Differential equations of quantum mechanics

DOI10.1090/qam/1611zbMath1504.35421arXiv2208.03774OpenAlexW4220824696WikidataQ115290641 ScholiaQ115290641MaRDI QIDQ5869466

Israel Michael Sigal

Publication date: 28 September 2022

Published in: Quarterly of Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2208.03774

zbMATH Keywords

quantum mechanicspartial differential equationsGinzburg-Landau equationsnonrelativistic QEDBogolubov-de Gennes systemHartree-Fock-Bogolubov system

Mathematics Subject Classification ID

Electromagnetic interaction; quantum electrodynamics (81V10) Statistical mechanics of superconductors (82D55) NLS equations (nonlinear Schrödinger equations) (35Q55) PDEs in connection with quantum mechanics (35Q40) General mathematical topics and methods in quantum theory (81Q99) Time-dependent Schrödinger equations and Dirac equations (35Q41) Ginzburg-Landau equations (35Q56) Fermionic systems in quantum theory (81V74) Resonances in quantum scattering theory (81U24) Bosonic systems in quantum theory (81V73)

Cites Work

• Dynamics of sound waves in an interacting Bose gas
• Effects of boundary curvature on surface superconductivity
• Multi-component Ginzburg-Landau theory: microscopic derivation and examples
• Pair excitations and the mean field approximation of interacting Bosons. I
• Mean-field models for disordered crystals
• The mass shell of the Nelson model without cut-offs
• On the Ginzburg-Landau functional in the surface superconductivity regime
• Asymptotic completeness for the massless spin-boson model
• Incompatibility of time-dependent Bogoliubov-de-Gennes and Ginzburg-Landau equations
• Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields
• On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms
• The transition to a giant vortex phase in a fast rotating Bose-Einstein condensate
• Geometric methods for nonlinear many-body quantum systems
• Ground states in the spin Boson model
• The mathematics of the Bose gas and its condensation.
• A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta
• Nucleation of bulk superconductivity close to critical magnetic field
• Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation
• Quantum Brownian motion in a simple model system
• Renormalized electron mass in nonrelativistic QED
• Infraparticle scattering states in non-relativistic QED. I: The Bloch-Nordsieck paradigm
• Effective dynamics of double solitons for perturbed mKdV
• The ground state energy of the massless spin-boson model
• Convergent expansions in non-relativistic QED: analyticity of the ground state
• Smoothness and analyticity of perturbation expansions in QED
• On pointwise decay of linear waves on a Schwarzschild black hole background
• Local decay in non-relativistic QED
• Spontaneous decay of resonant energy levels for molecules with moving nuclei
• Asymptotic completeness for Rayleigh scattering
• Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants
• The infrared behaviour in Nelson's model of a quantum particle coupled to a massless scalar field.
• Commutators, \(C_{0}\)-semigroups and resolvent estimates
• Maximal velocity of photons in non-relativistic QED
• The electronic structure of smoothly deformed crystals: Wannier functions and the Cauchy-Born Rule
• Analyticity of the ground state energy for massless Nelson models
• One-boson scattering processes in the massless spin-boson model. A non-perturbative formula
• Asymptotic completeness for N-body short-range quantum systems: A new proof
• Spectral analysis for optical fibres and stratified fluids. I: The limiting absorption principle
• Maximal speed of quantum propagation
• On the absence of excited eigenstates of atoms in QED
• Asymptotic completeness for Compton scattering
• A note on polarization vectors in quantum electrodynamics
• Vortices in the magnetic Ginzburg-Landau model
• On soliton dynamics in nonlinear Schrödinger equations
• The renormalized electron mass in non-relativistic quantum electrodynamics
• The translation invariant massive Nelson model. III: asymptotic completeness below the two-boson threshold
• Infrared renormalization in non-relativistic QED and scaling criticality
• Absence of ground states for a class of translation invariant models of non-relativistic QED
• Spectral methods in surface superconductivity
• Spectral theory for the standard model of non-relativistic QED
• Bulk superconductivity in type II superconductors near the second critical field
