@inproceedings{RungrottheeraChangSchulze2020, author = {Rungrottheera, Wannarut and Chang, Der-Chen and Schulze, Bert-Wolfgang}, title = {The edge calculus of singularity order >3}, series = {Journal of nonlinear and convex analysis : an international journal}, volume = {21}, booktitle = {Journal of nonlinear and convex analysis : an international journal}, number = {2}, publisher = {Yokohama Publishers}, address = {Yokohama}, issn = {1345-4773}, pages = {387 -- 401}, year = {2020}, abstract = {We study Mellin pseudo-differential algebras on singular straight cones and manifolds with singularity of order >= 3. Those are necessary to express parametrices of elliptic differential operators with a corresponding cornerdegenerate behavior, and we obtain regularity in weighted spaces.}, language = {en} } @article{FladFladHarutyunyanSchulze2020, author = {Flad, Heinz-J{\"u}rgen and Flad-Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Explicit Green operators for quantum mechanical Hamiltonians}, series = {Asian-European journal of mathematics : AEJM}, volume = {13}, journal = {Asian-European journal of mathematics : AEJM}, number = {7}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557120501223}, pages = {64}, year = {2020}, abstract = {We extend our approach of asymptotic parametrix construction for Hamiltonian operators from conical to edge-type singularities which is applicable to coalescence points of two particles of the helium atom and related two electron systems including the hydrogen molecule. Up to second-order, we have calculated the symbols of an asymptotic parametrix of the nonrelativistic Hamiltonian of the helium atom within the Born-Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behavior of the eigenfunctions near an edge.}, language = {en} } @article{ChangHedayatMahmoudiSchulze2017, author = {Chang, Der-Chen and Hedayat Mahmoudi, Mahdi and Schulze, Bert-Wolfgang}, title = {Singular degenerate operators}, series = {Applicable analysis : an international journal}, volume = {96}, journal = {Applicable analysis : an international journal}, number = {14}, publisher = {Routledge, Taylor \& Francis Group}, address = {Abingdon}, issn = {0003-6811}, doi = {10.1080/00036811.2017.1336546}, pages = {2434 -- 2456}, year = {2017}, abstract = {We outline some simplified and more general method for constructing parametrices on higher singular spaces. We also outline basic ideas on operators on manifolds with conical or edge singularities.}, language = {en} } @article{HedayatMahmoudiSchulze2016, author = {Hedayat Mahmoudi, Mahdi and Schulze, Bert-Wolfgang}, title = {Corner boundary value problems}, series = {Asian-European journal of mathematics}, volume = {10}, journal = {Asian-European journal of mathematics}, number = {1}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557117500541}, pages = {45}, year = {2016}, abstract = {The paper develops some crucial steps in extending the first-order cone or edge calculus to higher singularity orders. We focus here on order 2, but the ideas are motivated by an iterative approach for higher singularities.}, language = {en} } @article{KhalilSchulze2017, author = {Khalil, Sara and Schulze, Bert-Wolfgang}, title = {Boundary problems on a manifold with edge}, series = {Asian-European Journal of Mathematics}, volume = {10}, journal = {Asian-European Journal of Mathematics}, number = {2}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557117500875}, pages = {43}, year = {2017}, abstract = {We establish a calculus of boundary value problems (BVPs) on a manifold N with boundary and edge, based on Boutet de Monvel's theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices.}, language = {en} } @article{ChangSchulze2018, author = {Chang, Der-Chen and Schulze, Bert-Wolfgang}, title = {Corner spaces and Mellin quantization}, series = {Journal of nonlinear and convex analysis : an international journal}, volume = {19}, journal = {Journal of nonlinear and convex analysis : an international journal}, number = {2}, publisher = {Yokohama Publishers}, address = {Yokohama}, issn = {1345-4773}, pages = {179 -- 195}, year = {2018}, abstract = {Manifolds with corners in the present investigation are non-smooth configurations - specific stratified spaces - with an incomplete metric such as cones, manifolds with edges, or corners of piecewise smooth domains in Euclidean space. We focus here on operators on such "corner manifolds" of singularity order <= 2, acting in weighted corner Sobolev spaces. The corresponding corner degenerate pseudo-differential operators are formulated via Mellin quantizations, and they also make sense on infinite singular cones.