ISSN 0001-4346, Mathematical Notes, 2016, Vol. 100, No. 2, pp. 291–297. © Pleiades Publishing, Ltd., 2016.
On Sharp Asymptotic Formulas for the Sturm–Liouville
Operator with a Matrix Potential
*
F. Seref
**
and O. A. Veliev
***
Dogus University, Istanbul, Turkey
Received February 18, 2015
Abstract—In this article we obtain the sharp asymptotic formulas for the eigenvalues and eigen-
functions of the non-self-adjoint operators generated by a system of the Sturm–Liouville equations
with Dirichlet and Neumann boundary conditions. Using these asymptotic formulas, we find a
condition on the potential for which the root functions of these operators form a Riesz basis.
DOI: 10.1134/S0001434616070245
Keywords: differential operator, matrix potential, asymptotic formulas, Riesz basis.
1. INTRODUCTION
We consider the differential operators D
m
(Q) and N
m
(Q) generated in the space L
m
2
[0, 1] by the
differential expression
l(y)= −y
′′
(x)+ Q (x) y(x) (1.1)
with Dirichlet
y (1) = y (0) = 0, (1.2)
and Neumann
y
′
(1) = y
′
(0) = 0 (1.3)
boundary conditions respectively, where L
m
2
[0, 1] is the set of the vector functions f =(f
1
,f
2
,...,f
m
)
with f
k
∈ L
2
[0, 1] for k =1, 2,...,m and Q(x)=(q
i,j
(x)) is an m × m matrix with complex-valued
summable entries q
i,j
(x) . The norm ‖.‖ and inner product (., .) in L
m
2
[0, 1] are defined by
‖f ‖ =
1
ˆ
0
|f (x)|
2
dx
1/2
, (f,g)=
1
ˆ
0
〈f (x) ,g (x)〉 dx,
where |.| and 〈., .〉 are respectively the norm and the inner product in C
m
.
Note that general results concerning the Riesz basis property of ordinary differential operators of
higher order and more complicated boundary-value problems, when the equations and the boundary
conditions contain nonlinear functions of the spectral parameter, were obtained in the papers of A. A.
Shkalikov. In [1]–[5], he proved that the root functions (eigenfunctions and associated functions) of
the operators generated by an ordinary differential expression with summable matrix coefficients and
regular boundary conditions form a Riesz basis with parentheses and only the functions corresponding
to splitting eigenvalues should be included in the parentheses. In particular, if the boundary conditions
are strongly regular and there are no asymptotically splitting eigenvalues, then one has the ordinary
Riesz basis. Luzhina [6] generalized these results to boundary-value problems in which the coefficients
∗
The article was submitted by the authors for the English version of the journal.
**
E-mail: serefulya@gmail.com
***
E-mail: oveliev@dogus.edu.tr
291