TY - INPR A1 - Kapanadze, David A1 - Schulze, Bert-Wolfgang T1 - Boundary value problems on manifolds with exits to infinity N2 - We construct a new calculus of boundary value problems with the transmission property on a non-compact smooth manifold with boundary and conical exits to infinity. The symbols are classical both in covariables and variables. The operators are determined by principal symbol tuples modulo operators of lower orders and weights (such remainders are compact in weighted Sobolev spaces). We develop the concept of ellipticity, construct parametrices within the algebra and obtain the Fredholm property. For the existence of Shapiro-Lopatinskij elliptic boundary conditions to a given elliptic operator we prove an analogue of the Atiyah-Bott condition. T3 - Preprint - (2000) 06 KW - pseudo-differentialboundary value problems KW - elliptic operators on non-compact manifolds KW - Atiyah-Bott condition Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25727 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Asymptotics of solutions to elliptic equatons on manifolds with corners N2 - We show an explicit link between the nature of a singular point and behaviour of the coefficients of the equation, under which formal asymptotic expansions are still available. T3 - Preprint - (2000) 05 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25716 ER - TY - INPR A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Elliptic operators in subspaces N2 - We construct elliptic theory in the subspaces, determined by pseudodifferential projections. The finiteness theorem as well as index formula are obtained for elliptic operators acting in the subspaces. Topological (K-theoretic) aspects of the theory are studied in detail. T3 - Preprint - (2000) 04 KW - pseudodifferential subspaces KW - elliptic operators in subspaces KW - Fredholm property KW - index KW - K-theory KW - problem of classification Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25701 ER - TY - INPR A1 - Kapanadze, David A1 - Schulze, Bert-Wolfgang A1 - Witt, Ingo T1 - Coordinate invariance of the cone algebra with asymptotics N2 - The cone algebra with discrete asymptotics on a manifold with conical singularities is shown to be invariant under natural coordinate changes, where the symbol structure (i.e., the Fuchsian interior symbol, conormal symbols of all orders) follows a corresponding transformation rule. T3 - Preprint - (2000) 01 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25671 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - Operator algebras with symbol hierarchies on manifolds with singularities N2 - Problems for elliptic partial differential equations on manifolds M with singularities M' (here with piece-wise smooth geometry)are studied in terms of pseudo-differential algebras with hierarchies of symbols that consist of scalar and operator-valued components. Classical boundary value problems (with or without the transmission property) belong to the examples. They are a model for operator algebras on manifolds M with higher "polyhedral" singularities. The operators are block matrices that have upper left corners containing the pseudo-differential operators on the regular M\M' (plus certain Mellin and Green summands) and are degenerate (in streched coordinates) in a typical way near M'. By definition M' is again a manifold with singularities. The same is true of M'', and so on. The block matrices consist of trace, potential and Mellin and Green operators, acting between weighted Sobolev spaces on M(j) and M(k), with 0 ≤ j, k ≤ ord M; here M(0) denotes M, M(1) denotes M', etc. We generate these algebras, including their symbol hierarchies, by iterating so-called "edgifications" and "conifications" os algebras that have already been constructed, and we study ellipicity, parametrics and Fredholm property within these algebras. T3 - Preprint - (1999) 30 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25647 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - An algebra of boundary value problems not requiring Shapiro-Lopatinskil conditions N2 - We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro-Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces. T3 - Preprint - (1999) 24 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25596 ER - TY - INPR A1 - Nazaikinskii, Vladimir E. A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization methods in differential equations : Chapter 2: Quantization of Lagrangian modules N2 - In this chapter we use the wave packet transform described in Chapter 1 to quantize extended classical states represented by so-called Lagrangian sumbanifolds of the phase space. Functions on a Lagrangian manifold form a module over the ring of classical Hamiltonian functions on the phase space (with respect to pointwise multiplication). The quantization procedure intertwines this multiplication with the action of the corresponding quantum Hamiltonians; hence we speak of quantization of Lagrangian modules. The semiclassical states obtained by this quantization procedure provide asymptotic solutions to differential equations with a small parameter. Locally, such solutions can be represented by WKB elements. Global solutions are given by Maslov's canonical operator [2]; also see, e.g., [3] and the references therein. Here the canonical operator is obtained in the framework of the universal quantization procedure provided by the wave packet transform. This procedure was suggested in [4] (see also the references there) and further developed in [5]; our exposition is in the spirit of these papers. Some further bibliographical remarks can be found in the beginning of Chapter 1. T3 - Preprint - (1999) 22 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25582 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Nazaikinskii, Vladimir E. A1 - Sternin, Boris T1 - On the homotopy classification of elliptic operators on manifolds with singularities N2 - We study the homotopy classification of elliptic operators on manifolds