• Phase space analysis on some black hole manifolds
• Infrared-finite algorithms in QED. II. The expansion of the groundstate of an atom interacting with the quantized radiation field
• The BCS functional for general pair interactions
• Asymptotic electromagnetic fields in non-relativistic QED: The problem of existence revisited
• Ground state and resonances in the standard model of the non-relativistic QED
• Existence of minimizers for Kohn-Sham models in quantum chemistry
• Periodic minimizers of the anisotropic Ginzburg-Landau model
• A new proof of the Mourre estimate
• The N-particle scattering problem: asymptotic completeness for short- range systems
• Ground state(s) of the spin-boson Hamiltonian
• The classical field limit of scattering theory for non-relativistic many- boson systems. I
• The classical field limit of scattering theory for non-relativistic many- boson systems. II
• Absence of singular continuous spectrum for certain self-adjoint operators
• Propagation estimates for \(N\)-body Schrödinger operators
• Sharp propagation estimates for \(N\)-particle systems
• Compactification of complete Kähler surfaces with negative Ricci curvature
• Renormalization group analysis of spectral problems in quantum field theory
• Quantum electrodynamics of confined nonrelativistic particles
• The Thomas-Fermi theory of atoms, molecules and solids
• Asymptotic completeness for \(N\leq 4\) particle systems with the Coulomb-type interactions
• Asymptotic completeness of long-range \(N\)-body quantum systems
• Scattering theory for \(N\)-particle systems in constant magnetic fields
• Asymptotic completeness for particles in combined constant electric and magnetic fields. II.
• Structure of the resolvent for three-body potentials
• Ground state of a quantum particle coupled to a scalar Bose field
• Mathematical theory of non-equilibrium quantum statistical mechanics
• One non-relativistic particle coupled to a photon field
• Long-range scattering of three-body quantum systems: asymptotic completeness
• Semilinear wave equations on the Schwarzschild manifold. I: Local decay estimates.
• Adiabatic perturbation theory in quantum dynamics
• One-particle (improper) states in Nelson's massless model
• Effective dynamics for Bloch electrons: Peierls substitution and beyond
• Self-energy of one electron in non-relativistic QED.
• Soliton dynamics in a potential
• On the existence of ground states for massless Pauli-Fierz Hamiltonians
• The Bogoliubov free energy functional. II: The dilute limit
• Edge states in honeycomb structures
• Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii equation
• The Bogoliubov free energy functional. I: Existence of minimizers and phase diagram
• A new approach to continuous multi-scale analysis in nonrelativistic QED: ground states and photon number bounds for the spin-boson model with critical infrared singularity
• Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions \({d \leqslant 3}\)
• Reducibility of the quantum harmonic oscillator in \(d\)-dimensions with polynomial time-dependent perturbation
• Bogoliubov correction to the mean-field dynamics of interacting bosons
• On stability of Abrikosov vortex lattices
• Coulomb scattering in the massless Nelson model. III: Ground state wave functions and non-commutative recurrence relations
• Growth of Sobolev norms for time dependent periodic Schrödinger equations with sublinear dispersion
• Analyticity of resonances and eigenvalues and spectral properties of the massless spin-boson model
• Tosio Kato's work on non-relativistic quantum mechanics. I
• Existence of Hartree-Fock excited states for atoms and molecules
• The Dirac-Frenkel principle for reduced density matrices, and the Bogoliubov-de Gennes equations
• Semi-classical limit of the Levy-Lieb functional in density functional theory
• Binding energy for hydrogen-like atoms in the Nelson model without cutoffs
• The maximal excess charge in Müller density-matrix-functional theory
• Solutions of Hartree-Fock equations for Coulomb systems
• On the Kohn-Sham equations with periodic background potentials
• Enhanced binding in non-relativistic QED
• Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics
• Existence of atoms and molecules in non-relativistic quantum electrodynamics
• Free channel Fourier transform in the long-range \(N\)-body problem
• Asymptotic completeness for \(N\)-body Stark Hamiltonians