}, language = {en} } @article{RungrottheeraLyuSchulze2018, author = {Rungrottheera, Wannarut and Lyu, Xiaojing and Schulze, Bert-Wolfgang}, title = {Parameter-dependent edge calculus and corner parametrices}, series = {Journal of nonlinear and convex analysis : an international journal}, volume = {19}, journal = {Journal of nonlinear and convex analysis : an international journal}, number = {12}, publisher = {Yokohama Publishers}, address = {Yokohama}, issn = {1345-4773}, pages = {2021 -- 2051}, year = {2018}, abstract = {Let B be a compact manifold with smooth edge of dimension > 0. We study the interplay between parameter-dependent edge algebra algebra on B and operator families belonging to the corner calculus, and we characterize parametrices in the corner case.}, language = {en} } @article{HedayatMahmoudiSchulze2018, author = {Hedayat Mahmoudi, Mahdi and Schulze, Bert-Wolfgang}, title = {A new approach to the second order edge calculus}, series = {Journal of pseudo-differential operators and applications}, volume = {9}, journal = {Journal of pseudo-differential operators and applications}, number = {2}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-017-0191-2}, pages = {265 -- 300}, year = {2018}, abstract = {We establish essential steps of an iterative approach to operator algebras, ellipticity and Fredholm property on stratified spaces with singularities of second order. We cover, in particular, corner-degenerate differential operators. Our constructions are focused on the case where no additional conditions of trace and potential type are posed, but this case works well and will be considered in a forthcoming paper as a conclusion of the present calculus.}, language = {en} } @article{FladFladHarutyunyanSchulze2018, author = {Flad, Heinz-J{\"u}rgen and Flad-Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Ellipticity of the quantum mechanical Hamiltonians}, series = {Journal of pseudo-differential operators and applications}, volume = {9}, journal = {Journal of pseudo-differential operators and applications}, number = {3}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-017-0201-4}, pages = {451 -- 467}, year = {2018}, abstract = {In paper (Flad and Harutyunyan in Discrete Contin Dyn Syst 420-429, 2011) is shown that the Hamiltonian of the helium atom in the Born-Oppenheimer approximation, in the case if two particles coincide, is an edge-degenerate operator, which is elliptic in the corresponding edge calculus. The aim of this paper is an analogous investigation in the case if all three particles coincide. More precisely, we show that the Hamiltonian in the mentioned case is a corner-degenerate operator, which is elliptic as an operator in the corner analysis.}, language = {en} } @article{ChangMahmoudiSchulze2018, author = {Chang, Der-Chen and Mahmoudi, Mahdi Hedayat and Schulze, Bert-Wolfgang}, title = {Volterra operators in the edge-calculus}, series = {Analysis and Mathematical Physics}, volume = {8}, journal = {Analysis and Mathematical Physics}, number = {4}, publisher = {Springer}, address = {Basel}, issn = {1664-2368}, doi = {10.1007/s13324-018-0238-4}, pages = {551 -- 570}, year = {2018}, abstract = {We study the Volterra property of a class of anisotropic pseudo-differential operators on R x B for a manifold B with edge Y and time-variable t. This exposition belongs to a program for studying parabolicity in such a situation. In the present consideration we establish non-smoothing elements in a subalgebra with anisotropic operator-valued symbols of Mellin type with holomorphic symbols in the complex Mellin covariable from the cone theory, where the covariable t of t extends to symbolswith respect to t to the lower complex v half-plane. The resulting space ofVolterra operators enlarges an approach of Buchholz (Parabolische Pseudodifferentialoperatoren mit operatorwertigen Symbolen. Ph. D. thesis, Universitat Potsdam, 1996) by necessary elements to a new operator algebra containing Volterra parametrices under an appropriate condition of anisotropic ellipticity. Our approach avoids some difficulty in choosing Volterra quantizations in the edge case by generalizing specific achievements from the isotropic edge-calculus, obtained by Seiler (Pseudodifferential calculus on manifolds with non-compact edges, Ph. D. thesis, University of Potsdam, 1997), see also Gil et al. (in: Demuth et al (eds) Mathematical research, vol 100. Akademic Verlag, Berlin, pp 113-137, 1997; Osaka J Math 37: 221-260, 2000).