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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 -