TY - JOUR A1 - Flad, Heinz-Jürgen A1 - Flad-Harutyunyan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Explicit Green operators for quantum mechanical Hamiltonians BT - II. edge-type singularities of the helium atom JF - Asian-European journal of mathematics : AEJM N2 - 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. KW - Singular analysis KW - Schrodinger equation KW - many-electron systems KW - asymptotic properties of eigenfunctions Y1 - 2020 U6 - https://doi.org/10.1142/S1793557120501223 SN - 1793-5571 SN - 1793-7183 VL - 13 IS - 7 PB - World Scientific CY - Singapore ER - TY - CHAP A1 - Rungrottheera, Wannarut A1 - Chang, Der-Chen A1 - Schulze, Bert-Wolfgang T1 - The edge calculus of singularity order >3 T2 - Journal of nonlinear and convex analysis : an international journal N2 - 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. KW - Pseudo-differential algebras KW - symbols KW - singular manifolds KW - Mellin KW - operator calculus Y1 - 2020 SN - 1345-4773 SN - 1880-5221 VL - 21 IS - 2 SP - 387 EP - 401 PB - Yokohama Publishers CY - Yokohama ER - TY - JOUR A1 - Khalil, Sara A1 - Schulze, Bert-Wolfgang T1 - Calculus on a Manifold with Edge and Boundary JF - Complex analysis and operator theory N2 - 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. KW - algebra KW - Mellin quantization Y1 - 2019 U6 - https://doi.org/10.1007/s11785-018-0800-y SN - 1661-8254 SN - 1661-8262 VL - 13 IS - 6 SP - 2627 EP - 2670 PB - Springer CY - Basel ER - TY - JOUR A1 - Schulze, Bert-Wolfgang A1 - Seiler, Jörg T1 - Elliptic complexes on manifolds with boundary JF - The journal of geometric analysis N2 - 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. KW - Elliptic complexes KW - Manifolds with boundary KW - Atiyah-Bott obstruction KW - Toeplitz-type pseudodifferential operators Y1 - 2018 U6 - https://doi.org/10.1007/s12220-018-0014-6 SN - 1050-6926 SN - 1559-002X VL - 29 IS - 1 SP - 656 EP - 706 PB - Springer CY - New York ER - TY - JOUR A1 - Chang, Der-Chen A1 - Mahmoudi, Mahdi Hedayat A1 - Schulze, Bert-Wolfgang T1 - Volterra operators in the edge-calculus JF - Analysis and Mathematical Physics N2 - 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). KW - Volterra operator KW - Anisotropic pseudo-differential operators KW - Edge calculus KW - Operator-valued symbols of Mellin type Y1 - 2018 U6 - https://doi.org/10.1007/s13324-018-0238-4 SN - 1664-2368 SN - 1664-235X VL - 8 IS - 4 SP - 551 EP - 570 PB - Springer CY - Basel ER - TY - JOUR A1 - Flad, Heinz-Jürgen A1 - Flad-Harutyunyan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Ellipticity of the quantum mechanical Hamiltonians BT - corner singularity of the helium atom JF - Journal of pseudo-differential operators and applications N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1007/s11868-017-0201-4 SN - 1662-9981 SN - 1662-999X VL - 9 IS - 3 SP - 451 EP - 467 PB - Springer CY - Basel ER - TY - JOUR A1 - Hedayat Mahmoudi, Mahdi A1 - Schulze, Bert-Wolfgang T1 - A new approach to the second order edge calculus JF - Journal of pseudo-differential operators and applications N2 - 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. KW - Operators on singular manifolds KW - Mellin transform KW - Stratified spaces KW - Ellipticity and parametrices Y1 - 2018 U6 - https://doi.org/10.1007/s11868-017-0191-2 SN - 1662-9981 SN - 1662-999X VL - 9 IS - 2 SP - 265 EP - 300 PB - Springer CY - Basel ER - TY - JOUR A1 - Chang, Der-Chen A1 - Schulze, Bert-Wolfgang T1 - Corner spaces and Mellin quantization JF - Journal of nonlinear and convex analysis : an international journal N2 - 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. KW - Mellin quantizations KW - operator-valued symbols KW - weighted edge and corner spaces Y1 - 2018 SN - 1345-4773 SN - 1880-5221 VL - 19 IS - 2 SP - 179 EP - 195 PB - Yokohama Publishers CY - Yokohama ER - TY - JOUR A1 - Rungrottheera, Wannarut A1 - Lyu, Xiaojing A1 - Schulze, Bert-Wolfgang T1 - Parameter-dependent edge calculus and corner parametrices JF - Journal of nonlinear and convex analysis : an international journal N2 - 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. KW - Edge calculus KW - corner parametrices Y1 - 2018 SN - 1345-4773 SN - 1880-5221 VL - 19 IS - 12 SP - 2021 EP - 2051 PB - Yokohama Publishers CY - Yokohama ER - TY - JOUR A1 - Chang, Der-Chen A1 - Schulze, Bert-Wolfgang T1 - Ellipticity on spaces with higher singularities JF - Science China Mathematics N2 - 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. KW - pseudo-differential operators KW - operator-valued symbols KW - Fourier and Mellin transforms Y1 - 2017 U6 - https://doi.org/10.1007/s11425-016-0519-9 SN - 1674-7283 SN - 1869-1862 VL - 60 IS - 11 SP - 2053 EP - 2076 PB - Science China Press CY - Beijing ER - TY - JOUR A1 - Khalil, Sara A1 - Schulze, Bert-Wolfgang T1 - Boundary