}, language = {en} } @article{SchulzeSeiler2019, author = {Schulze, Bert-Wolfgang and Seiler, J{\"o}rg}, title = {Elliptic complexes on manifolds with boundary}, series = {The journal of geometric analysis}, volume = {29}, journal = {The journal of geometric analysis}, number = {1}, publisher = {Springer}, address = {New York}, issn = {1050-6926}, doi = {10.1007/s12220-018-0014-6}, pages = {656 -- 706}, year = {2019}, abstract = {We show that elliptic complexes of (pseudo) differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi, and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.}, language = {en} } @article{ChangSchulze2017, author = {Chang, Der-Chen and Schulze, Bert-Wolfgang}, title = {Ellipticity on spaces with higher singularities}, series = {Science China Mathematics}, volume = {60}, journal = {Science China Mathematics}, number = {11}, publisher = {Science China Press}, address = {Beijing}, issn = {1674-7283}, doi = {10.1007/s11425-016-0519-9}, pages = {2053 -- 2076}, year = {2017}, abstract = {We study corner-degenerate pseudo-differential operators of any singularity order and develop ellipticity based on the principal symbolic hierarchy, associated with the stratification of the underlying space. We construct parametrices within the calculus and discuss the aspect of additional trace and potential conditions along lower-dimensional strata.}, language = {en} } @article{KhalilSchulze2019, author = {Khalil, Sara and Schulze, Bert-Wolfgang}, title = {Calculus on a Manifold with Edge and Boundary}, series = {Complex analysis and operator theory}, volume = {13}, journal = {Complex analysis and operator theory}, number = {6}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-018-0800-y}, pages = {2627 -- 2670}, year = {2019}, abstract = {We study elements of the calculus of boundary value problems in a variant of Boutet de Monvel's algebra (Acta Math 126:11-51, 1971) on a manifold N with edge and boundary. If the boundary is empty then the approach corresponds to Schulze (Symposium on partial differential equations (Holzhau, 1988), BSB Teubner, Leipzig, 1989) and other papers from the subsequent development. For non-trivial boundary we study Mellin-edge quantizations and compositions within the structure in terms a new Mellin-edge quantization, compared with a more traditional technique. Similar structures in the closed case have been studied in Gil et al.}, language = {en} } @article{HarutyunyanSchulze2006, author = {Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {The Zaremba problem with singular interfaces as a corner boundary value problem}, series = {Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis}, volume = {25}, journal = {Potential analysis : an international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis}, publisher = {Springer}, address = {Dordrecht}, issn = {0926-2601}, doi = {10.1007/s11118-006-9020-6}, pages = {327 -- 369}, year = {2006}, abstract = {We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y, i.e., Au = f in int X, T (+/-) u = g(+/-) on int Y+/-, where Y is subdivided into subsets Y+/- with an interface Z and boundary conditions T+/- on Y+/- that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z subset of Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T- Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in Bull. Sci. Math. ( to appear). With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.}, language = {en} } @article{ChangSchulze2017, author = {Chang, D. -C. and Schulze, Bert-Wolfgang}, title = {Calculus on spaces with higher singularities}, series = {Journal of pseudo-differential operators and applications}, volume = {8}, journal = {Journal of pseudo-differential operators and applications}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-016-0180-x}, pages = {585 -- 622}, year = {2017}, abstract = {We establish extensions of the standard pseudo-differential calculus to specific classes of operators with operator-valued symbols occurring in symbolic hierarchies motivated by manifolds with higher singularities or stratified spaces.