• Adiabatic response for Lindblad dynamics
• Approach to ground state and time-independent photon bound for massless spin-boson models
• The translation invariant massive Nelson model: II. The continuous spectrum below the two-Boson threshold
• On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff
• Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field
• Gross-Pitaevskii limit of a homogeneous Bose gas at positive temperature
• Asymptotics in spin-boson type models
• Widths of highly excited resonances in multidimensional molecular predissociation
• Hamiltonian models of interacting fermion fields in quantum field theory
• Effective dynamics of a tracer particle interacting with an ideal Bose gas
• Minimal photon velocity bounds in non-relativistic quantum electrodynamics
• Coulomb scattering in the massless Nelson model. I. Foundations of two-electron scattering
• Thomas-Fermi approximation for coexisting two component Bose-Einstein condensates and nonexistence of vortices for small rotation
• Lattice solutions in a Ginzburg-Landau model for a chiral magnet
• Scaling limits of bosonic ground states, from many-body to non-linear Schrödinger
• Long-time behaviour of time-dependent density functional theory
• Correlation energy of a weakly interacting Fermi gas
• The Feshbach-Schur map and perturbation theory
• A survey on asymptotic stability of ground states of nonlinear Schrödinger equations. II
• The time-dependent Hartree-Fock-Bogoliubov equations for bosons
• One-boson scattering processes in the massive spin-boson model
• Continuum Schrödinger operators for sharply terminated graphene-like structures
• Co-rotational chiral magnetic skyrmions near harmonic maps
• A new method of construction of resonances that applies to critical models
• Vortex lattices and the Bogoliubov-de Gennes equations
• On existence of ground states in the spin boson model
• On Rayleigh scattering in non-relativistic quantum electrodynamics
• A new asymptotic perturbation theory with applications to models of massless quantum fields
• Continuous renormalization group analysis of spectral problems in quantum field theory
• Vortex analysis of the periodic Ginzburg-Landau model
• Molecular dynamics at an energy-level crossing
• Superconductivity between \(H_{C_2}\) and \(H_{C_3}\)
• Vortex patterns and sheets in segregated two component Bose-Einstein condensates
• The local density approximation in density functional theory
• Magnetic Schrödinger operators as the quasi-classical limit of Pauli-Fierz-type models
• Compactness of molecular reaction paths in quantum mechanics
• On effective PDEs of quantum physics
• Lower bounds on the growth of Sobolev norms in some linear time dependent Schrödinger equations
• The BCS critical temperature in a weak homogeneous magnetic field
• Quantum electrodynamics of atomic resonances
• Existence and construction of resonances for atoms coupled to the quantized radiation field
• Local density approximation for the almost-bosonic anyon gas
• Radiation conditions and scattering theory for \(N\)-particle Hamiltonians
• Local decay for weak interactions with massless particles
• Effective Maxwell equations from time-dependent density functional theory
• Relaxation of solitons in nonlinear Schrödinger equations with potential
• Triviality of Bloch and Bloch-Dirac bundles
• Fast soliton scattering by delta impurities
• Fiber Hamiltonians in non-relativistic quantum electrodynamics
• Rayleigh scattering at atoms with dynamical nuclei
• Lowest Landau level approach in superconductivity for the Abrikosov lattice close to \(H_{{c}_{2}}\)
• Scattering of an infraparticle: The one particle sector in Nelson's massless model
• Long time motion of NLS solitary waves in a confining potential
• Vortices in Bose-Einstein condensates
• Global stability of the normal state of superconductors in the presence of a strong electric current
• Solitary wave dynamics in an external potential
• Spectral theory of massless Pauli-Fierz models
• The translation invariant massive Nelson model. I: The bottom of the spectrum
• Existence of ground state eigenvalues for the spin-boson model with critical infrared divergence and multiscale analysis
• Long time dynamics of Schrödinger and wave equations on flat tori
• Growth of Sobolev norms for abstract linear Schrödinger equations
• Growth of Sobolev norms for unbounded perturbations of the Schrödinger equation on flat tori
