Inverse Problems for Differential Operators with Indefinite Discontinuous Weights
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Inverse Problems for Differential Operators with Indefinite Discontinuous Weights V. Yurko Abstract. Non-self-adjoint second-order differential operators on a finite interval with indefinite discontinuous weights are studied. Properties of spectral characteristics are established and inverse problems of recovering operators from their spectral characteristics are investigated. For these class of nonlinear inverse problems algorithms for constructing the global solutions are developed, and uniqueness theorems are proved. Mathematics Subject Classification. 34A55, 34B24, 47E05. Keywords. Differential operators, indefinite discontinuous weights, inverse spectral problems, method of spectral mappings.
1. Introduction We consider the following differential equation − y (x) + q(x)y(x) = λr(x)y(x),
0 < x < π.
2
(1) 2
Here λ is the spectral parameter, r(x) = −ω for x < a, r(x) = α for x > a, and α > 0, ω > 0, a ∈ (0, π). The function q(x) is complex-valued, and q(x) ∈ L(0, π). Denote by L the non-self-adjoint boundary value problem for Eq. (1) with boundary conditions U (y) := y (0) − hy(0) = 0,
V (y) := y (π) + Hy(π) = 0,
(2)
and with jump conditions y(a + 0) = a1 y(a − 0), y (a + 0) = y (a − 0)/a1 + a2 y(a − 0), a1 = 0, (3) here h, H, a1 , a2 are complex numbers, and a± := a1 ± iω/(αa1 ) = 0. The function r(x) changes sign in the interior point. The point x = a is called the turning point. In this paper we establish properties of spectral characteristics of 0123456789().: V,-vol
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V. Yurko
Results Math
L, and study inverse problems of recovering L from its spectral characteristics. For definiteness, let Re a1 > 0 or Re a1 = 0, Im a1 > 0. Differential equations with turning points and indefinite weights arise in various areas of mathematics as well as in applications (see [3,9,10] for details). In particular, they appear in elasticity, optics, geophysics, electronics and other branches of natural sciences and engineering. Moreover, a wide class of differential equations with Bessel-type singularities and their perturbations can be reduced to differential equations with turning points and indefinite weights. It is also known that inverse spectral problems play an important role for solving nonlinear integrable evolution equations (KdV equation and others). Inverse problems for equations with turning points and indefinite weights help to study the blow-up behavior of solutions for such nonlinear equations. Inverse spectral problems for the classical Sturm–Liouville equation −y + q(x)y = λy, have been studied fairly completely (see [5,7,8] and the references therein). The presence of turning points and indefinite weights in the differential equation produces essential qualitative modifications in the investigations of inverse problems. Some aspects of the inverse problem theory for such equations on a finite interval without jump conditions were studied in [2,4,11] and other works. In [2,11] uniqueness theorems were proved for the self-adjoint case in different statements; in
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