}, language = {en} } @article{ChangViahmoudiSchulze2016, author = {Chang, D. -C. and Viahmoudi, M. Hedayat and Schulze, Bert-Wolfgang}, title = {PSEUDO-DIFFERENTIAL ANALYSIS WITH TWISTED SYMBOLIC STRUCTURE}, series = {Journal of nonlinear and convex analysis : an international journal}, volume = {17}, journal = {Journal of nonlinear and convex analysis : an international journal}, publisher = {Yokohama Publishers}, address = {Yokohama}, issn = {1345-4773}, pages = {1889 -- 1937}, year = {2016}, abstract = {This paper is devoted to pseudo-differential operators and new applications. We establish necessary extensions of the standard calculus to specific classes of operator-valued symbols occurring in principal symbolic hierarchies of operators on manifolds with singularities or stratified spaces.}, language = {en} } @article{LyuSchulze2016, author = {Lyu, Xiaojing and Schulze, Bert-Wolfgang}, title = {Mellin Operators in the Edge Calculus}, series = {Complex analysis and operator theory}, volume = {10}, journal = {Complex analysis and operator theory}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-015-0511-6}, pages = {965 -- 1000}, year = {2016}, abstract = {A manifold M with smooth edge Y is locally near Y modelled on X-Delta x Omega for a cone X-Delta := ( (R) over bar (+) x X)/({0} x X) where Xis a smooth manifold and Omega subset of R-q an open set corresponding to a chart on Y. Compared with pseudo-differential algebras, based on other quantizations of edge-degenerate symbols, we extend the approach with Mellin representations on the r half-axis up to r = infinity, the conical exit of X-boolean AND = R+ x X (sic) (r, x) at infinity. The alternative description of the edge calculus is useful for pseudo-differential structures on manifolds with higher singularities.}, language = {en} } @article{FladHarutyunyanSchulze2016, author = {Flad, H. -J. and Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Asymptotic parametrices of elliptic edge operators}, series = {Journal of pseudo-differential operators and applications}, volume = {7}, journal = {Journal of pseudo-differential operators and applications}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-016-0159-7}, pages = {321 -- 363}, year = {2016}, abstract = {We study operators on singular manifolds, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. We introduce asymptotic parametrices, using tools from cone and edge pseudo-differential algebras. Our structures are motivated by models of many-particle physics with singular Coulomb potentials that contribute higher order singularities in Euclidean space, determined by the number of particles.}, language = {en} } @article{FladHarutyunyanSchulze2015, author = {Flad, Heinz-J{\"u}rgen and Harutyunyan, Gohar and Schulze, Bert-Wolfgang}, title = {Singular analysis and coupled cluster theory}, series = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, volume = {17}, journal = {Physical chemistry, chemical physics : a journal of European Chemical Societies}, number = {47}, publisher = {Royal Society of Chemistry}, address = {Cambridge}, issn = {1463-9076}, doi = {10.1039/c5cp01183c}, pages = {31530 -- 31541}, year = {2015}, abstract = {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.}, language = {en} } @article{MahmoudiSchulzeTepoyan2015, author = {Mahmoudi, Mahdi Hedayat and Schulze, Bert-Wolfgang and Tepoyan, Liparit}, title = {Continuous and variable branching asymptotics}, series = {Journal of pseudo-differential operators and applications}, volume = {6}, journal = {Journal of pseudo-differential operators and applications}, number = {1}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-015-0110-3}, pages = {69 -- 112}, year = {2015}, abstract = {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.}, language = {en} } @article{ChangQianSchulze2015, author = {Chang, Der-Chen and Qian, Tao and Schulze, Bert-Wolfgang}, title = {Corner Boundary Value Problems}, series = {Complex analysis and operator theory}, volume = {9}, journal = {Complex analysis and operator theory}, number = {5}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-014-0424-9}, pages = {1157 -- 1210}, year = {2015}, abstract = {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.