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 - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Elliptic complexes of pseudodifferential operators on manifolds with edges N2 - On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof. T3 - Preprint - (1998) 14 KW - manifolds with singularities KW - pseudodifferential operators KW - elliptic complexes KW - Hodge theory Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25257 ER - TY - INPR A1 - Buchholz, Thilo A1 - Schulze, Bert-Wolfgang T1 - Volterra operators and parabolicity : anisotropic pseudo-differential operators N2 - Parabolic equations on manifolds with singularities require a new calculus of anisotropic pseudo-differential operators with operator-valued symbols. The paper develops this theory along the lines of sn abstract wedge calculus with strongly continuous groups of isomorphisms on the involved Banach spaces. The corresponding pseodo-diferential operators are continuous in anisotropic wedge Sobolev spaces, and they form an alegbra. There is then introduced the concept of anisotropic parameter-dependent ellipticity, based on an order reduction variant of the pseudo-differential calculus. The theory is appled to a class of parabolic differential operators, and it is proved the invertibility in Sobolev spaces with exponential weights at infinity in time direction. T3 - Preprint - (1998) 11 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25231 ER - TY - INPR A1 - Nacinovich, Mauro A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - On carleman formulas for the dolbeault cohomology N2 - We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C n with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology. T3 - Preprint - (1998) 10 KW - ∂-operator KW - cohomology KW - integral formulas Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25224 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Euler solutions of pseudodifferential equations N2 - We consider a homogeneous pseudodifferential equation on a cylinder C = IR x X over a smooth compact closed manifold X whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators over X. When assuming the symbol to be independent on the variable t element IR, we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points since C can be thought of as a 'stretched' manifold with conical points at t = -infinite and t = infinite. T3 - Preprint - (1998) 09 KW - pseudodifferential operator KW - meromorphic family KW - residue Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25211 ER - TY - INPR A1 - Paneah, Boris A1 - Schulze, Bert-Wolfgang T1 - On the existence of smooth solutions of the dirichlet problem for hyperbolic : differential equations T3 - Preprint - (1998) 05 Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25179 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A remark on the index of symmetric operators N2 - We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol. T3 - Preprint - (1998) 04 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25169 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The index of higher order operators on singular surfaces N2 - The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol. T3 - Preprint - (1998) 03 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25127 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Lefschetz fixed point formula in the relative elliptic theory N2 - A version of the classical Lefschetz fixed point formula is proved for the cohomology of the cone of a cochain mapping of elliptic complexes. As a particular case we show a Lefschetz formula for the relative de Rham cohomology. T3 - Preprint - (1998) 01 KW - elliptic complexes KW - relative cohomology KW - Lefschetz number Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25159 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris A1 - Shatalov, Victor T1 - Spectral boundary value problems and elliptic equations on singular manifolds N2 - For elliptic operators on manifolds with boundary, we define spectral boundary value problems, which generalize the Atiyah-Patodi-Singer problem to the case of nonhomogeneous boundary conditions, operators of arbitrary order, and nonself-adjoint conormal symbols. The Fredholm property is proved and equivalence with certain elliptic equations on manifolds with conical singularities is established. T3 - Preprint - (1997) 36 KW - APS problem KW - spectral resolution KW - nonhomogeneous boundary value problems KW - Fredholm property Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25147 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - On general boundary value problems for elliptic equations N2 - We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved. T3 - Preprint - (1997) 35 KW - elliptic boundary value problems KW - Atiyah-Bott condition KW - Calderón projections KW - Cauchy Riemann operator KW - Euler operator Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25138 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - On the index formula for singular surfaces N2 - In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators. T3 - Preprint - (1997) 31 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25116 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The index of elliptic operators on manifolds with conical points N2 - For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero. T3 - Preprint - (1997) 24 KW - manifolds with singularities KW - pseudodifferential operators KW - elliptic operators KW - index Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25096 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - Quantization of symplectic transformations on manifolds with conical singularities N2 - The structure of symplectic (canonical) transformations on manifolds with conical singularities is established. The operators associated with these transformations are defined in the weight spaces and their properties investigated. T3 - Preprint - (1997) 23 KW - manifolds with conical singularities KW - symplectic (canonical) transformations KW - quantization KW - Mellin transform Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25084 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - A Lefschetz fixed point theorem for manifolds with conical singularities N2 - We establish an Atiyah-Bott-Lefschetz formula for elliptic operators on manifolds with conical singular points. T3 - Preprint - (1997) 20 KW - elliptic operator KW - Fredholm property KW - conical singularities KW - pseudo-diferential operators KW - Lefschetz fixed point formula KW - regularizer Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25073 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - Operator algebras on singular manifolds. IV, V T3 - Preprint - (1997) 19 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25062 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The Riemann-Roch theorem for manifolds with conical singularities N2 - The classical Riemann-Roch theorem is extended to solutions of elliptic equations on manifolds with conical points. T3 - Preprint - (1997) 18 KW - manifolds with singularities KW - elliptic operators KW - divisors Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25051 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - Operator algebras on singular manifolds. I T3 - Preprint - (1997) 16 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25011 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - Nonstationary problems for equations of Borel-Fuchs type N2 - In the paper, the nonstationary problems for equations of Borel-Fuchs type are investigated. The asymptotic expansion are obtained for different orders of degeneration of operators in question. The approach to nonstationary problems based on the asymptotic theory on abstract algebras is worked out. T3 - Preprint - (1997) 11 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-24973 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - On the index of differential operators on manifolds with conical singularities N2 - The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point. T3 - Preprint - (1997) 10 KW - conical singularities KW - Mellin transform KW - pseudodiferential operators KW - ellipticity KW - Fredholm operators KW - regularizers KW - analytic index Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-24965 ER - TY - INPR A1 - Rabinovich, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A calculus of boundary value problems in domains with Non-Lipschitz Singular Points N2 - The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points. T3 - Preprint - (1997) 09 KW - pseudodifferential operators KW - boundary value problems KW - manifolds with cusps Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-24957 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Lefschetz theory on manifolds with edges : introduction N2 - The aim of this book is to develop the Lefschetz fixed point theory for elliptic complexes of pseudodifferential operators on manifolds with edges. The general Lefschetz theory contains the index theory as a special case, while the case to be studied is much more easier than the index problem. The main topics are: - The calculus of pseudodifferential operators on manifolds with edges, especially symbol structures (inner as well as edge symbols). - The concept of ellipticity, parametrix constructions, elliptic regularity in Sobolev spaces. - Hodge theory for elliptic complexes of pseudodifferential operators on manifolds with edges. - Development of the algebraic constructions for these complexes, such as homotopy, tensor products, duality. - A generalization of the fixed point formula of Atiyah and Bott for the case of simple fixed points. - Development of the fixed point formula also in the case of non-simple fixed points, provided that the complex consists of diferential operarators only. - Investigation of geometric complexes (such as, for instance, the de Rham complex and the Dolbeault complex). Results in this direction are desirable because of both purely mathe matical reasons and applications in natural sciences. N2 - Ziel des Buches ist es, die Lefschetz-Theorie der Fixpunkte für elliptische Komplexe von Pseudodifferentialoperatoren auf Mannigfaltigkeiten mit Kanten zu gewinnen. Die allgemeine Lefschetz-Theorie enthält die Index-Theorie als Spezialfall, aber der Fall, den wir analysieren werden, ist viel leichter als das Index-Problem. Ergebnisse in dieser Richtung sind wünschenswert, einerseits aus innermathematischen Gründen, aber auch im Hinblick auf Anwendungen in den Naturwissenschaften. T3 - Preprint - (1997) 08 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-24948 ER -