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On Sturm-Liouville operators with discontinuity conditions inside an interval. (English) Zbl 1168.34019

The author considers, in the space L 2 (0,π), the following problem

-y '' +q(x)y=k 2 y,0<x<π,(1)

with boundary conditions

y ' (0)=0,y(π)=0,(2)

and with the jump conditions

y(d+0)=ay(d-0),y ' (d+0)=by ' (d-0),(3)

where q(x) and a are real, d(π 2,π), a>0, a1, qL 2 (0,π).

The author studies only the case

b=a -1 (4)

for the jump condition (3). Some references about mechanics, physics, etc., problems which generate boundary-value problems with discontinuities inside the interval are given. As the potential q(x) and the number a are real due to the condition (4) the eigenfunctions are orthogonal.

If, k=1,2,, λ n =k n 2 denotes the eigenvalues of the problem (1)–(3), then

k n =k n o +c n k n o ,c n =O(1),n,

where the values k n o correspond to the case q(x)0, i.e. k n o are the roots of the equations

a+1 acoskπ+a-1 acosk(2d-π)=0·

Note that inf n,m |k n o -k m o |>0.

The proof is based on the transformation operators (the author uses the methods of V. A. Marchenko [Sturm-Liouville operators and applications, Translated from the Russian by A. Iacob, Basel: Birkhäuser (1986; Zbl 0592.34011)]. Later the author proves the uniqueness for the inverse problems: the reconstruction of the boundary-value problem (1)–(3) from the Weyl function, from spectral data and from two spectra.

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