TY - BOOK
AB - This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem that have been obtained by the authors in the past seven years. It addresses a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. The book provides a pedagogical entry into an active area of ongoing research for both graduate students and researchers. It is an outgrowth of a course given by the authors for graduate students and post-doctoral researchers at the Oberwolfach Research Institute in 2004. The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.
AU - Lieb, Élliott H
AU - Robert Seiringer
AU - Solovej, Jan P
AU - Yngvason, Jakob
ID - 2335
T2 - The mathematics of the Bose gas and its condensation
TI - The mathematics of the Bose gas and its condensation
VL - 34
ER -
TY - CHAP
AB -
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4–5 decades ago, and to explore new regimes not treated before. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous — and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πħ2 aρ/m where a is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N 7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, several other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the ‘traps’ actually used in the experiments. For this system it is also possible to prove complete Bose condensation and superfluidity as we have shown. On the frontier of experimental developments is the possibility that a dilute gas in an elongated trap will behave like a one-dimensional system; we have proved this mathematically. Another topic is a proof that Foldy’s 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy; using this we can also prove the N 7/5 formula for the ground state energy of the two-component charged Bose gas proposed by Dyson in 1967. All of this is quite recent work and it is hoped that the mathematical methodology might be useful, ultimately, to solve more complex problems connected with these interesting systems.
AU - Lieb, Élliott H
AU - Robert Seiringer
AU - Solovej, Jan P
AU - Yngvason, Jakob
ED - Benedicks, Michael
ED - Jones, Peter W
ED - Smirnov, Stanislav
ED - Winckler, Björn
ID - 2336
T2 - Perspectives in Analysis
TI - The quantum-mechanical many-body problem: The Bose gas
VL - 27
ER -
TY - JOUR
AB - The validity of substituting a c-number z for the k = 0 mode operator a0 is established rigorously in full generality, thereby verifying one aspect of Bogoliubov's 1947 theory. This substitution not only yields the correct value of thermodynamic quantities such as the pressure or ground state energy, but also the value of |z|2 that maximizes the partition function equals the true amount of condensation in the presence of a gauge-symmetry-breaking term. This point had previously been elusive.
AU - Lieb, Élliott H
AU - Robert Seiringer
AU - Yngvason, Jakob
ID - 2359
IS - 8
JF - Physical Review Letters
TI - Justification of c-number substitutions in bosonic hamiltonians
VL - 94
ER -
TY - JOUR
AB - The strong subadditivity of entropy plays a key role in several areas of physics and mathematics. It states that the entropy S[±]=- Tr(Ï±lnÏ±) of a density matrix Ï±123 on the product of three Hilbert spaces satisfies S[Ï±123]- S[Ï±12]≤S[Ï±23]-S[Ï±2]. We strengthen this to S[Ï±123]-S[Ï±12] ≤αnα(S[Ï±23α]-S[Ï±2α]), where the nα are weights and the Ï±23α are partitions of Ï±23. Correspondingly, there is a strengthening of the theorem that the map A|Trexp[L+lnA] is concave. As applications we prove some monotonicity and convexity properties of the Wehrl coherent state entropy and entropy inequalities for quantum gases.
AU - Lieb, Élliott H
AU - Robert Seiringer
ID - 2361
IS - 6
JF - Physical Review A - Atomic, Molecular, and Optical Physics
TI - Stronger subadditivity of entropy
VL - 71
ER -
TY - JOUR
AB - Recent developments in the physics of low-density trapped gases make it worthwhile to verify old, well-known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground-state energy of a dilute gas of N fermions interacting with a short-range, positive potential of scattering length a. For spin-12 fermions, this is E∼E0+(22m)2πNa, where E0 is the energy of the noninteracting system and is the density. A similar formula holds in two dimensions (2D), with a replaced by ln(a2). Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.
