VORTICES IN BOSE-EINSTEIN CONDENSATES

By JESSICA SANCHEZ

CONTENTS THE PHYSICAL EXPERIMENTS AND THEIR MATHEMATICAL MODELLING A HINT ON THE EXPERIMENTS BRIEF HISTORICAL SUMMARY HOW TO ACHIEVE A BEC EXPERIMENTALLY EXPERIMENTAL OBSERVATIONS THE MATHEMATICAL FRAMEWORK THE GROSS–PITAEVSKII ENERGY THE THOMAS–FERMI REGIME REMARKS ON THE ORIGINAL PROBLEM MEAN-?ELD QUANTUM HALL REGIME FLOW AROUND AN OBSTACLE THE MATHEMATICAL SETTING: A SURVEY OF THE MAIN THEOREMS SMALL ? PROBLEM THE TWO-DIMENSIONAL SETTING THE THREE-DIMENSIONAL SETTING VORTEX LATTICE TWO-DIMENSIONAL MODEL FOR A ROTATING CONDENSATE MAIN RESULTS SINGLE-VORTEX SOLUTION AND LOCATION OF VORTICES IDEAS OF THE PROOF PRELIMINARIES DETERMINING THE DENSITY PRO?LE EXISTENCE OF A MINIMIZER OF E? SPLITTING THE ENERGY BOUNDED NUMBER OF VORTICES FIRST ENERGY BOUND VORTEX BALLS THE ROTATION TERM A LOWER BOUND EXPANSION RE?NED STRUCTURE OF VORTICES SOME LOCAL ESTIMATES BAD DISCS NO DEGREE-ZERO VORTEX PROOF OF PROPOSITION 3 12 LOWER BOUND UPPER BOUND FINAL EXPANSION AND PROPERTIES OF VORTICES VORTICES HAVE DEGREE ONE THE SUBCRITICAL CASE THE SUPERCRITICAL CASE OPEN QUESTIONS VORTICES IN THE REGION OF LOW DENSITY OTHER TRAPPING POTENTIALS INTERMEDIATE TIME-DEPENDENT PROBLEM OTHER TRAPPING POTENTIALS NON RADIAL HARMONIC POTENTIAL QUARTIC POTENTIAL GIANT VORTEX CIRCLE OF VORTICES OPEN QUESTIONS GIANT VORTEX OR ISOLATED VORTICES HIGH-VELOCITY AND QUANTUM HALL REGIME INTRODUCTION LOWEST LANDAU LEVEL CONSTRUCTION OF AN UPPER BOUND PROPERTIES OF THE MINIMIZER REGULAR LATTICE DISTORTED LATTICE IN?NITE NUMBER OF ZEROS LOWER BOUND AND ? CONVERGENCE RESTRICTION TO THE LLL REDUCTION TO A TWO-DIMENSIONAL PROBLEM MEAN ?ELD MODEL THREE-DIMENSIONAL ROTATING CONDENSATE NUMERICAL SIMULATIONS FORMAL DERIVATION OF THE REDUCED ENERGY E[? ] THE SOLUTION WITHOUT VORTICES DECOUPLING THE ENERGY ESTIMATE OF G?(V?) ESTIMATE OF I?(V?) FINAL ESTIMATE FOR THE ENERGY ? CONVERGENCE RESULTS MAIN IDEAS IN THE PROOF SINGLE VORTEX LINE, STUDY OF E[? ] SETTING OF MINIMIZATION OF E[? ] THE BENT VORTEX PROPERTIES OF CRITICAL POINTS A FEW OPEN QUESTIONS SMALL VELOCITY CRITICAL POINTS OF E[?] FINITE NUMBER OF VORTICES WHOLE SPACE PROBLEM DECAY OF VORTICES SUPER?UID FLOW AROUND AN OBSTACLE MATHEMATICAL SETTING TWO-DIMENSIONAL ?OW THREE-DIMENSIONAL ?OW AROUND A CONDENSATE PROOF OF THEOREM 7 1 SOLUTIONS AT C = 0 EXISTENCE OF A SOLUTION TO IR BOUNDS ON THE SOLUTIONS OF IR ESTIMATING THE MOMENTUM PROOF OF THEOREM 7 4 LIMIT AT IN?NITY PROOF OF THEOREM 7 2 PROOF OF THEOREM 7 14 PROOF OF THEOREM 7 16 FURTHER OPEN PROBLEMS SETTING IN THE WHOLE SPACE FOR THE THOMAS–FERMI REGIME THREE-DIMENSIONAL PROBLEM TWO-DIMENSIONAL PROBLEM PAINLEV´E BOUNDARY LAYER VORTICES IN THE HOLE OTHER SCALINGS OTHER MODELS OPTICAL LATTICES MULTICOMPONENT CONDENSATES CONDENSATE AND NONCONDENSED GAS FERMI GASES REFERENCES