TY - JOUR
A1 - Flad, Heinz-Jürgen
A1 - Harutyunyan, Gohar
A1 - Schulze, Bert-Wolfgang
T1 - Singular analysis and coupled cluster theory
JF - Physical chemistry, chemical physics : a journal of European Chemical Societies
N2 - The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the asymptotic behaviour of wavefunctions near these singularities. In the present work, we want to introduce the mathematical framework of singular analysis and discuss a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems. This corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator. The Green operator encodes essential asymptotic information and we present as our main result an explicit asymptotic formula for this operator. First applications to many-particle models in quantum chemistry are presented in order to demonstrate the feasibility of our approach. The focus is on the asymptotic behaviour of ladder diagrams, which provide the dominant contribution to short-range correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation.
Y1 - 2015
U6 - http://dx.doi.org/10.1039/c5cp01183c
SN - 1463-9076 (print)
SN - 1463-9084 (online)
VL - 17
IS - 47
SP - 31530
EP - 31541
PB - Royal Society of Chemistry
CY - Cambridge
ER -
TY - JOUR
A1 - Mahmoudi, Mahdi Hedayat
A1 - Schulze, Bert-Wolfgang
A1 - Tepoyan, Liparit
T1 - Continuous and variable branching asymptotics
JF - Journal of pseudo-differential operators and applications
N2 - The regularity of solutions to elliptic equations on a manifold with singularities, say, an edge, can be formulated in terms of asymptotics in the distance variable r > 0 to the singularity. In simplest form such asymptotics turn to a meromorphic behaviour under applying the Mellin transform on the half-axis. Poles, multiplicity, and Laurent coefficients form a system of asymptotic data which depend on the specific operator. Moreover, these data may depend on the variable y along the edge. We then have y-dependent families of meromorphic functions with variable poles, jumping multiplicities and a discontinuous dependence of Laurent coefficients on y. We study here basic phenomena connected with such variable branching asymptotics, formulated in terms of variable continuous asymptotics with a y-wise discrete behaviour.
KW - Asymptotics of solutions
KW - Weighted edge spaces
KW - Edge symbols
Y1 - 2015
U6 - http://dx.doi.org/10.1007/s11868-015-0110-3
SN - 1662-9981 (print)
SN - 1662-999X (online)
VL - 6
IS - 1
SP - 69
EP - 112
PB - Springer
CY - Basel
ER -
TY - JOUR
A1 - Chang, Der-Chen
A1 - Qian, Tao
A1 - Schulze, Bert-Wolfgang
T1 - Corner Boundary Value Problems
JF - Complex analysis and operator theory
N2 - Boundary value problems on a manifold with smooth boundary are closely related to the edge calculus where the boundary plays the role of an edge. The problem of expressing parametrices of Shapiro-Lopatinskij elliptic boundary value problems for differential operators gives rise to pseudo-differential operators with the transmission property at the boundary. However, there are interesting pseudo-differential operators without the transmission property, for instance, the Dirichlet-to-Neumann operator. In this case the symbols become edge-degenerate under a suitable quantisation, cf. Chang et al. (J Pseudo-Differ Oper Appl 5(2014):69-155, 2014). If the boundary itself has singularities, e.g., conical points or edges, then the symbols are corner-degenerate. In the present paper we study elements of the corresponding corner pseudo-differential calculus.
KW - Corner pseudo-differential operators
KW - Ellipticity of corner-degenerate operators
KW - Meromorphic operator-valued symbols
Y1 - 2015
U6 - http://dx.doi.org/10.1007/s11785-014-0424-9
SN - 1661-8254 (print)
SN - 1661-8262 (online)
VL - 9
IS - 5
SP - 1157
EP - 1210
PB - Springer
CY - Basel
ER -
TY - JOUR
A1 - Lyu, Xiaojing
A1 - Qian, Tao
A1 - Schulze, Bert-Wolfgang
T1 - Order filtrations of the edge algebra
JF - Journal of pseudo-differential operators and applications
N2 - By edge algebra we understand a pseudo-differential calculus on a manifold with edge. The operators have a two-component principal symbolic hierarchy which determines operators up to lower order terms. Those belong to a filtration of the corresponding operator spaces. We give a new characterisation of this structure, based on an alternative representation of edge amplitude functions only containing holomorphic edge-degenerate Mellin symbols.
