The coupled nonlinear Maxwell–Schrödinger system describes a quantum wave function generating its own electromagnetic field. Komech's lecture notes (arXiv:1907.05786) conjecture that this conservative Hamiltonian system has a global attractor consisting of symmetry-group orbits: every finite-energy solution should converge, locally in L2L^2 and H1H^1, to a stationary orbit (A±(x),e−iω±tψ±(x))(\mathbf A_\pm(x),e^{-i\omega_\pm t}\psi_\pm(x)) in the presence of static external potentials (Eq. (12.1)), and, in the translation-invariant case, should decompose into finitely many moving solitons plus dispersive radiation (Eq. (15.3)). This would give a dynamical underpinning of Bohr's transition postulate and of wave–particle duality. No proof or refutation is known: well-posedness is settled, solitary waves are known to exist, the analogous attraction has been proved only for model systems (Maxwell–Lorentz with a classical spinning charge, U(1)-invariant equations with concentrated nonlinearities, damped-driven finite-dimensional Maxwell–Bloch models), and the authors' own 2023 survey states the asymptotics 'have not been proved for the Maxwell–Schrödinger system'. The problem is open.

Background

The Maxwell–Schrödinger system couples the Schrödinger equation for a charged quantum particle to the classical electromagnetic field that the particle's charge and current densities themselves generate (self-consistent, or semiclassical, electrodynamics). In the source lecture notes, the system is written, for the wave function ψ(x,t)\psi(x,t) and the magnetic vector potential A(x,t)\mathbf A(x,t) with scalar potential A0=(−Δ)−1ρA_0=(-\Delta)^{-1}\rho, as

iℏψ˙(x,t)=12m[−iℏ∇−ec(A+Aext)]2ψ+e(A0+A0ext)ψ,i\hbar\dot\psi(x,t)=\frac{1}{2m}\Bigl[-i\hbar\nabla-\frac{e}{c}(\mathbf A+\mathbf A^{ext})\Bigr]^2\psi+e(A_0+A_0^{ext})\psi,

together with the Maxwell equations for A\mathbf A which, in the Coulomb gauge div⁡A(x,t)≡0\operatorname{div}\mathbf A(x,t)\equiv 0, take the form (source Eqs. (10.2)–(10.3))

1c2A¨(x,t)=ΔA(x,t)+1cPj(x,t),ΔA0(x,t)=−ρ(x,t),\frac{1}{c^{2}}\ddot{\mathbf A}(x,t)=\Delta\mathbf A(x,t)+\frac{1}{c}P\mathbf j(x,t),\qquad \Delta A_0(x,t)=-\rho(x,t),

where PP is the orthogonal projection onto divergence-free vector fields, and which are sourced by the charge density ρ=e∣ψ∣2\rho=e|\psi|^2 and the current j=emRe⁡{ψˉ[−iℏ∇−ec(A+Aext)]ψ}\mathbf j=\frac{e}{m}\operatorname{Re}\{\bar\psi[-i\hbar\nabla-\frac{e}{c}(\mathbf A+\mathbf A^{ext})]\psi\}; here e,m,c,ℏe,m,c,\hbar are the electron charge, electron mass, speed of light, and Planck constant, and Aext,A0ext\mathbf A^{ext},A_0^{ext} are static external potentials (for an atom, the nuclear Coulomb potential A0ext=−Ze/∣x∣A_0^{ext}=-Ze/|x|). The system is Hamiltonian (conservative): for static external potentials its energy — the Hamilton functional (source Eq. (10.6))