problems on a manifold with edge JF - Asian-European Journal of Mathematics N2 - 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. KW - manifolds with edge and boundary KW - distribution with asymptotics KW - ellipticity KW - Fredholm property Y1 - 2017 U6 - https://doi.org/10.1142/S1793557117500875 SN - 1793-5571 SN - 1793-7183 VL - 10 IS - 2 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Chang, Der-Chen A1 - Hedayat Mahmoudi, Mahdi A1 - Schulze, Bert-Wolfgang T1 - Singular degenerate operators JF - Applicable analysis : an international journal N2 - 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. KW - Operators on singular cones KW - Mellin symbols with values in the edge calculus KW - parametrices of elliptic operators Y1 - 2017 U6 - https://doi.org/10.1080/00036811.2017.1336546 SN - 0003-6811 SN - 1563-504X VL - 96 IS - 14 SP - 2434 EP - 2456 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Chang, D. -C. A1 - Schulze, Bert-Wolfgang T1 - Calculus on spaces with higher singularities JF - Journal of pseudo-differential operators and applications N2 - 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. KW - Pseudo-differential operators KW - Operator-valued symbols KW - Fourier and Mellin transform Y1 - 2017 U6 - https://doi.org/10.1007/s11868-016-0180-x SN - 1662-9981 SN - 1662-999X VL - 8 SP - 585 EP - 622 PB - Springer CY - Basel ER - TY - JOUR A1 - Hedayat Mahmoudi, Mahdi A1 - Schulze, Bert-Wolfgang T1 - Corner boundary value problems JF - Asian-European journal of mathematics N2 - 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. KW - Mellin operators KW - Mellin oscillatory integrals KW - exit calculus KW - weighted Sobolev spaces Y1 - 2016 U6 - https://doi.org/10.1142/S1793557117500541 SN - 1793-5571 SN - 1793-7183 VL - 10 IS - 1 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Flad, H. -J. A1 - Harutyunyan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Asymptotic parametrices of elliptic edge operators JF - Journal of pseudo-differential operators and applications N2 - 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. KW - Cone and edge pseudo-differential operators KW - Ellipticity of edge-degenerate operators KW - Meromorphic operator-valued symbols KW - Asymptotics of solutions Y1 - 2016 U6 - https://doi.org/10.1007/s11868-016-0159-7 SN - 1662-9981 SN - 1662-999X VL - 7 SP - 321 EP - 363 PB - Springer CY - Basel ER - TY - JOUR A1 - Lyu, Xiaojing A1 - Schulze, Bert-Wolfgang T1 - Mellin Operators in the Edge Calculus JF - Complex analysis and operator theory N2 - 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. KW - Edge degenerate operators KW - Mellin and Green operators edge symbols Y1 - 2016 U6 - https://doi.org/10.1007/s11785-015-0511-6 SN - 1661-8254 SN - 1661-8262 VL - 10 SP - 965 EP - 1000 PB - Springer CY - Basel ER - TY - JOUR A1 - Chang, D. -C. A1 - Viahmoudi, M. Hedayat A1 - Schulze, Bert-Wolfgang T1 - PSEUDO-DIFFERENTIAL ANALYSIS WITH TWISTED SYMBOLIC STRUCTURE JF - Journal of nonlinear and convex analysis : an international journal N2 - 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. KW - Pseudo-differential operators KW - boundary value problems KW - operator valued symbols KW - Fourier transform Y1 - 2016 SN - 1345-4773 SN - 1880-5221 VL - 17 SP - 1889 EP - 1937 PB - Yokohama Publishers CY - Yokohama ER - TY - GEN A1 - Flad, Heinz-Jürgen A1 - Harutyunyan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Singular analysis and coupled cluster theory 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 shortrange correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 302 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102306 SP - 31530 EP - 31541 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 - https://doi.org/10.1007/s11785-014-0424-9 SN - 1661-8254 SN - 1661-8262 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 - https://doi.org/10.1007/s11868-015-0126-8 SN - 1662-9981 SN - 1662-999X VL - 6 IS - 3 SP - 279 EP - 305 PB - Springer CY - Basel ER - 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 - https://doi.org/10.1039/c5cp01183c SN - 1463-9076 SN - 1463-9084 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 - https://doi.org/10.1007/s11868-015-0110-3 SN - 1662-9981 SN - 1662-999X VL - 6 IS - 1 SP - 69 EP - 112 PB - Springer CY - Basel ER - TY - JOUR A1 - Rungrottheera, Wannarut 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 - https://doi.org/10.1080/17476933.2013.876416 SN - 1747-6933 SN - 1747-6941 VL - 59 IS - 12 SP - 1706 EP - 1738 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Rungrottheera, Wannarut 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 - https://doi.org/10.1007/s11868-014-0100-x SN - 1662-9981 SN - 1662-999X VL - 5 IS - 4 SP - 455 EP - 479 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 - https://doi.org/10.1007/s11785-013-0289-3 