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 - 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 - 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 - 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 -