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Discontinuous Sturm-Liouville problems containing eigenparameter in the boundary conditions. (English) Zbl 1112.34070

The authors consider the following discontinuous Sturm-Liouville eigenvalue problems with eigenvalue parameters both in the equation and in one of the boundary conditions:

τu:=-a(x)u '' +q(x)u=λu,x[-1,0)(0,1]

with the boundary conditions at x=±1

L 1 u:=α 1 u(-1)+α 2 u ' (-1)=0;L 4 (λ)u:=λ(β 1 ' u(1)-β 2 ' u ' (1))+((β 1 u(1)-β 2 u ' (1))=0,

and the transition conditions

L 2 u:=γ 1 u(-0)-δ 1 u(+0)=0,L 3 u:=γ 2 u ' (-0)-δ 2 u ' (+0)=0,

where a(x)=a 1 2 >0 for x[-1,0), a(x)=a 2 2 >0 for x(0,1]; q(x) is a given real-valued function which is continuous in [-1,0] and in [0,1]; α i ,β i ,β i ' ,γ i ,δ i ,i=1,2, are real numbers with |α 1 |+|α 2 |0,|β 1 ' |+|β 2 ' |+|β 1 |+|β 2 |0,|γ 1 |+|δ 1 |0,|γ 2 |+|δ 2 |0, and β 1 ' β 2 -β 1 β 2 ' >0. With an operator theoretic approach, some classical properties and asymptotic approximate formulae for eigenvalues and normalized eigenfunctions are obtained.

MSC:
34L20Asymptotic distribution of eigenvalues for OD operators
34B24Sturm-Liouville theory
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