Y1 - 2015
U6 - http://dx.doi.org/10.1007/s11868-015-0126-8
SN - 1662-9981 (print)
SN - 1662-999X (online)
VL - 6
IS - 3
SP - 279
EP - 305
PB - Springer
CY - Basel
ER -
TY - JOUR
A1 - Rungrottheera, W.
A1 - Schulze, Bert-Wolfgang
T1 - Weighted spaces on corner manifolds
JF - Complex variables and elliptic equations
N2 - We study spaces on manifolds with double weights and iterated discrete and continuous asymptotics, and their relationship with corner pseudo-differential operators.
KW - manifolds with corners
KW - iterated asymptotics
KW - operators with corner symbols
KW - 35J70
KW - 47G30
KW - 58J40
Y1 - 2014
U6 - http://dx.doi.org/10.1080/17476933.2013.876416
SN - 1747-6933 (print)
SN - 1747-6941 (online)
VL - 59
IS - 12
SP - 1706
EP - 1738
PB - Routledge, Taylor & Francis Group
CY - Abingdon
ER -
TY - JOUR
A1 - Chang, Der-Chen
A1 - Habal, Nadia
A1 - Schulze, Bert-Wolfgang
T1 - The edge algebra structure of the Zaremba problem
JF - Journal of pseudo-differential operators and applications
N2 - We study mixed boundary value problems, here mainly of Zaremba type for the Laplacian within an edge algebra of boundary value problems. The edge here is the interface of the jump from the Dirichlet to the Neumann condition. In contrast to earlier descriptions of mixed problems within such an edge calculus, cf. (Harutjunjan and Schulze, Elliptic mixed, transmission and singular crack problems, 2008), we focus on new Mellin edge quantisations of the Dirichlet-to-Neumann operator on the Neumann side of the boundary and employ a pseudo-differential calculus of corresponding boundary value problems without the transmission property at the interface. This allows us to construct parametrices for the original mixed problem in a new and transparent way.
Y1 - 2014
U6 - http://dx.doi.org/10.1007/s11868-013-0088-7
SN - 1662-9981 (print)
SN - 1662-999X (online)
VL - 5
IS - 1
SP - 69
EP - 155
PB - Springer
CY - Basel
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Wei, Y.
T1 - The Mellin-edge quantisation for corner operators
JF - Complex analysis and operator theory
N2 - We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold with second order singularities. The typical ingredients come from the "most singular" stratum of which is a second order edge where the infinite transversal cone has a base that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over . In this respect our result is formally analogous to a quantisation rule of (Osaka J. Math. 37:221-260, 2000) for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over that tend to infinity.
Y1 - 2014
U6 - http://dx.doi.org/10.1007/s11785-013-0289-3
SN - 1661-8254 (print)
SN - 1661-8262 (online)
VL - 8
IS - 4
SP - 803
EP - 841
PB - Springer
CY - Basel
ER -
TY - JOUR
A1 - Rungrottheera, W.
A1 - Schulze, Bert-Wolfgang
A1 - Wong, M. W.
T1 - Iterative properties of pseudo-differential operators on edge spaces
JF - Journal of pseudo-differential operators and applications
N2 - Pseudo-differential operators with twisted symbolic estimates play a large role in the calculus on manifolds with edge singularities. We study here aspects of the underlying abstract concept and establish a new result on iteration of quantizations.
KW - Pseudo-differential operators
KW - Twisted symbolic estimates
KW - Quantizations
Y1 - 2014
U6 - http://dx.doi.org/10.1007/s11868-014-0100-x
SN - 1662-9981 (print)
SN - 1662-999X (online)
VL - 5
IS - 4
SP - 455
EP - 479
PB - Springer
CY - Basel
ER -
TY - JOUR
A1 - Flad, Heinz-Jürgen
A1 - Harutyunyan, Gohar
A1 - Schneider, Reinhold
A1 - Schulze, Bert-Wolfgang
T1 - Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom
JF - Manuscripta mathematica
N2 - We study a new approach to determine the asymptotic behaviour of quantum many-particle systems near coalescence points of particles which interact via singular Coulomb potentials. This problem is of fundamental interest in electronic structure theory in order to establish accurate and efficient models for numerical simulations. Within our approach, coalescence points of particles are treated as embedded geometric singularities in the configuration space of electrons. Based on a general singular pseudo-differential calculus, we provide a recursive scheme for the calculation of the parametrix and corresponding Green operator of a nonrelativistic Hamiltonian. In our singular calculus, the Green operator encodes all the asymptotic information of the eigenfunctions. Explicit calculations and an asymptotic representation for the Green operator of the hydrogen atom and isoelectronic ions are presented.