• The quantum N-body problem
• The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases
• The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties
• Fully coupled Pauli-Fierz systems at zero and positive temperature
• Stability of Abrikosov lattices under gauge-periodic perturbations
• Microscopic derivation of Ginzburg-Landau theory
• A mathematical perspective on density functional perturbation theory
• Effective Evolution Equations from Quantum Dynamics
• Topologically protected states in one-dimensional continuous systems and Dirac points
• Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature
• A note on 2D focusing many-boson systems
• Pair excitations and the mean field approximation of interacting Bosons, II
• Mass renormalization in nonrelativistic quantum electrodynamics
• The uncertainty principle
• Dynamics of KdV Solitons in the Presence of a Slowly Varying Potential
• A variational formulation of schrödinger-poisson systems in dimension d ≤ 3
• Dynamics of Charged Particles and their Radiation Field
• Decay for the wave and Schrödinger evolutions on manifolds with conical ends, Part II
• Atomism and quantization
• Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities
• Fluctuations around Hartree states in the mean-field regime
• The stability of matter: from atoms to stars
• Soliton Interaction with Slowly Varying Potentials
• Thomas-fermi and related theories of atoms and molecules
• Breathing patterns in nonlinear relaxation
• Mathematical issues in the modelling of supersolids
• Infraparticle scattering states in nonrelativistic quantum electrodynamics. II. Mass shell properties
• The Stability of Matter in Quantum Mechanics
• Asymptotic completeness for n–body schrödinger operators with short–range interactions
• Spectral analysis for optical fibres and stratifled fluids II: Absence of Eigenvalues
• A time-dependent approach to the completeness of multiparticle quantum systems
• Spectral Analysis of N-Body Systems Coupled to a Bosonic Field
• Asymptotic completeness for Rayleigh scattering
• ON THE SCATTERING THEORY OF MASSLESS NELSON MODELS
• Quantum Theory of Radiation
• ASYMPTOTIC COMPLETENESS IN QUANTUM IN FIELD THEORY: MASSIVE PAULI–FIERZ HAMILTONIANS
• Reducibility of 1-d Schrödinger equation with time quasiperiodic unbounded perturbations. I
• Van der Waals Interactions Between Two Hydrogen Atoms: The Slater--Kirkwood Method Revisited
• Large Outgoing Solutions to Supercritical Wave Equations
• Coulomb scattering in the massless Nelson model II. Regularity of ground states
• Dispersive analysis of charge transfer models
• Minimal escape velocities
• Scattering theory forN—particle systems in constant magnetic fields II. Long-range interactions
• RADIATIVE DECAY: NONPERTURBATIVE APPROACHES
• Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field
• The Ground State Energy of the Three Dimensional Ginzburg-Landau Functional Part I: Bulk Regime
• Electronic structure of smoothly deformed crystals: Cauchy-born rule for the nonlinear tight-binding model
• The Kohn-Sham equation for deformed crystals
• Mathematical analysis of quantum fields—Historical survey and a new asymptotic perturbation theory
• Scattering theory for mathematical models of the weak interaction
• Mathematical Concepts of Quantum Mechanics
• A short proof of the ionization conjecture in Müller theory
• SCATTERING THEORY FOR THE WAVE EQUATION ON NON-COMPACT MANIFOLDS
• Quantum Dynamical Systems
• Long‐Range Behavior of the van der Waals Force
• Interface layer of a two-component Bose–Einstein condensate
• Generalization of Lieb's variational principle to Bogoliubov–Hartree–Fock theory
• Renormalization Group and Problem of Radiation
• On the shape of interlayer vortices in the Lawrence--Doniach model
• The time-dependent Born-Oppenheimer approximation
• From atoms to crystals: a mathematical journey
• Scattering and small data completeness for the critical nonlinear Schrödinger equation
• ON THE SELF-ADJOINTNESS AND DOMAIN OF PAULI–FIERZ TYPE HAMILTONIANS
• On the Ginzburg‐Landau critical field in three dimensions
• Ginzburg-Landau vortices
• Ground states in non-relativistic quantum electrodynamics.
• Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field
• Boundary behavior of the Ginzburg-Landau order parameter in the surface superconductivity regime
• The external field dependence of the BCS critical temperature
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