@article{ChangKhalilSchulze2021,
  author    = {Chang, Der-Chen and Khalil, Sara and Schulze, Bert-Wolfgang},
  title     = {Analysis on regular corner spaces},
  series = {The journal of geometric analysis},
  volume    = {31},
  journal   = {The journal of geometric analysis},
  number    = {9},
  publisher = {Springer},
  address   = {New York},
  issn      = {1050-6926},
  doi       = {10.1007/s12220-021-00614-3},
  pages     = {9199 -- 9240},
  year      = {2021},
  abstract  = {We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.},
  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}
}
@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{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{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{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{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{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{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{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}
}