Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces - Jan Maas; van Neerven, Jan M (Journal of Functional Analysis)
Let (E, H, μ) be an abstract Wiener space and let DV : = V D, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space under(H, {combining low line}). Given a bounded operator B on under(H, {combining low line}), coercive on the range over(R (V), -), we consider the operators A : = V* B V in H and under(A, {combining low line}) : = V V* B in under(H, {combining low line}), as well as the realisations of the operators L : = DV* B DV and under(L, {combining low line}) : = DV DV* B in Lp (E, μ) and Lp (E, μ ; under(H, {combining low line})) respectively, where 1 < p < ∞. Our main result asserts that the following four assertions are equivalent: (1)D (sqrt(L)) = D (DV) with {norm of matrix} sqrt(L) f norm of matrixp {minus tilde} {norm of matrix} DV f norm of matrixp for f ∈ D (sqrt(L));(2)under(L, {combining low line}) admits a bounded H∞-functional calculus on over(R (DV), -);(3)D (sqrt(A)) = D (V) with {norm of matrix} sqrt(A) h {norm of matrix} {minus tilde} {norm of matrix} V h {norm of matrix} for h ∈ D (sqrt(A));(4)under(A, {combining low line}) admits a bounded H∞-functional calculus on over(R (V), -). Moreover, if these conditions are satisfied, then D (L) = D (DV2) ∩ D (DA). The equivalence (1)-(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where under(H, {combining low line}) = H, V = I, B = frac(1, 2) I). A one-sided version of (1)-(4), giving Lp-boundedness of the Riesz transform DV / sqrt(L) in terms of a square function estimate, is also obtained. As an application let -A generate an analytic C0-contraction semigroup on a Hilbert space H and let -L be the Lp-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an Lp-domain characterisation for the operator L.

Local atomic structure of niobates and titanates from X-ray absorption spectroscopic data - Vedrinskiǐ, Rostislav V; Kraǐzman, V. L; Mikhail Lemeshko; Nazarenko, Elena S; Novakovich, Alexander A; Reznichenko, Larisa A; Fokin, Vladimir N; Shuvaeva, Victoria A
The local atomic structure of PbTiO3, BaTiO3, and KNbO3 perovskite-type crystals and K x Na1 − x NbO3 solid solutions in different phases is investigated using the angular dependence of the pre-edge structure of the Ti and Nb K X-ray absorption spectra and the EXAFS data. In noncubic phases, a considerable deviation of the local structure from the structure determined from diffraction data is observed only for the tetragonal phase of the BaTiO3 crystal. It is revealed that, in the cubic phase of niobates, the niobium atoms are characterized by significant displacements from the centrosymmetric positions along the threefold axes, so that they are close in the magnitude and the direction to the displacements in the low-temperatures rhombohedral phases.

Rotational structure of weakly bound molecular ions - Mikhail Lemeshko; Frierich, Bretislav (Journal of Atomic and Molecular Sciences)
Relying on the quantization rule of Raab and Friedrich [Phys. Rev. A (2009) in press], we derive simple and accurate formulae for the number of rotational states supported by a weakly bound vibrational level of a diatomic molecular ion. We also provide analytic estimates of the rotational constants of any such levels up to threshold for dissociation and obtain a criterion for determining whether a given weakly bound vibrational level is rotationless. The results depend solely on the long-range part of the molecular potential.

Collisions of paramagnetic molecules in magnetic fields: An analytic model based on Fraunhofer diffraction of matter waves - Mikhail Lemeshko; Friedrich, Břetislav (Physical Review A - Atomic, Molecular, and Optical Physics)
We investigate the effects of a magnetic field on the dynamics of rotationally inelastic collisions of open-shell molecules (Σ2, Σ3, and Π2) with closed-shell atoms. Our treatment makes use of the Fraunhofer model of matter wave scattering and its recent extension to collisions in electric [M. Lemeshko and B. Friedrich, J. Chem. Phys. 129, 024301 (2008)] and radiative fields [M. Lemeshko and B. Friedrich, Int. J. Mass. Spec. 280, 19 (2009)]. A magnetic field aligns the molecule in the space-fixed frame and thereby alters the effective shape of the diffraction target. This significantly affects the differential and integral scattering cross sections. We exemplify our treatment by evaluating the magnetic-field-dependent scattering characteristics of the He-CaH (XΣ+2), He-O2 (XΣ–3), and He-OH (XΠΩ2) systems at thermal collision energies. Since the cross sections can be obtained for different orientations of the magnetic field with respect to the relative velocity vector, the model also offers predictions about the frontal-versus-lateral steric asymmetry of the collisions. The steric asymmetry is found to be almost negligible for the He-OH system, weak for the He-CaH collisions, and strong for the He-O2. While odd ΔM transitions dominate the He-OH [J=3/2,f→J′,e/f] integral cross sections in a magnetic field parallel to the relative velocity vector, even ΔM transitions prevail in the case of the He-CaH (X2Σ+) and He-O2 (XΣ−3) collision systems. For the latter system, the magnetic field opens inelastic channels that are closed in the absence of the field. These involve the transitions N=1,J=0→N′, J′ with J′=N′.