SN - 1661-8254 SN - 1661-8262 VL - 8 IS - 4 SP - 803 EP - 841 PB - Springer CY - Basel 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 - https://doi.org/10.1007/s11868-013-0088-7 SN - 1662-9981 SN - 1662-999X VL - 5 IS - 1 SP - 69 EP - 155 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 BT - 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 - https://doi.org/10.1007/s00229-011-0429-x SN - 0025-2611 VL - 135 IS - 3-4 SP - 497 EP - 519 PB - Springer CY - New York ER - TY - INPR A1 - Hovhannisyan, A. H. A1 - Schulze, Bert-Wolfgang T1 - On a method for solution of the ordinary differential equations connected with Huygens' equations T3 - Preprint - (2010) 01 Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-45381 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 - https://doi.org/10.1007/s00605-008-0058-y SN - 1437-739X 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 - https://doi.org/10.1016/j.jfa.2008.07.024 SN - 0022-1236 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 - https://doi.org/10.1007/s10455-008-9143-7 SN - 0232-704X ER - TY - INPR A1 - Ma, L. A1 - Schulze, Bert-Wolfgang T1 - Operators on manifolds with conical singularities N2 - We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges. T3 - Preprint - (2009) 07 KW - Operators on manifolds with conical singularities KW - conormal symbols KW - ellipticity of cone operators Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-36608 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - Boundary value problems with the transmission property N2 - We give a survey on the calculus of (pseudo-differential) boundary value problems with the transmision property at the boundary, and ellipticity in the Shapiro-Lopatinskij sense. Apart from the original results of the work of Boutet de Monvel we present an approach based on the ideas of the edge calculus. In a final section we introduce symbols with the anti-transmission property. T3 - Preprint - (2009) 03 Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30377 ER - TY - INPR A1 - Abed, Jamil A1 - Schulze, Bert-Wolfgang T1 - Edge-degenerate families of ΨDO’s on an infinite cylinder N2 - We establish a parameter-dependent pseudo-differential calculus on an infinite cylinder, regarded as a manifold with conical exits to infinity. The parameters are involved in edge-degenerate form, and we formulate the operators in terms of operator-valued amplitude functions. T3 - Preprint - (2009) 01 KW - Edge-degenerate operators KW - parameter-dependent pseudodifferential operators KW - norm estimates with respect to a parameter Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30365 ER - TY - BOOK A1 - Abed, Jamil A1 - Schulze, Bert-Wolfgang T1 - Operators with Corner-degenerate Symbols T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2008 SN - 1437-739X PB - Univ. CY - Potsdam ER - TY - BOOK A1 - Schulze, Bert-Wolfgang A1 - Wei, Ya-wei T1 - Edge-boundary problems with singular trace conditions T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2008 SN - 1437-739X PB - Univ. CY - Potsdam ER - TY - BOOK A1 - Schulze, Bert-Wolfgang T1 - On a paper of Krupchyk, Tarkhanov and Tuomela T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2008 SN - 1437-739X PB - Univ. CY - Potsdam ER - TY - INPR A1 - Abed, Jamil A1 - Schulze, Bert-Wolfgang T1 - Operators with corner-degenerate symbols N2 - We establish elements of a new approch to ellipticity and parametrices within operator algebras on a manifold with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaces. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The “full” calculus is voluminous; so we content ourselves here with some typical aspects such as symbols in terms of order reducing families, classes of relevant examples, and operators near the conical exit to infinity. T3 - Preprint - (2008) 01 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30299 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Wei, Y. 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 (σψ; σ∂), 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 σ∧, 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 [6] on closed' manifolds with edge. Here we concentrate on the phenomena in combination with boundary conditions and edge problem. T3 - Preprint - (2008) 04 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30317 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - The iterative structure of corner operators N2 - We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27 - 31, 2008, at the MSRI, University of Berkeley. T3 - Preprint - (2008) 08 KW - Categories of stratified spaces KW - ellipticity of corners operators KW - principal symbolic hierarchies KW - boundary value problems Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30353 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - On a paper of Krupchyk, Tarkhanov, and Tuomela T3 - Preprint - (2008) 05 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30325 ER - TY - BOOK A1 - Flad, H.