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Completeness of eigenfunctions of Sturm-Liouville problems with transmission conditions. (English) Zbl 1210.34123

This work considers the discontinuous boundary value problem with the eigenvalue parameter appearing linearly in the boundary conditions for a second order ordinary differential equation:

-(a(x)u ' ) ' +q(x)u=λu,x-1,00,1
α 1 u-1+α 2 u ' -1=0
β 1 u1-β 2 u ' 1=λβ 2 ' u ' 1-β 1 ' u1
u0+-α 3 u0--β 3 u ' 0-=0,
u ' 0+-α 4 u0--β 4 u ' 0-=0,

where λ is a spectral parameter, ax=a 1 2 for -1x0 and ax=a 2 1 for 0x1, a i 0 i=1,2 are real numbers, qxL 1 -1,1, α i , β i , β j ' (i=1,2,3,4, j=1,2) and α 3 β 4 -α 4 β 3 >0, β 1 ' β 2 -β 1 β 2 ' >0, α 1 +α 2 0·

Firstly, the given boundary value problem is reduced to an linear operator A in a special Hilbert space H. Then, it is shown that the operator A is selfadjoint in the space H. It is proved that the operator A has only point spectrum and the corresponding eigenfunctions form an orthonormal basis in H.

There are many studies about boundary problems with discontinuous coefficients and transmission conditions, completeness, minimality and basis property of the eigenfunctions of this boundary value problem [for example, see A. Gomilko and V. Pivovarcik, Math. Nachr. 245, 72–93 (2002; Zbl 1023.34023); V. Pivovarcik, Asymptotic Analysis., 26 (2001)]; R. Kh. Amirov, A. Z. Ozkan and B. Keskin, Integral Transforms Spec. Funct. 20, No. 7–8, 607–618 (2009; Zbl 1181.34019) and others].

MSC:
34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
47E05Ordinary differential operators
34B08Parameter dependent boundary value problems for ODE
34B24Sturm-Liouville theory