We give a description of finite-zone PT-potentials in terms of explicit theta functional formulas.

Определяются спектральные многообразия (Блоха) многомерных дифференциальных операторов на неодносвязных многообразиях. С их помощью дается описание аналитической зависимости спектров магнитных лапласианов на неодносвязных многообразиях от значений потоков Ааронова-Бома и строятся аналоги спектральных кривых для двумерных операторов Дирака на римановых поверхностях и тем самым новые конформные инварианты погружений римановых поверхностей в трех- и четырехмерные евклидовы пространства.

Дано описание конечнозонных $\mathcal{PT}$-потенциалов в терминах явных тета-функциональных формул.

Излагаются основные физические и математические идеи (П. Кюри, Дарбу, Пуанкаре, Дирак), приведшие к понятию магнитного заряда, общая конструкция магнитных лапласианов для магнитных монополей на римановых многообразиях и результаты Ю. А. Кордюкова и автора по квазиклассическому приближению для собственных сечений этих операторов.

We discuss the mechanism of formation of singularities of solutions to the Novikov–Veselov, modified Novikov–Veselov, and Davey–Stewartson II (DSII) equations obtained by the Moutard type transformations. These equations admit the L, A, B-triple presentation, the generalization of the L, A-pairs for 2+1-soliton equations. We relate the blow-up of solutions to the non-conservation of the zero level of discrete spectrum of the L-operator. We also present a class of exact solutions, of the DSII system, which depend on two functional parameters, and show that all possible singularities of solutions to DSII equation obtained by the Moutard transformation are indeterminancies, i.e., points when approaching which in different spatial directions the solution has different limits.

Описаны связи уравнений Эйлера на центральных расширениях алгебр Ли с уравнениями Эйлера на исходных, расширяемых, алгебрах. Рассмотрена специальная бесконечная последовательность центральных расширений нильпотентных алгебр Ли, строящихся по алгебре Ли формальных векторных полей на прямой, и описаны орбиты коприсоединенных представлений для этих алгебр. С помощью компактных нильмногообразий, построенных по этим алгебрам И.К. Бабенко и автором, показано, что накрывающие группы Ли для симплектических нильмногообразий могут иметь любой ранг как разрешимые группы Ли.

It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already – an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy G_2 and Spin(7). In this article we provide a method of how to confirm that the famous Joyce examples of holonomy G_2 indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy Spin(7)). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kähler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly Kähler" naturally generalize compact Kähler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.

Spectral (Bloch) varieties of multidimensional differential operators on non-simply connected manifolds are defined. In their terms it is given a description of the analytical dependence of the spectra of magnetic Laplacians on non-simply connected manifolds on the values of the Aharonov-Bohm fluxes and a construction of analogues of spectral curves for two-dimensional Dirac operators on Riemann surfaces and, thereby, new conformal invariants of immersions of surfaces into 3- and 4-dimensional Euclidean spaces.

It is a prominent conjecture (relating Riemannian geom-etry and algebraic topology) that all simply-connected compact man-ifolds of special holonomy should be formal spaces, i.e., their rationalhomotopy type should be derivable from their rational cohomology al-gebra already - an as prominent as particular property in rational ho-motopy theory. Special interest now lies on exceptional holonomy G2and Spin(7).In this article we provide a method of how to confirm that the famousJoyce examples of holonomy G2indeed are formal spaces; we concretelyexert this computation for one example, which may serve as a blueprintfor the remaining Joyce examples (potentially also of holonomy Spin(7)).These considerations are preceded by another result identifying theformality of manifolds admitting special structures: we prove the for-mality of nearly K ahler manifolds.A connection between these two results can be found in the factthat both "special holonomy" and "nearly K ahler" naturally generalizecompact K ahler manifolds, whose formality is a classical and celebratedtheorem by Deligne-Griffiths-Morgan-Sullivan.

We consider the Schrödinger operator with regular short range complex-valued potential in dimension d≥ 1 . We show that, for d≥ 2 , the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of the potential (that is Hermiticity of Hamiltonian). In contrast, for d=1 , we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for d=3 . Some directions for further research are formulated.

We present the basic physical and mathematical ideas (P. Curie, Darboux, Poincare, Dirac) that led to the concept of magnetic charge, the general construction of magnetic Laplacians for magnetic monopoles on Riemannian manifolds, and the results of Yu.A. Kordyukov and the author on the quasi-classical approximation for the eigensections of these operators.

Using the generalization of the multidimensional WKB method to magnetic Laplacians corresponding to monopoles, which we proposed earlier, we obtain explicit formulas for quasi-classical approximations of eigenfunctions for the Dirac monopole.

Методами конечнозонного интегрирования построены ортогональные криволинейные системы координат в евклидовом пространстве, отвечающие пучкам ранга один без кручения над приводимыми сингулярными спектральными кривыми. Библиография: 10 названий.