TAILIEUCHUNG - Determination of a fractional-linear pencil of Sturm-Liouville operators by two of its spectra
In this paper we consider the Sturm-Liouville equations on a finite interval which is fractional-linear in the spectral parameter. The inverse spectral problem consisting of the recovering of the operator from the two spectra is investigated and a uniqueness theorem for solution of the inverse problem is proved. | Turk J Math 28 (2004) , 307 – 311. ¨ ITAK ˙ c TUB Determination of a Fractional-Linear Pencil of Sturm-Liouville Operators by Two of Its Spectra R. T. Pashayev Abstract In this paper we consider the Sturm-Liouville equations on a finite interval which is fractional-linear in the spectral parameter. The inverse spectral problem consisting of the recovering of the operator from the two spectra is investigated and a uniqueness theorem for solution of the inverse problem is proved. Key Words: Eigenvalue, scattering function. 1. Introduction Consider the second order differential equation ( 00 −y + q(x)y = λy, a b we have f(x, k) = eikx if x > b. Therefore the continuity of f(x, k) and f 0 (x, k) at x = b yields eikb = c1 ϕ(b, k) + c2 ψ(b, k), ikeikb = c1 ϕ0 (b, k) + c2 ψ0 (b, k) from which we find the coefficients c1 and c2 : 309 PASHAYEV c1 = 1 ikb e {ikϕ(b, k) − ϕ0 (b, k)}, h c2 = 1 ikb 0 e {ϕ (b, k) − ikϕ(b, k)}. h Hence f(x, k) = 1 1 ikb e {ikϕ(b, k) − ϕ0 (b, k)}ϕ(x, k) + eikb {ϕ0 (b, k) − ikϕ(b, k)}ψ(x, k) h h and the formula (7) follows. The lemma is proved. 2 Theorem 2 The function q(x) and number h are uniquely determined by two spectra {λn } and {µn }. Proof. It is known that the scattering function of problem (5),(6) has the form ([1],[2]): S(k) = f 0 (0, −k) − hf(0, −k) . f 0 (0, k) − hf(0, k) So by virtue of (7) we get S(k) = ϕ0 (b, k) + ikϕ(b, k) −2kbi e . ϕ0 (b, k) − ikϕ(b, k) Hence, by (4) we find S(k) = e−2kbi Let us set 310 Φ2 (k) + ikΦ1 (k) . Φ2 (k) − ikΦ1 (k) (8) PASHAYEV F (k) = Φ2 (k) − ikΦ1 (k). We can prove in a standard way that {λn } and {µn } are intermittent, so that the argument of F (k) as k describes the real axis once is zero (see [3]). Hence, F (k) is non-zero in the upper halfplane, . problem (5),(6) has no eigenvalues. Thus the scattering data of problem (5),(6) consist of the function S(k) only. On the other hand in [2] it has been proved that the coefficient qe(x) of Equation (5) and the number h
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