}, language = {en} } @article{LyuQianSchulze2015, author = {Lyu, Xiaojing and Qian, Tao and Schulze, Bert-Wolfgang}, title = {Order filtrations of the edge algebra}, series = {Journal of pseudo-differential operators and applications}, volume = {6}, journal = {Journal of pseudo-differential operators and applications}, number = {3}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-015-0126-8}, pages = {279 -- 305}, year = {2015}, abstract = {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.}, language = {en} } @article{RungrottheeraSchulze2014, author = {Rungrottheera, Wannarut and Schulze, Bert-Wolfgang}, title = {Weighted spaces on corner manifolds}, series = {Complex variables and elliptic equations}, volume = {59}, journal = {Complex variables and elliptic equations}, number = {12}, publisher = {Routledge, Taylor \& Francis Group}, address = {Abingdon}, issn = {1747-6933}, doi = {10.1080/17476933.2013.876416}, pages = {1706 -- 1738}, year = {2014}, abstract = {We study spaces on manifolds with double weights and iterated discrete and continuous asymptotics, and their relationship with corner pseudo-differential operators.}, language = {en} } @article{ChangHabalSchulze2014, author = {Chang, Der-Chen and Habal, Nadia and Schulze, Bert-Wolfgang}, title = {The edge algebra structure of the Zaremba problem}, series = {Journal of pseudo-differential operators and applications}, volume = {5}, journal = {Journal of pseudo-differential operators and applications}, number = {1}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-013-0088-7}, pages = {69 -- 155}, year = {2014}, abstract = {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.}, language = {en} } @article{SchulzeWei2014, author = {Schulze, Bert-Wolfgang and Wei, Y.}, title = {The Mellin-edge quantisation for corner operators}, series = {Complex analysis and operator theory}, volume = {8}, journal = {Complex analysis and operator theory}, number = {4}, publisher = {Springer}, address = {Basel}, issn = {1661-8254}, doi = {10.1007/s11785-013-0289-3}, pages = {803 -- 841}, year = {2014}, abstract = {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.}, language = {en} } @article{RungrottheeraSchulzeWong2014, author = {Rungrottheera, Wannarut and Schulze, Bert-Wolfgang and Wong, M. W.}, title = {Iterative properties of pseudo-differential operators on edge spaces}, series = {Journal of pseudo-differential operators and applications}, volume = {5}, journal = {Journal of pseudo-differential operators and applications}, number = {4}, publisher = {Springer}, address = {Basel}, issn = {1662-9981}, doi = {10.1007/s11868-014-0100-x}, pages = {455 -- 479}, year = {2014}, abstract = {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.}, language = {en} } @article{FladHarutyunyanSchneideretal.2011, author = {Flad, Heinz-J{\"u}rgen and Harutyunyan, Gohar and Schneider, Reinhold and Schulze, Bert-Wolfgang}, title = {Explicit Green operators for quantum mechanical Hamiltonians}, series = {Manuscripta mathematica}, volume = {135}, journal = {Manuscripta mathematica}, number = {3-4}, publisher = {Springer}, address = {New York}, issn = {0025-2611}, doi = {10.1007/s00229-011-0429-x}, pages = {497 -- 519}, year = {2011}, abstract = {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.}, language = {en} } @article{SchulzeWei2009, author = {Schulze, Bert-Wolfgang and Wei, Ya-wei}, title = {Edge-boundary problems with singular trace conditions}, issn = {0232-704X}, doi = {10.1007/s10455-008-9143-7}, year = {2009}, abstract = {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.}, language = {en} } @article{Schulze2009, author = {Schulze, Bert-Wolfgang}, title = {On a paper of Krupchyk, Tarkhanov, and Tuomela}, issn = {0022-1236}, doi = {10.1016/j.jfa.2008.07.024}, year = {2009}, abstract = {We compare the above-mentioned article with the content of a previous publication}, language = {en} } @article{DinesLiuSchulze2009, author = {Dines, Nicoleta and Liu, Xiaochun and Schulze, Bert-Wolfgang}, title = {Edge quantisation of elliptic operators}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell}, issn = {1437-739X}, doi = {10.1007/s00605-008-0058-y}, year = {2009}, abstract = {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.