INVERSE PROBLEM OF THE STORM-LIUVILLE OPERATOR WITH NON-SEPARATED BOUNDARY VALUE CONDITIONS AND SYMMETRIC POTENTIAL
Ключевые слова:
Sturm-Liouville operator, spectrum, inverse Sturm-Liouville problem, Borg theorem, Hambardzumyan theorem, Levinson theorem, non-separated boundary value conditions, symmetric potential, invariant subspaces, differential operators, inverse spectral problems.Аннотация
Under the inverse tasks of spectral analysis understand tasks reconstruction of a linear operator from one or another of its spectral characteristics. The first significant result in this direction was obtained in 1929 by V.A. Ambartsumian. He proved the following theorem. The first mathematician who drew attention to the importance of this result of Ambartsumian was the Swedish mathematician Borg. He performed the first systematic study of one of the important inverse problems, namely, the inverse problem for the classical Sturm - Liouville operator of the form (1.1) with respect to spectra. Borg showed that in the general case one spectrum of the Sturm - Liouville operator does not determine it, so the result of Ambartsumian is an exception to the general rule. In the same work, Borg shows that two spectra of the Sturm - Liouville operator (under various boundary conditions) uniquely determine it. More precisely, Borg proved the following theorem. The inverse problems for differential operators with decaying boundary conditions are fairly well understood. More difficult inverse problems for Sturm - Liouville operators with unseparated boundary conditions have also been studied. In particular, the periodic boundary-value problem was considered in a number of papers. I.V. Stankevich proposed the formulation of the inverse problem and proved the corresponding uniqueness theorem.
The present work is devoted to a generalization of the theorems of Ambartsumian and Levinson, in particular, our results contain the results of these authors. In the paper, a uniqueness theorem is proved, for one spectrum, for the Sturm-Liouville operator with unseparated boundary conditions, a real continuous and symmetric potential. The research method differs from all previously known methods, and is based on the internal symmetry of the operator generated by invariant subspaces.
Note that the operator we are considering is non-self-adjoint, although the potential is real and symmetric, this moment plays an essential role for our method, because we construct a pair of Borg operators through the operator and its adjoint one. Other authors use the Leibenzon mapping method.
