Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

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

doi:10.1007/s00229-011-0429-x

Cited by 13 publications

(14 citation statements)

References 20 publications

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“…Remark 6.23. It is important to notice that finding good compactifications of X related to the N -body problem is useful for the problem of approximating numerically the eigenvalues and eigenfunctions of N -body Hamiltonians [1,16,17,18,19,50]. In particular, this gives a further justification for trying to find the structure of the character space of E(X).…”

Section: )mentioning

confidence: 99%

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

Georgescu

Nistor

2017

J. Operator Theory

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ABSTRACT. We determine the essential spectrum of Hamiltonians with N -body type interactions that have radial limits at infinity. This extends the HVZ-theorem, which treats perturbations of the Laplacian by potentials that tend to zero at infinity. Our proof involves C * -algebra techniques that allows one to treat large classes of operators with local singularities and general behavior at infinity. In our case, the configuration space of the system is a finite dimensional, real vector space X, and we consider the algebra E(X) of functions on X generated by functions of the form v • π Y , where Y runs over the set of all linear subspaces of X, π Y is the projection of X onto the quotient X/Y , and v : X/Y → C is a continuous function that has uniform radial limits at infinity. The group X acts by translations on E(X), and hence the crossed product E (X) := E(X) ⋊ X is well defined; the Hamiltonians that are of interest to us are the self-adjoint operators affiliated to it. We determine the characters of E(X). This then allows us to describe the quotient of E (X) with respect to the ideal of compact operators, which in turn gives a formula for the essential spectrum of any self-adjoint operator affiliated to E (X).

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Section: )mentioning

confidence: 99%

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

Georgescu

Nistor

2017

J. Operator Theory

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“…[47,Theorem 6.8.1]) that u is analytic on R 3N \ S. In this case a strong local regularity result was obtained in [24] in the neighborhood of the simple coalescence points, where it was shown that locally u(x) = u 1 (x) + |l(x)|u 2 (x) with u 1 and u 2 real analytic and l linear. See also [8,10,11,16,20,22,21,23,25,36,38,51,54,55,56] and references therein for more results on the regularity of the eigenfunctions of Schrödinger operators. Related is [19] which was circulated after this article has been submitted.…”

Section: Introductionmentioning

confidence: 99%

Regularity for Eigenfunctions of Schrödinger Operators

Ammann

Carvalho²,

Nistor

2012

Lett Math Phys

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We prove a regularity result in weighted Sobolev (or Babuška-Kondratiev) spaces for the eigenfunctions of certain Schrödinger-type operators. Our results apply, in particular, to a non-relativistic Schrödinger operator of an N -electron atom in the fixed nucleus approximation. More precisely, let K m a (R 3N , r S ) be the weighted Sobolev space obtained by blowing up the set of singular points of the potentiala for all sm ∈ Z + and all a ≤ 0. Our result extends to the case when b j and c ij are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a < 3/2. Contents 1. Introduction 1 2. Differential structure of blow-ups 5 3. Lie structure at infinity 12 4. Regularity of eigenfunctions 24 Appendix A. b-tangent bundle and partial b-structure on [M : X] 28 References 29

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“…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: Singular Analysis Meets Quantum Chemistrymentioning

confidence: 99%

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

Flad

Flad-Harutyunyan

Schulze

2019

Asian-European J. Math.

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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 (...

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Explicit Green operators for quantum mechanical Hamiltonians. I. The hydrogen atom

Cited by 13 publications

References 20 publications

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

Regularity for Eigenfunctions of Schrödinger Operators

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

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