Singular components of spectral measures for ergodic Jacobi matrices

For ergodic 1d Jacobi operators we prove that the random singular components of any spectral measure are almost surely mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's equation this yields the first rigorous proof of the Thouless' formula for the Lyapunov exponent in the dual regions.Comment: to appear in the Journal of Mathematical Physics, vol 52 (2011

Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two

We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential V(\x). We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i}$ at the high energy region. Second, the isoenergetic curves in the space of momenta \k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.Comment: 125 pages, 4 figures. arXiv admin note: incorporates arXiv:1205.118

Bound States at Threshold resulting from Coulomb Repulsion

The eigenvalue absorption for a many-particle Hamiltonian depending on a parameter is analyzed in the framework of non-relativistic quantum mechanics. The long-range part of pair potentials is assumed to be pure Coulomb and no restriction on the particle statistics is imposed. It is proved that if the lowest dissociation threshold corresponds to the decay into two likewise non-zero charged clusters then the bound state, which approaches the threshold, does not spread and eventually becomes the bound state at threshold. The obtained results have applications in atomic and nuclear physics. In particular, we prove that atomic ion with atomic critical charge $Z_{cr}$ and $N_e$ electrons has a bound state at threshold given that $Z_{cr} \in (N_e -2, N_e -1)$, whereby the electrons are treated as fermions and the mass of the nucleus is finite.Comment: This is a combined and updated version of the manuscripts arXiv:math-ph/0611075v2 and arXiv:math-ph/0610058v

Pauli-Fierz model with Kato-class potentials and exponential decays

Generalized Pauli-Fierz Hamiltonian with Kato-class potential \KPF in nonrelativistic quantum electrodynamics is defined and studied by a path measure. \KPF is defined as the self-adjoint generator of a strongly continuous one-parameter symmetric semigroup and it is shown that its bound states spatially exponentially decay pointwise and the ground state is unique.Comment: We deleted Lemma 3.1 in vol.

On Singularity formation for the L^2-critical Boson star equation

We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\partial_t u = \sqrt{-\Delta+m^2} \, u - (|x|^{-1} \ast |u|^2) u$ in 3 space dimensions. Under the sole assumption that the solution blows up in $H^{1/2}$ at finite time, we show that $u(t)$ has a unique weak limit in $L^2$ and that $|u(t)|^2$ has a unique weak limit in the sense of measures. Moreover, we prove that the limiting measure exhibits minimal mass concentration. A central ingredient used in the proof is a "finite speed of propagation" property, which puts a strong rigidity on the blowup behavior of $u$. As the second main result, we prove that any radial finite-time blowup solution $u$ converges strongly in $L^2$ away from the origin. For radial solutions, this result establishes a large data blowup conjecture for the $L^2$-critical boson star equation, similar to a conjecture which was originally formulated by F. Merle and P. Raphael for the $L^2$-critical nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704]. We also discuss some extensions of our results to other $L^2$-critical theories of gravitational collapse, in particular to critical Hartree-type equations.Comment: 24 pages. Accepted in Nonlinearit

On the Geometry of Supersymmetric Quantum Mechanical Systems

We consider some simple examples of supersymmetric quantum mechanical systems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.Comment: 18 page

Extended States for Polyharmonic Operators with Quasi-periodic Potentials in Dimension Two

We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential V(\x). We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i}$ at the high energy region. Second, the isoenergetic curves in the space of momenta \k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.Comment: This is an announcement only. Text with the detailed proof is under preparation. 11 pages, 4 figures. arXiv admin note: text overlap with arXiv:math-ph/0601008, arXiv:0711.4404, arXiv:1008.463

Existence of the Stark-Wannier quantum resonances

In this paper we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.Comment: 30 pages, 1 figur

Shape invariance through Crum transformation

We show in a rigorous way that Crum's result on equal eigenvalue spectrum of Sturm-Liouville problems can be obtained iteratively by successive Darboux transformations. It can be shown that all neighbouring Darboux-transformed potentials of higher order, u_{k} and u_{k+1}, satisfy the condition of shape invariance provided the original potential u does. We use this result to proof that under the condition of shape invariance the n-th iteration of the original Sturm-Liouville problem defined through shape invariance is equal to the n-th Crum transformationComment: 26 pp, one more reference, J.-M. Sparenberg and D. Baye, J. Phys. A 28, 5079 (1995), has been added as Ref. 18 in the published version, which has 47 ref

Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field

Based on the Mehler heat kernel of the Schroedinger operator for a free electron in a constant magnetic field an estimate for the kernel of E_A is derived, where E_A represents the kinetic energy of a Dirac electron within the pseudorelativistic no-pair Brown-Ravenhall model. This estimate is used to provide the bottom of the essential spectrum for the two-particle Brown-Ravenhall operator, describing the motion of the electrons in a central Coulomb field and a constant magnetic field, if the central charge is restricted to Z below or equal 86