@unpublished{KrainerSchulze2001,
  author    = {Krainer, Thomas and Schulze, Bert-Wolfgang},
  title     = {On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder [Part 1: Chapter 1+2]},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25987},
  year      = {2001},
  abstract  = {We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.},
  language  = {en}
}
@unpublished{KrainerSchulze2001,
  author    = {Krainer, Thomas and Schulze, Bert-Wolfgang},
  title     = {On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder [Part 2: Chapter 3-5]},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25992},
  year      = {2001},
  abstract  = {We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.},
  language  = {en}
}
@unpublished{KrainerSchulze2001,
  author    = {Krainer, Thomas and Schulze, Bert-Wolfgang},
  title     = {On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder [Part 3: Chapter 6+7]},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26000},
  year      = {2001},
  abstract  = {We consider general parabolic systems of equations on the infinite time interval in case of the underlying spatial configuration is a closed manifold. The solvability of equations is studied both with respect to time and spatial variables in exponentially weighted anisotropic Sobolev spaces, and existence and maximal regularity statements for parabolic equations are proved. Moreover, we analyze the long-time behaiour of solutions in terms of complete asymptotic expansions. These results are deduced from a pseudodifferential calculus that we construct explicitly. This algebra of operators is specifically designed to contain both the classical systems of parabolic equations of general form and their inverses, parabolicity being reflected purely on symbolic level. To this end, we assign t = ∞ the meaning of an anisotropic conical point, and prove that this interprtation is consistent with the natural setting in the analysis of parabolic PDE. Hence, major parts of this work consist of the construction of an appropriate anisotropiccone calculus of so-called Volterra operators. In particular, which is the most important aspect, we obtain the complete characterization of the microlocal and the global kernel structure of the inverse of parabolicsystems in an infinite space-time cylinder. Moreover, we obtain perturbation results for parabolic equations from the investigation of the ideal structure of the calculus.},
  language  = {en}
}
@unpublished{Schulze2001,
  author    = {Schulze, Bert-Wolfgang},
  title     = {Operators with symbol hierarchies and iterated asymptotics},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25948},
  year      = {2001},
  abstract  = {Contents: Introduction 1 Edge calculus with parameters 1.1 Cone asymptotics and Green symbols 1.2 Mellin edge symbols 1.3 The edge symbol algebra 1.4 Operators on a manifold with edges 2 Corner symbols and iterated asymptotics 2.1 Holomorphic corner symbols 2.2 Meromorphic corner symbols and ellipicity 2.3 Weighted corner Sobolev spaces 2.4 Iterated asymptotics 3 The edge corner algebra with trace and potential conditions 3.1 Green corner operators 3.2 Smoothing Mellin corner operators 3.3 The edge corner algebra 3.4 Ellipicity and regularity with asymptotics 3.5 Examples and remarks},
  language  = {en}
}
@unpublished{KapanadzeSchulze2001,
  author    = {Kapanadze, David and Schulze, Bert-Wolfgang},
  title     = {Symbolic calculus for boundary value problems on manifolds with edges},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-26046},
  year      = {2001},
  abstract  = {Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.},
  language  = {en}
}
@unpublished{NazaikinskiiSchulzeSternin2000,
  author    = {Nazaikinskii, Vladimir and Schulze, Bert-Wolfgang and Sternin, Boris},
  title     = {Quantization methods in differential equations : Chapter 11: Noncommutative analysis and high-frequency asymptotics},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25857},
  year      = {2000},
  abstract  = {Content: Chapter 11: Noncommutative Analysis and High-Frequency Asymptotics 11.1 Statement of the Problem 11.2 Mixed Asymptotics: the General Scheme 11.3 The Asymptotic Solution of Main Problem 11.4 Analysis of the Asymptotic Solution},
  language  = {en}
}
@unpublished{KytmanovMyslivetsSchulzeetal.2001,
  author    = {Kytmanov, Aleksandr and Myslivets, Simona and Schulze, Bert-Wolfgang and Tarkhanov, Nikolai Nikolaevich},
  title     = {Elliptic problems for the Dolbeault complex},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-25979},
  year      = {2001},
  abstract  = {The inhomogeneous ∂-equations is an inexhaustible source of locally unsolvable equations, subelliptic estimates and other phenomena in partial differential equations. Loosely speaking, for the anaysis on complex manifolds with boundary nonelliptic problems are typical rather than elliptic ones. Using explicit integral representations we assign a Fredholm complex to the Dolbeault complex over an arbitrary bounded domain in C up(n).},
  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{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}
}