AU - Lieb, Élliott H
AU - Robert Seiringer
AU - Solovej, Jan P
ID - 2362
IS - 5
JF - Physical Review A - Atomic, Molecular, and Optical Physics
TI - Ground state energy of the low density Fermi gas
VL - 71
ER -
TY - JOUR
AB - The plant hormone auxin elicits many specific context-dependent developmental responses. Auxin promotes degradation of Aux/IAA proteins that prevent transcription factors of the auxin response factor (ARF) family from regulating auxin-responsive target genes. Aux/IAAs and ARFs are represented by large gene families in Arabidopsis. Here we show that stabilization of BDL/IAA12 or its sister protein IAA13 prevents MP/ARF5-dependent embryonic root formation whereas stabilized SHY2/IAA3 interferes with seedling growth. Although both bdl and shy2-2 proteins inhibited MP/ARF5-dependent reporter gene activation, shy2-2 was much less efficient than bdl to interfere with embryonic root initiation when expressed from the BDL promoter. Similarly, MP was much more efficient than ARF16 in this process. When expressed from the SHY2 promoter, both shy2-2 and bdl inhibited cell elongation and auxin-induced gene expression in the seedling hypocotyl. By contrast, gravitropism and auxin-induced gene expression in the root, which were promoted by functionally redundant NPH4/ARF7 and ARF19 proteins, were inhibited by shy2-2, but not by bdl protein. Our results suggest that auxin signals are converted into specific responses by matching pairs of coexpressed ARF and Aux/IAA proteins.
AU - Weijers, Dolf
AU - Eva Benková
AU - Jäger, Katja E
AU - Schlereth, Alexandra
AU - Hamann, Thorsten
AU - Kientz, Marika
AU - Wilmoth, Jill C
AU - Reed, Jason W
AU - Jürgens, Gerd
ID - 2867
IS - 10
JF - EMBO Journal
TI - Developmental specificity of auxin response by pairs of ARF and Aux/IAA transcriptional regulators
VL - 24
ER -
TY - JOUR
AB - Transmission of signals within the brain is essential for cognitive function, but it is not clear how neural circuits support reliable and accurate signal propagation over a sufficiently large dynamic range. Two modes of propagation have been studied: synfire chains, in which synchronous activity travels through feedforward layers of a neuronal network, and the propagation of fluctuations in firing rate across these layers. In both cases, a sufficient amount of noise, which was added to previous models from an external source, had to be included to support stable propagation. Sparse, randomly connected networks of spiking model neurons can generate chaotic patterns of activity. We investigate whether this activity, which is a more realistic noise source, is sufficient to allow for signal transmission. We find that, for rate-coded signals but not for synfire chains, such networks support robust and accurate signal reproduction through up to six layers if appropriate adjustments are made in synaptic strengths. We investigate the factors affecting transmission and show that multiple signals can propagate simultaneously along different pathways. Using this feature, we show how different types of logic gates can arise within the architecture of the random network through the strengthening of specific synapses.
AU - Vogels, Tim P
AU - Abbott, L. F.
ID - 8028
IS - 46
JF - Journal of Neuroscience
SN - 0270-6474
TI - Signal propagation and logic gating in networks of integrate-and-fire neurons
VL - 25
ER -
TY - JOUR
AB - For any n ≧ 2, let F ∈ ℤ [ x 1, … , xn ] be a form of degree d≧ 2, which produces a geometrically irreducible hypersurface in ℙn–1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F; B) = O(B n− 2+ ε ), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε.
AU - Timothy Browning
AU - Heath-Brown, Roger
ID - 212
IS - 584
JF - Journal fur die Reine und Angewandte Mathematik
TI - Counting rational points on hypersurfaces
ER -
TY - JOUR
AB - Given an absolutely irreducible ternary form F, the purpose of this paper is to produce better upper bounds for the number of integer solutions to the equation F=0, that are restricted to lie in very lopsided boxes. As an application of the main result, a new paucity estimate is obtained for equal sums of two like powers.
AU - Timothy Browning
AU - Heath-Brown, Roger
ID - 214
IS - 2
JF - Mathematische Zeitschrift
TI - Plane curves in boxes and equal sums of two powers
VL - 251
ER -
TY - CHAP
AB - The paper surveys the mirror symmetry conjectures of Hausel-Thaddeus and Hausel-Rodriguez-Villegas concerning the equality of certain Hodge numbers of SL(n, ℂ) vs. PGL(n, ℂ) flat connections and character varieties for curves, respectively. Several new results and conjectures and their relations to works of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan are explained. These use the representation theory of finite groups of Lie-type via the arithmetic of character varieties and lead to an unexpected conjecture for a Hard Lefschetz theorem for their cohomology.
AU - Tamas Hausel
ID - 1444
T2 - Geometric Methods in Algebra and Number Theory
TI - Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve
VL - 235
ER -