}, language = {en} } @article{FedosovSchulzeTarkhanov2001, author = {Fedosov, Boris and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich}, title = {A general index formula on toric manifolds with conical point}, year = {2001}, language = {en} } @article{Schulze2001, author = {Schulze, Bert-Wolfgang}, title = {Operator algebras with symbol hierarchies on manifolds with singularities}, year = {2001}, language = {en} } @book{SchulzeDemuthAlbeverioetal.2001, author = {Schulze, Bert-Wolfgang and Demuth, Michael and Albeverio, Sergio and Schrohe, Elmar}, title = {Advances in Partial differential equations}, publisher = {Birkh{\"a}user}, address = {Basel}, year = {2001}, language = {en} } @book{AlbeverioDemuthSchroheetal.2002, author = {Albeverio, Sergio and Demuth, Michael and Schrohe, Elmar and Schulze, Bert-Wolfgang}, title = {Parabolicity, volterra calculus, and conical singularities : a volume of advances in partial differential equations}, series = {Operator theory : advances and applications}, volume = {138}, journal = {Operator theory : advances and applications}, publisher = {Birkh{\"a}user Verl.}, address = {Basel}, isbn = {3-7643-6906-x}, pages = {358 S.}, year = {2002}, language = {en} } @book{KapanadzeSchulze2003, author = {Kapanadze, David and Schulze, Bert-Wolfgang}, title = {Crack theory and edge singularities}, series = {Mathematics and its applications}, volume = {561}, journal = {Mathematics and its applications}, publisher = {Kluwer Acad. Publ}, address = {Dordrecht}, isbn = {1-4020-1524-0}, pages = {485 S.}, year = {2003}, language = {en} } @article{SchroheSchulze1994, author = {Schrohe, Elmar and Schulze, Bert-Wolfgang}, title = {Boundary value problems in Boutet de Monvel{\"i}s algebra for manifolds with conical singularities I}, year = {1994}, language = {en} } @article{Schulze1994, author = {Schulze, Bert-Wolfgang}, title = {The variable discrete asymptotics in pseudo-differential boundary value problems}, year = {1994}, language = {en} } @article{Schulze1994, author = {Schulze, Bert-Wolfgang}, title = {Pseudo-differential operators, ellipticity and asymptotics on manifolds with edges}, year = {1994}, language = {en} } @article{SchulzeSterninSatalov1994, author = {Schulze, Bert-Wolfgang and Sternin, Boris Ju. and Satalov, Viktor E.}, title = {Resurgent analysis and differential equations with singularities}, year = {1994}, language = {en} } @book{Schulze1994, author = {Schulze, Bert-Wolfgang}, title = {Pseudo-differential boundary value problems, conical singularities, and asymptotics}, series = {Mathematical topics}, volume = {4}, journal = {Mathematical topics}, publisher = {Akad.-Verl.}, address = {Berlin}, isbn = {3-05-501597-5}, pages = {580 S.}, year = {1994}, language = {en} } @article{DorschfeldtSchulze1994, author = {Dorschfeldt, Christoph and Schulze, Bert-Wolfgang}, title = {Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach}, year = {1994}, language = {en} } @article{SchulzeSterninSatalov1995, author = {Schulze, Bert-Wolfgang and Sternin, Boris Ju. and Satalov, Viktor E.}, title = {Resurgent analysis in the theory of differential equations with singularities}, year = {1995}, language = {en} } @article{Schulze1995, author = {Schulze, Bert-Wolfgang}, title = {The variable discrete asymptotics in pseudo-differential boundary value problems II}, year = {1995}, language = {en} } @article{SchroheSchulze1995, author = {Schrohe, Elmar and Schulze, Bert-Wolfgang}, title = {Mellin quantization in the cone calculus for Boutet de Monvel{\"i}s algebra}, year = {1995}, language = {en} } @article{Schulze1995, author = {Schulze, Bert-Wolfgang}, title = {Transmission algebras on singular spaces with components of different dimensions}, year = {1995}, language = {en} } @article{SchroheSchulze1995, author = {Schrohe, Elmar and Schulze, Bert-Wolfgang}, title = {Boundary value problems in Boutet de Monvel's algebra for manifolds with conical singularities II}, year = {1995}, language = {en} } @article{Schulze1995, author = {Schulze, Bert-Wolfgang}, title = {The variable discrete asymptotics in pseudo-differential boundary value problems II}, year = {1995}, language = {en} } @article{FedosovSchulze1996, author = {Fedosov, Boris V. and Schulze, Bert-Wolfgang}, title = {On the index elliptic operators on a cone}, year = {1996}, language = {en} } @article{SchulzeTarchanov1996, author = {Schulze, Bert-Wolfgang and Tarchanov, Nikolaj N.}, title = {Coordinate invariance of the wedge Sobolev spaces = Wedge Sobelev spaces}, year = {1996}, language = {en} } @article{DorschfeldtGriemeSchulze1997, author = {Dorschfeldt, Christoph and Grieme, Ulrich and Schulze, Bert-Wolfgang}, title = {Pseudo-differential calculus in the Fourieredge approach on non-compact manifolds}, year = {1997}, language = {en} }