The indefinite Sturm-Liouville operator A = (sgn x)(−d 2 /dx 2 + q(x)) is studied. It is proved that similarity of A to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of Weyl functions are given. For operators with a finite-zone potential, the components A ess and A disc of A corresponding to essential and discrete spectrums, respectively, are considered. A criterion of similarity of A ess to a selfadjoint operator is given in terms of Weyl functions for the Sturm-Liouville operator −d 2 /dx 2 + q(x) with a finite-zone potential q. Jordan structure of the operator A disc is described. We present an example of the operator A = (sgn x)(−d 2 /dx 2 + q(x)) such that A is nondefinitizable and A is similar to a normal operator.
The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given α ∈ R a medium that generates a resonance on the line α+iR with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints 0 ≤ b 1 ≤ B(x) ≤ b 2 with certain constants b 1,2 . The paper gives an accurate definition of optimal structures that ensures their existence. We prove that optimal structures are piecewise constant functions taking only two extreme possible values b 1 and b 2 . This result explains an effect recently observed in numerical experiments. Then we show that intervals of constancy of an optimal structure are tied to the phase of the corresponding resonant mode and write this connection as a nonlinear eigenvalue problem.MSC-classes: 49R05, 78M50, 35P25, 47N50, 47A55
We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh m-functions. Also we obtain necessary conditions for regularity of the critical points 0 and ∞ of J-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular critical point zero. In particular, it is shown that 0 is a singular critical point of the operator − (sgn x) (3|x|+1) −4/3 d 2 dx 2 acting in the Hilbert space L 2 (R, (3|x| + 1) −4/3 dx) and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type (sgn x)(−d 2 /dx 2 + q(x)) with the same properties.
Sufficient conditions for the similarity of the operator A := 1 r(x) (− d 2 dx 2 + q(x)) with an indefinite weight r(x) = (sgn x)|r(x)| are obtained. These conditions are formulated in terms of Titchmarsh-Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J -nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r(x) = sgn x and q ∈ L 1 (R, (1 + |x|) dx), we prove that A is similar to a self-adjoint operator if and only if A is J -nonnegative. The latter condition on q is sharp, i.e., we construct q ∈ γ <1 L 1 (R, (1 + |x|) γ dx) such that A is J -nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J -positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q ≡ 0, we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for "forward-backward" diffusion equations. (I.M. Karabash), duzer80@mail.ru (A.S. Kostenko), mmm@telenet.dn.ua (M.M. Malamud).
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