Asymptotic parametrices of elliptic edge operators

Flad, Heinz-Jürgen; Harutyunyan, Gohar; Schulze, Bert-Wolfgang

doi:10.1007/s11868-016-0159-7

Cited by 4 publications

(11 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This actually represents the penultimate step in the asymptotic parametrix construction discussed in Ref. [11], cf. Corollary 2.24 and Theorem 2.26 therein.…”

Section: Resultsmentioning

confidence: 96%

“…Furthermore, it requires an appropriate notion of ellipticity, which implies existence of a parametrix for an elliptic operator. For the sake of a concise presentation, we will state in the following only some basic notations and refer to [11] as well as the monographs [39] for further details. A pseudo-differential operator P , in the edge-degenerate operator class L µ (Y × R 3 , g), for weight data g = (γ, γ − µ, Θ), γ, µ ∈ R, is given in the general form…”

Section: Pseudo-differential Operators On Manifolds With Edge Singulamentioning

confidence: 99%

Section: Pseudo-differential Operators On Manifolds With Edge Singula...mentioning

confidence: 99%

“…, see e.g. [11] for further details. Here, M µ O (X, R 3 ) is the set of all holomorphic Mellin symbols with values in L µ cl (X, R 3 ), the space of classical parameter-dependent pseudo differential operators on the base X.…”

Section: Pseudo-differential Operators On Manifolds With Edge Singula...mentioning

confidence: 99%

“…The basic idea is to construct an asymptotic parametrix for a Hamiltonian which encodes all the required asymptotic information. Details concerning the general concept of an asymptotic parametrix has been presented elsewhere [11], and a first application to the Hamiltonian of the hydrogen atom was given in [9]. In the present work, we want to establish an explicit construction of an asymptotic local parametrix for the Hamiltonian of two-electron systems, in particular the helium series and hydrogen molecule, near coalescence points of two particles, i.e., two electrons or an electron and a nucleus.…”

Section: Introduction 1singular Analysis Meets Quantum Chemistrymentioning

confidence: 99%

See 4 more Smart Citations

Explicit Green operators for quantum mechanical Hamiltonians. II. Edge-type singularities of the helium atom

Flad

Flad-Harutyunyan

Schulze

2019

Asian-European J. Math.

View full text Add to dashboard Cite

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 nonrelativisic Hamiltonian of the helium atom within the Born-Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behaviour of the eigenfunctons near an edge. 1 We do not consider spin degrees of freedom or equivalent permutational symmetries of the electron coordinates in our discussion.2 The manifold M \ M0 actually correponds to the mathematical notion of a configuration space of N ordered particles in R 3 . more generally, as the inner part of an open smooth manifold with boundary. Next, let us consider the subset M 1 ⊂ M 0 of all coalescence points of more than two particles. The stratum M 0 \ M 1 is an open smooth manifold representing edges of M. Correspondingly, we denote M \ M 1 as a singular manifold with edges. Higher order strata can be constructed along the same lines, e.g., let M 2 ⊂ M 1 denote the set of coalescence points of more than three particles. Again the stratum M 1 \M 2 is an open smooth manifold representing the lowest order type of corners in M. Therefore, M \ M 2 is a singular manifold with edges and corners. In this way the configuration space can be decomposed into its strata, i.e.,The singular operator calculus associates classes of degenerate differential operators to the singular manifolds M \ M i , i = 0, 1, . . . , and a corresponding hierarchy of symbols to the strata.Within the present work we want to study edge singularities corresponding to coalescence points of two particles, i.e., two electrons or an electron and a nucleus. Higher order corner sigularities are subject of our future work.Near an edge, M \ M 1 is identified with a wedgewith smooth base X, homeomorphic to S 2 , the unit sphere 3 , and edge Y . The class Diff µ deg (W) of edge-degenerate Fuchs-type differential operators of order µ ∈ Nis defined on the associated open stretched wedgeHere r ∈ R + denotes the distance to the open edge Y ⊂ R q and y is a q-dimensional variable, varying on Y (for a Coulomb system consisting of N electrons the dimension q of Y equals 3N − 3). The coefficients a jα (r, y) take values in differential operators of order µ − (j + |α|) on the base X of the cone and are smooth in the respective variables up to r = 0. On an open stretched wedge it is the distance variable r ∈ R + which carries the asymptotic information. In order to incorporate asymptotics into Sobolev spaces let us proceed in a recursive manner. Weighted Sobolev spaces K s,γ (X ∧ ) on an open stretched cone with base X are defined with respect to the corresponding polar coordinatesx → (r, x) viafor a cut-off function ω, i.e., ω ∈ C ∞ 0 (R + ) such that ω(r) = 1 near r = 0. Here H s,γ (X ∧ ) = r γ H s,0 (...

show abstract

“…This actually represents the penultimate step in the asymptotic parametrix construction discussed in Ref. [11], cf. Corollary 2.24 and Theorem 2.26 therein.…”

Section: Resultsmentioning

confidence: 96%

Section: Pseudo-differential Operators On Manifolds With Edge Singulamentioning

confidence: 99%

Section: Pseudo-differential Operators On Manifolds With Edge Singula...mentioning

confidence: 99%

Section: Pseudo-differential Operators On Manifolds With Edge Singula...mentioning

confidence: 99%

Section: Introduction 1singular Analysis Meets Quantum Chemistrymentioning

confidence: 99%

See 3 more Smart Citations