Michael Melgaard

Eigenvalue Asymptotics for Weakly Perturbed Dirac and Schrödinger Operators with Constant Magnetic Fields of Full Rank

Abstract The even-dimensional Dirac and Schrödinger operators with a constant magnetic field of full rank have purely essential spectrum consisting of isolated eigenvalues, so-called Landau levels. For a sign-definite electric potential Vwhich tends to zero at infinity, not too fast, it is known for the Schrödinger operator that the number of eigenvalues near each Landau level is infinite and their leading (quasi-classical) asymptotics depends on the rate of decay for V. We show, both for Schrödinger and Dirac operators, that, for anysign-definite, bounded Vwhich tends to zero at infinity, there still are an infinite number of eigenvalues near each Landau level. For compactly supported V, we establish the non-classicalformula, not depending on V, describing how the eigenvalues converge to the Landau levels asymptotically.

STIEFEL AND GRASSMANN MANIFOLDS IN QUANTUM CHEMISTRY

We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slater type variational spaces in many-particle Hartree–Fock theory and beyond. In particular, we prove that they are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on the existence of solutions to Hartree–Fock type equations.

Perturbation of eigenvalues embedded at a threshold

Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small off-diagonal perturbations. The results are based on given asymptotic expansions of the component Hamiltonians.

Schrödinger Operators with Singular Potentials

We describe classical and recent results on the spectral theory of Schrodinger and Pauli operators with singular electric and magnetic potentials.

Complex Absorbing Potential Method for the Perturbed Dirac Operator

The Complex Absorbing Potential (CAP) method is widely used to compute resonances in quantum chemistry, both for nonrelativistic and relativistic Hamiltonians. In the semiclassical limit ℏ → 0 we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudodifferential operator theory and microlocal analysis.

Spectral properties in the low-energy limit of one-dimensional Schrödinger operators H = –d2/dx2 + V. The Case 〈1, V1〉 ≠ 0

In this paper we consider the Schrodinger operator H = –d2/dx2+ V in L2(ℝ), where V satisfies an abstract short-range condition and the (solvability) condition 〈1, V1〉 = 0. Spectral properties of H in the low-energy limit are analyzed. Asymptotic expansions for R(ζ) = (H – ζ)–1 and the S-matrix S(λ) are deduced for ζ → 0 and λ ↓ 0, respectively. Depending on the zero-energy properties of H, the expansions of R(ζ) take different forms. Generically, the expansions of R(ζ) do not contain negative powers; the appearance of negative powers in ζ1/2 is due to the possible presence of zero-energy resonances (half-bound states) or the eigenvalue zero of H (bound state), or both. It is found that there are at most two zero resonances modulo L2-functions.

Threshold properties of matrix-valued Schrödinger operators

We present some results on the perturbation of eigenvalues embedded at a threshold for a two-channel Hamiltonian with three-dimensional Schrodinger operators as entries and with a small off-diagonal perturbation. In particular, we show how the threshold eigenvalue gives rise to discrete eigenvalues below the threshold and, moreover, we establish a criterion on existence of half-bound states associated with embedded pseudo eigenvalues.

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian Hm+Hom+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (Hm, Hom) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.

On bound states for systems of weakly coupled Schrödinger equations in one space dimension

We establish the Birman–Schwinger relation for a class of Schrodinger operators −d2/dx2⊗1H+V on L2(R,H), where H is an auxiliary Hilbert space and V is an operator-valued potential. As an application we give an asymptotic formula for the bound states which may arise for a weakly coupled Schrodinger operator with a matrix potential (having one or more thresholds). In addition, for a two-channel system with eigenvalues embedded in the continuous spectrum we show that, under a small perturbation, such eigenvalues turn into resonances.

On the maximal ionization for the atomic Pauli operator

Within the Born–Oppenheimer approximation Lieb proved that the number of non-relativistic, spin- particles that can be bound to an atom of nuclear charge Z in the presence of an external magnetic field satisfies Nmax<2Z+1, provided the magnetic field tends to zero at infinity and the coupling between the magnetic field and the spin is ignored. Assuming that the magnetic field is generic, we prove an upper bound which holds when the spin-field coupling is included; the set of generic magnetic fields contains an open, dense subset of .