The effect of a nonresonant radiative field on low-energy rotationally inelastic Na+ + N2 collisions - Mikhail Lemeshko; Friedrich, Břetislav (International Journal of Mass Spectrometry)
We examine the effects of a linearly polarized nonresonant radiative field on the dynamics of rotationally inelastic Na+ + N2 collisions at eV collision energies. Our treatment is based on the Fraunhofer model of matter wave scattering and its recent extension to collisions in electric fields [M. Lemeshko, B. Friedrich, J. Chem. Phys. 129 (2008) 024301]. The nonresonant radiative field changes the effective shape of the target molecule by aligning it in the space-fixed frame. This markedly alters the differential and integral scattering cross-sections. As the cross-sections can be evaluated for a polarization of the radiative field collinear or perpendicular to the relative velocity vector, the model also offers predictions about steric asymmetry of the collisions.

Rotational and rotationless states of weakly bound molecules - Mikhail Lemeshko; Friedrich, Břetislav (Physical Review A - Atomic, Molecular, and Optical Physics)
By making use of the quantization rule of Raab and Friedrich [Phys. Rev. A 78, 022707 (2008)], we derive simple and accurate formulae for the number of rotational states supported by a weakly bound vibrational level of a diatomic molecule and the rotational constants of any such levels up to the threshold, and provide a criterion for determining whether a given weakly bound vibrational level is rotationless. The results depend solely on the long-range part of the molecular potential and are applicable to halo molecules.

Model analysis of rotationally inelastic Ar + H2O scattering in an electric field - Mikhail Lemeshko; Friedrich, Břetislav (Journal of Physical Chemistry A)
We develop an analytic model of thermal state-to-state rotationally inelastic collisions of asymmetric-top molecules with closed-shell atoms in electric fields and apply it to the Ar-H2O collision system. The predicted cross sections as well as the steric asymmetry of the collisions show at fields up to 150 kV/cm characteristic field-dependent features which can be experimentally tested. Particularly suitable candidates for such tests are the 000 → 220 and 101→ 221 channels, arising from the relaxation of the field-free selection rules due to the hybridization of J states by the field. Averaging over the M' product channels is found to largely obliterate the orientation effects brought about by the field.

Probing weakly bound molecules with nonresonant light - Mikhail Lemeshko; Friedrich, Břetislav (Physical Review Letters)
We show that weakly bound molecules can be probed by "shaking" in a pulsed nonresonant laser field. The field introduces a centrifugal term which expels the highest vibrational level from the potential that binds it. Our numerical simulations applied to the Rb2 and KRb Feshbach molecules indicate that shaking by feasible laser pulses can be used to accurately recover the square of the vibrational wave function and, by inversion, also the long-range part of the molecular potential.

Manin's conjecture for a quartic del Pezzo surface with A4 singularity - Timothy Browning; Derenthal, Ulrich (Annales de l'Institut Fourier)
The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A4.

Quantitative Arithmetic of Projective Varieties - Browning, Timothy D
Winner of the Ferran Sunyer i Balaguer Prize 2009. First attempt to systematically survey the range of available tools from analytic number theory that can be applied to study the density of rational points on projective varieties. Designed to rapidly guide the reader to the many areas of ongoing research in the domain. Provides an extensive bibliography.

Rational points on quartic hypersurfaces - Timothy Browning; Heath-Brown, Roger (Journal fur die Reine und Angewandte Mathematik)
Let X be a projective non-singular quartic hypersurface of dimension 39 or more, which is defined over . We show that X() is non-empty provided that X() is non-empty and X has p-adic points for every prime p.

Linear growth for Châtelet surfaces - Timothy Browning (Mathematische Annalen)
An upper bound of the expected order of magnitude is established for the number of ℚ-rational points of bounded height on Châtelet surfaces defined over ℚ.