Y1 - 2011
U6 - http://dx.doi.org/10.1007/s00229-011-0429-x
SN - 0025-2611 (print)
VL - 135
IS - 3-4
SP - 497
EP - 519
PB - Springer
CY - New York
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Wei, Ya-wei
T1 - Edge-boundary problems with singular trace conditions
N2 - The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure (sigma(psi), sigma(partial derivative)), consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary, we have a third symbolic component, namely, the edge symbol sigma(boolean AND), referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions 'in integral form' there may exist singular trace conditions, investigated in Kapanadze et al., Internal Equations and Operator Theory, 61, 241-279, 2008 on 'closed' manifolds with edge. Here, we concentrate on the phenomena in combination with boundary conditions and edge problem.
Y1 - 2009
UR - http://www.springerlink.com/content/100233
U6 - http://dx.doi.org/10.1007/s10455-008-9143-7
SN - 0232-704X
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - On a paper of Krupchyk, Tarkhanov, and Tuomela
N2 - We compare the above-mentioned article with the content of a previous publication
Y1 - 2009
UR - http://www.sciencedirect.com/science/journal/00221236
U6 - http://dx.doi.org/10.1016/j.jfa.2008.07.024
SN - 0022-1236
ER -
TY - JOUR
A1 - Dines, Nicoleta
A1 - Liu, Xiaochun
A1 - Schulze, Bert-Wolfgang
T1 - Edge quantisation of elliptic operators
JF - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell
N2 - The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy sigma = (sigma(psi), sigma(boolean AND)), where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol sigma(boolean AND) which includes the (in general not explicity known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems.
Y1 - 2009
UR - http://www.springerlink.com/content/103082
U6 - http://dx.doi.org/10.1007/s00605-008-0058-y
SN - 1437-739X
ER -
TY - JOUR
A1 - Fedosov, Boris
A1 - Schulze, Bert-Wolfgang
A1 - Tarkhanov, Nikolai Nikolaevich
T1 - A general index formula on toric manifolds with conical point
Y1 - 2001
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - Operator algebras with symbol hierarchies on manifolds with singularities
Y1 - 2001
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
A1 - Demuth, Michael
A1 - Albeverio, Sergio
A1 - Schrohe, Elmar
T1 - Advances in Partial differential equations
Y1 - 2001
PB - Birkhäuser
CY - Basel
ER -
TY - BOOK
A1 - Albeverio, Sergio
A1 - Demuth, Michael
A1 - Schrohe, Elmar
A1 - Schulze, Bert-Wolfgang
T1 - Parabolicity, volterra calculus, and conical singularities : a volume of advances in partial differential equations
T3 - Operator theory : advances and applications
Y1 - 2002
SN - 3-7643-6906-x
VL - 138
PB - Birkhäuser Verl.
CY - Basel
ER -
TY - BOOK
A1 - Kapanadze, David
A1 - Schulze, Bert-Wolfgang
T1 - Crack theory and edge singularities
T3 - Mathematics and its applications
Y1 - 2003
SN - 1-4020-1524-0
VL - 561
PB - Kluwer Acad. Publ
CY - Dordrecht
ER -
TY - JOUR
A1 - Schrohe, Elmar
A1 - Schulze, Bert-Wolfgang
T1 - Boundary value problems in Boutet de Monvelïs algebra for manifolds with conical singularities I
Y1 - 1994
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - The variable discrete asymptotics in pseudo-differential boundary value problems
Y1 - 1994
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - Pseudo-differential operators, ellipticity and asymptotics on manifolds with edges
Y1 - 1994
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - Resurgent analysis and differential equations with singularities
Y1 - 1994
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
T1 - Pseudo-differential boundary value problems, conical singularities, and asymptotics
T3 - Mathematical topics
Y1 - 1994
SN - 3-05-501597-5
VL - 4
PB - Akad.-Verl.
