Differential Equations of Quantum Mechanics

Abstract

We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.

References in corpus (93)

• Solitary Wave Dynamics in an External Potential
• Fast soliton scattering by delta impurities
• Ground States in Non-relativistic Quantum Electrodynamics
• Triviality of Bloch and Bloch-Dirac bundles
• Effective dynamics for Bloch electrons: Peierls substitution and beyond
• The Electronic Ground State Energy Problem: a New Reduced Density Matrix Approach
• Geometric methods for nonlinear many-body quantum systems
• Pair excitations and the mean field approximation of interacting Bosons, I
• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta
• Reducibility of the Quantum Harmonic Oscillator in $d$-dimensions with Polynomial Time Dependent Perturbation
• On time dependent Schr{ö}dinger equations: global well-posedness and growth of Sobolev norms
• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• Phase Space Analysis on some Black Hole Manifolds
• Scattering and small data completeness for the critical nonlinear Schroediger equation
• Long time motion of NLS solitary waves in a confining potential
• Adiabatic response for Lindblad dynamics
• The BCS Functional for General Pair Interactions
• Infraparticle Scattering States in Non-Relativistic QED: I. The Bloch-Nordsieck Paradigm
• On the self-adjointness and domain of Pauli-Fierz type Hamiltonians
• Infraparticle Scattering States in Non-Relativistic QED: II. Mass Shell Properties
• The time-dependent Born-Oppenheimer approximation
• The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate
• Pair excitations and the mean field approximation of interacting Bosons, II
• Ground States in the Spin Boson Model
• Enhanced Binding in non-relativistic QED
• The BCS functional of superconductivity and its mathematical properties
• Renormalized Electron Mass in Nonrelativistic QED
• The Local Density Approximation in Density Functional Theory
• Vortex-peak interaction and lattice shape in rotating two-component Bose-Einstein condensates
• Bogoliubov correction to the mean-field dynamics of interacting bosons
• The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
• Mean-field limit of Bose systems: rigorous results
• Absence of Ground States for a Class of Translation Invariant Models of Non-relativistic QED
• The Bogoliubov free energy functional II. The dilute limit
• Correlation Energy of a Weakly Interacting Fermi Gas
• Mass renormalization in nonrelativistic QED
• Approach to ground state and time-independent photon bound for massless spin-boson models
• Analyticity of The Ground State Energy For Massless Nelson Models
• Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation
• Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d\le 3$
• Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime
• Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation
• Breathing patterns in nonlinear relaxation
• The Bogoliubov free energy functional I. Existence of minimizers and phase diagram
• On the Ginzburg--Landau Functional in the Surface Superconductivity Regime
• Local density approximation for the almost-bosonic anyon gas
• Gross-Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature
• Ground State and Resonances in the Standard Model of Non-relativistic QED
• Vortex patterns in the almost-bosonic anyon gas
• A Note on Polarization Vectors in Quantum Electrodynamics
• On Existence of Ground States in the Spin Boson Model
• One non-relativistic particle coupled to a photon field
• Magnetic Schrödinger Operators as the Quasi-Classical Limit of Pauli-Fierz-type Models
• Multi-component Ginzburg-Landau theory: microscopic derivation and examples
• The external field dependence of the BCS critical temperature
• The translation invariant massive Nelson model: III. Asymptotic completeness below the two-boson threshold
• Quantum Brownian Motion in a Simple Model System
• The Ground State Energy of The Massless Spin-Boson Model
• Stability of Abrikosov lattices under gauge-periodic perturbations
• Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature
• Coulomb scattering in the massless Nelson model I. Foundations of two-electron scattering
• Global stability of the normal state of superconductors in the presence of a strong electric current
• Long-time behaviour of time-dependent density functional theory
• Maximal Speed of Quantum Propagation
• Effects of boundary curvature on surface superconductivity
• Existence of Hartree-Fock excited states for atoms and molecules
• Spontaneous decay of resonant energy levels for molecules with moving nuclei
• Incompatibility of time-dependent Bogoliubov--de-Gennes and Ginzburg--Landau equations
• Coulomb scattering in the massless Nelson model III. Ground state wave functions and non-commutative recurrence relations
• Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture
• Analyticity of Resonances and Eigenvalues and Spectral Properties of the massless Spin-Boson Model
• Compactness of molecular reaction paths in quantum mechanics
• The BCS critical temperature in a weak homogeneous magnetic field
• The translation Invariant Massive Nelson Model: II. The Continuous Spectrum Below the Two-boson Threshold
• Quantum Electrodynamics of Atomic Resonances
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
• Coherent infrared representations in non-relativistic QED
• Mass Renormalization and Energy Level Shift in Non-Relativistic QED
• Lattice solutions in a Ginzburg-Landau model for a chiral magnet
• A survey on asymptotic stability of ground states of nonlinear Schrodinger equations II
• Continuum Schroedinger operators for sharply terminated graphene-like structures
• Generalized One-Particle Density Matrices and Quasifree States
• Microscopic Derivation of the Ginzburg-Landau Model
• Hamiltonian models of interacting fermion fields in Quantum Field Theory
• Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon
• The Mass Shell of the Nelson Model without Cut-Offs
• The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime
• On the Bogolubov-de Gennes Equations
• Dynamics of interacting bosons: a compact review
• Co-rotational chiral magnetic skyrmions near harmonic maps
• Instability of electroweak homogeneous vacua in strong magnetic fields
• Van der Waals interactions between two hydrogen atoms: The next orders
• Molecular dynamics at an energy-level crossing