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 -