with singularities and establish necessary and sufficient conditions under which the classification splits into terms corresponding to the principal symbol and the conormal symbol. T3 - Preprint - (1999) 21 KW - elliptic operators KW - homotopy classification KW - manifold with singularities KW - Atiyah-Singer theorem Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25574 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Savin, Anton T1 - The homotopy classification and the index of boundary value problems for general elliptic operators N2 - We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition. T3 - Preprint - (1999) 20 KW - elliptic boundary value problems KW - Atiyah-Bott condition KW - index theory KW - K-theory KW - homotopy classification Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25568 ER - TY - INPR A1 - Rabinovich, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Boundary value problems in domains with corners N2 - We describe Fredholm boundary value problems for differential equations in domains with intersecting cuspidal edges on the boundary. T3 - Preprint - (1999) 19 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25552 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A general index formula on tropic manifolds with conical points N2 - We solve the index problem for general elliptic pseudodifferential operators on toric manifolds with conical points. T3 - Preprint - (1999) 15 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25501 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Savin, Anton A1 - Sternin, Boris T1 - Elliptic operators in subspaces and the eta invariant N2 - The paper deals with the calculation of the fractional part of the η-invariant for elliptic self-adjoint operators in topological terms. The method used to obtain the corresponding formula is based on the index theorem for elliptic operators in subspaces obtained in [1], [2]. It also utilizes K-theory with coefficients Zsub(n). In particular, it is shown that the group K(T*M,Zsub(n)) is realized by elliptic operators (symbols) acting in appropriate subspaces. T3 - Preprint - (1999) 14 KW - index of elliptic operators in subspaces KW - K-theory KW - eta-invariant KW - mod k index KW - Atiyah-Patodi-Singer theory Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25496 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Regularisation of mixed boundary problems N2 - We show an application of the spectral theorem in constructing approximate solutions of mixed boundary value problems for elliptic equations. T3 - Preprint - (1999) 09 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25454 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization and the wave packet transform T3 - Preprint - (1999) 08 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25447 ER - TY - INPR A1 - Schrohe, Elmar A1 - Schulze, Bert-Wolfgang T1 - Edge-degenerate boundary value problems on cones N2 - We consider edge-degenerate families of pseudodifferential boundary value problems on a semi-infinite cylinder and study the behavior of their push-forwards as the cylinder is blown up to a cone near infinity. We show that the transformed symbols belong to a particularly convenient symbol class. This result has applications in the Fredholm theory of boundary value problems on manifolds with edges. T3 - Preprint - (1999) 06 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25436 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Ellipticity and parametrices on manifolds with caspidal edges T3 - Preprint - (1999) 04 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25411 ER - TY - INPR A1 - Rabinovich, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Boundary value problems in cuspidal wedges N2 - The paper is devoted to pseudodifferential boundary value problems in domains with cuspidal wedges. Concerning the geometry we even admit a more general behaviour, namely oscillating cuspidal wedges. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to edges. T3 - Preprint - (1998) 24 KW - pseudodifferential operators KW - boundary value problems KW - manifolds with edges Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25363 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - A semiclassical quantization on manifolds with singularities and the Lefschetz Formula for Elliptic Operators N2 - For general endomorphisms of elliptic complexes on manifolds with conical singularities, the semiclassical asymptotics of the Atiyah-Bott-Lefschetz number is calculated in terms of fixed points of the corresponding canonical transformation of the symplectic space. T3 - Preprint - (1998) 19 KW - elliptic operator KW - Fredholm property KW - conical singularities KW - pseudodiferential operators KW - Lefschetz fixed point formula KW - regularizer Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25296 ER - TY - INPR A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - On the invariant index formulas for spectral boundary value problems N2 - In the paper we study the possibility to represent the index formula for spectral boundary value problems as a sum of two terms, the first one being homotopy invariant of the principal symbol, while the second depends on the conormal symbol of the problem only. The answer is given in analytical, as well as in topological terms. T3 - Preprint - (1998) 18 KW - spectral boundary value problems KW - Fredholm property KW - index of elliptic operator KW - Chern character KW - Atiyah-Patodi-Singer theory Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25285 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - The index of quantized contact transformations on manifolds with conical singularities N2 - The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator. T3 - Preprint - (1998) 16 KW - manifolds with conical singularities KW - contact transformations KW - quantization KW - ellipticity KW - Fredholm operators KW - regularizers KW - index formulas Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25276 ER -