-J. A1 - Schneider, R. A1 - Schulze, Bert-Wolfgang T1 - Asymptotic Regularity of Solutions of Hartree-Fock Equations with Coulomb Potential T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2007 SN - 1437-739X PB - Univ. CY - Potsdam ER - TY - INPR A1 - Flad, Heinz-Jürgen A1 - Schneider, Reinhold A1 - Schulze, Bert-Wolfgang T1 - Asymptotic regularity of solutions of Hartree-Fock equations with coulomb potential N2 - We study the asymptotic regularity of solutions of Hartree-Fock equations for Coulomb systems. In order to deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo-differential operators on conical manifolds. First, the non-self-consistent-field case is considered which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator which guarantee ellipticity for the local part of the Fock operator on the open stretched cone R+ × S². This proves existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed-point approach based on Cances and Le Bris analysis of the level-shifting algorithm, we show via another bootstrap argument, that the corresponding self-consistent-field solutions of the Hartree-Fock equation have the same type of asymptotic regularity. T3 - Preprint - (2007) 05 Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30268 ER - TY - BOOK A1 - Kapanadze, David A1 - Schulze, Bert-Wolfgang A1 - Seiler, Jörg T1 - Operators with singular trace conditions on a manifold with edges T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2006 SN - 1437-739X PB - Univ. CY - Potsdam ER - TY - BOOK A1 - Schulze, Bert-Wolfgang T1 - The structure of operators on manifolds with polyhedral singularities T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2006 SN - 1437-739X PB - Univ. CY - Potsdam ER - TY - BOOK A1 - Schulze, Bert-Wolfgang T1 - Elliptic differential operators on Manifolds with Edges T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2006 SN - 1437-739X PB - Univ. CY - Potsdam ER - TY - JOUR A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - The relative index for corner singularities N2 - We study pseudo-differential operators on a cylinder R x B where B has conical singularities. Configurations of that kind are the local model of corner singularities with cross section B. Operators in our calculus are assumed to have symbols a which are meromorphic in the complex covariable with values in the algebra of all cone operators on B. We show an explicit formula for solutions of the homogeneous equation if a is independent of the axial variable t is an element of R. Each non-bijectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula Y1 - 2006 UR - http://www.springerlink.com/content/300422 U6 - https://doi.org/10.1007/s00020-005-1367-3 SN - 0378-620X ER - TY - JOUR A1 - Schulze, Bert-Wolfgang A1 - Seiler, Jörg T1 - Edge operators with conditions of Toeplitz type N2 - Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of 2 X 2-block matrix operators with trace and potential operators on the edges. The picture is similar to the pseudodifferential analysis of boundary-value problems. The extra conditions satisfy an analogue of the Shapiro-Lopatinskij condition, provided a topological obstruction for the elliptic edge-degenerate operator in the upper left corner vanishes; this is an analogue of a condition of Atiyah and Bott in boundary-value problems. In general, however, we need global projection data, similarly to global boundary conditions, known for Dirac operators or other geometric operators. The present paper develops a new calculus with global projection data for operators on manifolds with edges. In particular, we show the Fredholm property in a suitable scale of spaces and construct parametrices within the calculus Y1 - 2006 ER - TY - JOUR A1 - Coriasco, Sandro A1 - Schulze, Bert-Wolfgang T1 - Edge problems on configurations with model cones of different dimensions N2 - Elliptic equations on configurations W = W-1 boolean OR (. . .) boolean OR W-N with edge Y and components W-j of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here edges. Starting from edge-degenerate operators on Wj, j = 1, . . . , N, we construct an algebra with extra 'transmission' conditions on Y that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator- valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on WY. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics Y1 - 2006 UR - http://projecteuclid.org/Dienst/UI/1.0/Journal?authority=euclid.ojm SN - 0030-6126 ER - TY - BOOK A1 - Schulze, Bert-Wolfgang T1 - Pseudo-differential calculus on Manifolds with geometric singularities T3 - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell Y1 - 2006 SN - 1437-739X PB - Univ. CY - Potsdam ER -