
estimates hold
,
.
Below 
.
Assumption
1. Let 
. Then the following equality holds
(1)
Proof. The condition 
is equivalent the
inequality 
.
For all 
And in the view of absolutely continuity of
indefinite integral 
(2)
If in (2) we suppose that 
, then
And there exists 
, 
, 
such that
.
But in this case 
, which is
impossible.
Assumption
2. Let 
. Then the
function 
is continuous on half-axle 
.
Proof.
From simple estimate
, 
It follows that
for all 
.
In the view of equality (1) for all 
(3)
Since 
, we study
continuity of function 
. Now we assume
that the function 
is not continuity in some
point 
. But then
there exists 
and sequence 
, 
, such that
, (4)
where 
, 
. Since 
we have
And we could choose number 
such that the following estimates hold:
if 
(5)
if 
(6)
Further, assuming that 
we consider all possible
cases in (4).
If 
, then
according to (5), (6)
,
which is impossible.
If 
, then
according to (5) and equality (1) for 
,
which is also impossible.
The author thanks Professor L.K.Kusainova
for discussion of the presented results and for generous pieces of advice,
which have improved the final version of this paper.
References
Otelbaev Ì.Î. Estimates of spectrum of Sturm-Liouville operator. –
Alma-Ata: Gylym, 1990. – 191c.