CY - Berlin
ER -
TY - JOUR
A1 - Dorschfeldt, Christoph
A1 - Schulze, Bert-Wolfgang
T1 - Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach
Y1 - 1994
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - Resurgent analysis in the theory of differential equations with singularities
Y1 - 1995
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - The variable discrete asymptotics in pseudo-differential boundary value problems II
Y1 - 1995
ER -
TY - JOUR
A1 - Schrohe, Elmar
A1 - Schulze, Bert-Wolfgang
T1 - Mellin quantization in the cone calculus for Boutet de Monvelïs algebra
Y1 - 1995
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - Transmission algebras on singular spaces with components of different dimensions
Y1 - 1995
ER -
TY - JOUR
A1 - Schrohe, Elmar
A1 - Schulze, Bert-Wolfgang
T1 - Boundary value problems in Boutet de Monvel's algebra for manifolds with conical singularities II
Y1 - 1995
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - The variable discrete asymptotics in pseudo-differential boundary value problems II
Y1 - 1995
ER -
TY - JOUR
A1 - Fedosov, Boris V.
A1 - Schulze, Bert-Wolfgang
T1 - On the index elliptic operators on a cone
Y1 - 1996
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Tarchanov, Nikolaj N.
T1 - Coordinate invariance of the wedge Sobolev spaces = Wedge Sobelev spaces
BT - Wedge Sobolev spaces
Y1 - 1996
ER -
TY - JOUR
A1 - Dorschfeldt, Christoph
A1 - Grieme, Ulrich
A1 - Schulze, Bert-Wolfgang
T1 - Pseudo-differential calculus in the Fourieredge approach on non-compact manifolds
Y1 - 1997
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
T1 - Boundary value problems and edges pseudo-differential operators
Y1 - 1997
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - Nonstationary problems for Borel-Fuchs type equations
Y1 - 1997
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - Operator algebras associated with resurgent transforms and differential equations on manifolds with singularities
Y1 - 1997
ER -
TY - JOUR
A1 - Gil, J. B.
A1 - Schulze, Bert-Wolfgang
A1 - Seiler, Jörg
T1 - Holomorphic operator-valued symbols for edgedegenerate pseudo-differential operators
Y1 - 1997
ER -
TY - JOUR
A1 - Buchholz, Thilo
A1 - Schulze, Bert-Wolfgang
T1 - Anisotropic edges pseudo-differential operators withdiscrete asymptotics
Y1 - 1997
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - Structure rings of singularities and differential equations
Y1 - 1997
ER -
TY - JOUR
A1 - Schulze, Bert-Wolfgang
A1 - Tarchanov, Nikolaj N.
T1 - The index of elliptic operators on manifolds with cups
Y1 - 1997
ER -
TY - BOOK
A1 - Egorov, Jurij V.
A1 - Schulze, Bert-Wolfgang
T1 - Pseudo-differential operators, singularities, applicatons
T3 - Operator theory
Y1 - 1997
SN - 3-7643-5484-4
VL - 93
PB - Birkhäuser
CY - Basel
ER -
TY - JOUR
A1 - Schrohe, Elmar
A1 - Schulze, Bert-Wolfgang
T1 - A symbol algebra for pseudodifferential boundary value problems on manifolds with edges
Y1 - 1997
ER -
TY - BOOK
A1 - Fedosov, Boris V.
A1 - Schulze, Bert-Wolfgang
A1 - Tarchanov, Nikolaj N.
T1 - The index of elliptic operators on manifolds with conical points
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 24
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Fedosov, Boris V.
A1 - Schulze, Bert-Wolfgang
A1 - Tarchanov, Nikolaj N.
T1 - On the index formula for singular surfaces
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 31
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
A1 - Tarchanov, Nikolaj N.
T1 - The Rieman-Roch theorem for manifolds with conical singularities
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 18
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - On general boundary value problems for elliptic equations
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 35
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
A1 - Schrohe, Elmar
T1 - Arbeitsgruppe "Partielle Differentialgleichungen und Komplexe Analysis" (seit 1992)
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 12
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
A1 - Tarchanov, Nikolaj N.
T1 - Lefschetz theory on manifolds with edges : introduction
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 08
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - On the index of differential operators on manifolds with conical singularities
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 10
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Schulze, Bert-Wolfgang
A1 - Sternin, Boris Ju.
A1 - Satalov, Viktor E.
T1 - Nonstationary problems for equations of Borel-Fuchs type
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 11
PB - Univ.
CY - Potsdam
ER -
TY - BOOK
A1 - Rabinovich, Vladimir
A1 - Schulze, Bert-Wolfgang
A1 - Tarchanov, Nikolaj N.
T1 - A calculus of boundary value problems in domains with Non-Lipschitz singular points
T3 - Preprint / Universität Potsdam, Institut für Mathematik
Y1 - 1997
VL - 1997, 09
PB - Univ.
CY - Potsdam
ER -