Manin's conjecture for a cubic surface with D 5 singularity - Timothy Browning; Derenthal, Ulrich (International Mathematics Research Notices)
The Manin conjecture is established for a split singular cubic surface in ℙ 3, with singularity type D 5.

The Stability of Matter in Quantum Mechanics - Lieb, Élliott; Seiringer, Robert
Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. A unique account of the subject, this book provides a complete, self-contained description of research on the stability of matter problem. It introduces the necessary quantum mechanics to mathematicians, and aspects of functional analysis to physicists. The topics covered include electrodynamics of classical and quantized fields, Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, gravitational stability of stars, basics of equilibrium statistical mechanics, and the existence of the thermodynamic limit. The book is an up-to-date account for researchers, and its pedagogical style makes it suitable for advanced undergraduate and graduate courses in mathematical physics.

A linear criterion for solutions of non-linear equations, with application to the BCS gap equation - Hainzl, Christian; Robert Seiringer

The ground state energy of the weakly interacting Bose gas at high density - Giuliani, Alessandro; Robert Seiringer (Journal of Statistical Physics)
We prove the Lee-Huang-Yang formula for the ground state energy of the 3D Bose gas with repulsive interactions described by the exponential function, in a simultaneous limit of weak coupling and high density. In particular, we show that the Bogoliubov approximation is exact in an appropriate parameter regime, as far as the ground state energy is concerned.

Yrast line of a rapidly rotating Bose gas: Gross-Pitaevskii regime - Lieb, Élliott H; Robert Seiringer; Yngvason, Jakob (Physical Review A - Atomic, Molecular, and Optical Physics)
We consider an ultracold rotating Bose gas in a harmonic trap close to the critical angular velocity, so that the system can be considered to be confined to the lowest Landau level. With this assumption we prove that the Gross-Pitaevskii energy functional accurately describes the ground-state energy of the corresponding N -body Hamiltonian with contact interaction provided the total angular momentum L is much less than N2. While the Gross-Pitaevskii energy is always an obvious variational upper bound to the ground-state energy, a more refined analysis is needed to establish it as an exact lower bound. We also discuss the question of Bose-Einstein condensation in the parameter range considered. Coherent states together with inequalities in spaces of analytic functions are the main technical tools.

Rigorous upper bound on the critical temperature of dilute Bose gases - Robert Seiringer; Ueltschi, Daniel (Physical Review B - Condensed Matter and Materials Physics)
We prove exponential decay of the off-diagonal correlation function in the two-dimensional homogeneous Bose gas when a2 ρ is small and the temperature T satisfies T> 4πρ ln | ln (a2 ρ) |. Here, a is the scattering length of the repulsive interaction potential and ρ is the density. To the leading order in a2 ρ, this bound agrees with the expected critical temperature for superfluidity. In the three-dimensional Bose gas, exponential decay is proved when T- Tc (0) Tc (0) >5 a ρ1/3, where Tc (0) is the critical temperature of the ideal gas. While this condition is not expected to be sharp, it gives a rigorous upper bound on the critical temperature for Bose-Einstein condensation.

Strongly correlated phases in rapidly rotating Bose gases - Lewin, Mathieu; Robert Seiringer (Journal of Statistical Physics)
We consider a system of trapped spinless bosons interacting with a repulsive potential and subject to rotation. In the limit of rapid rotation and small scattering length, we rigorously show that the ground state energy converges to that of a simplified model Hamiltonian with contact interaction projected onto the Lowest Landau Level. This effective Hamiltonian models the bosonic analogue of the Fractional Quantum Hall Effect (FQHE). For a fixed number of particles, we also prove convergence of states; in particular, in a certain regime we show convergence towards the bosonic Laughlin wavefunction. This is the first rigorous justification of the effective FQHE Hamiltonian for rapidly rotating Bose gases. We review previous results on this effective Hamiltonian and outline open problems.

Probabilistic coherence and proper scoring rules - Predd, Joel B; Robert Seiringer; Lieb, Élliott H; Osherson, Daniel N; Poor, Harold V; Kulkarni, Sanjeev R (IEEE Transactions on Information Theory)
This paper provides self-contained proof of a theorem relating probabilistic coherence of forecasts to their non-domination by rival forecasts with respect to any proper scoring rule. The theorem recapitulates insights achieved by other investigators, and clarifies the connection of coherence and proper scoring rules to Bregman divergence.