H(Π,A,ψ)=12[∥1cΠ∥2+∥rot⁡A∥2]+(ψ,HAψ),HA=12m[−iℏ∇−ec(A+Aext)]2+e(12A0+A0ext),\mathcal H(\bm\Pi,\mathbf A,\psi)=\frac{1}{2}\Bigl[\Bigl\|\frac{1}{c}\bm\Pi\Bigr\|^{2}+\|\operatorname{rot}\mathbf A\|^{2}\Bigr]+(\psi,H_{\mathbf A}\psi),\qquad H_{\mathbf A}=\frac{1}{2m}\Bigl[-i\hbar\nabla-\frac{e}{c}(\mathbf A+\mathbf A^{ext})\Bigr]^{2}+e\Bigl(\frac{1}{2}A_0+A_0^{ext}\Bigr),

with field momentum Π=A˙\bm\Pi=\dot{\mathbf A}, L2L^2 norm ∥⋅∥\|\cdot\| and inner product (⋅,⋅)(\cdot,\cdot), and with the factor 12A0\frac12 A_0 in HAH_{\mathbf A} accounting for the self-interaction of the induced charge density — is conserved, so there is no dissipation that could drive relaxation; throughout, 'finite-energy solution' means a solution with H<∞\mathcal H<\infty. It is also invariant under the unitary group U(1)U(1) acting as ψ↦eiθψ\psi\mapsto e^{i\theta}\psi (phase gauge symmetry), which by Noether's theorem implies charge conservation.

Because of the U(1)U(1) symmetry, the natural long-time limits are not static states but stationary orbits: finite-energy solutions of the form (A(x),e−iωtψ(x))(\mathbf A(x),e^{-i\omega t}\psi(x)) with ω∈R\omega\in\mathbb R, i.e. time-periodic-up-to-phase solutions generalizing the bound states of the linear Schrödinger atom (Bohr's stationary orbits). Such stationary orbits are known to exist: radial solitary waves with fixed L2L^2-norm for the 3D system were constructed by Coclite–Georgiev (2004), and bound states in one dimension by Chapling (2016); for the translation-invariant system without external potentials, moving solitons carry an additional space–time phase factor (see item (ii) of the Problem Statement), the standing (v=0v=0) case being the one explicitly constructed in these works. Since the 1990s, Komech and collaborators have developed a theory of global attractors for Hamiltonian partial differential equations, in which relaxation is caused not by damping but by energy radiation: nonlinear frequency mixing transfers energy from bound frequencies to the continuous spectrum, from which it disperses to infinity. With this mechanism, global attraction to stationary orbits (or to solitons) was proved for a list of model systems: a wave equation coupled to nonlinear oscillators and to a charged particle (Maxwell–Lorentz type, i.e. the classical equations of motion of a charged point particle coupled to the electromagnetic field), U(1)-invariant Klein–Gordon, Dirac, and wave equations with concentrated nonlinearities (nonlinear terms supported at a single point, modelling the particle as a point source), and their discrete space-time approximations; the analogous soliton attraction for the Maxwell–Lorentz equations with a rotating classical charge in 2D and 3D was proved in 2023. Notably, even in these models the proved statement is attraction to the whole solitary manifold (the union of all stationary orbits), while convergence to an individual orbit remains open even there. The 2019 lecture notes and the 2020–2023 surveys by the same group elevate these results to Conjecture A: for 'generic' GG-invariant Hamiltonian nonlinear PDEs, the set of GG-orbits of stationary states should form a global attractor for all finite-energy solutions, convergence being measured in local seminorms (i.e. on every bounded spatial region). For the physical Maxwell–Schrödinger system this conjecture is the motivation of the whole program, and its proof is explicitly stated to be an open problem; the same asymptotics would give, in the translation-invariant case, a soliton interpretation of wave–particle duality and a possible dynamical route to the Born probabilistic rule, which the notes likewise leave unresolved.

Problem Statement

Prove or disprove the global-attraction asymptotics conjectured for the coupled nonlinear Maxwell–Schrödinger system (Komech, Lectures on Quantum Mechanics for mathematicians, arXiv:1907.05786, Eqs. (12.1) and (15.3); Conjecture A specialized to the symmetry group U(1)U(1)):

(i) Static external potentials (atomic regime, Eq. (12.1)): show that every finite-energy solution (A,A0,ψ)(\mathbf A,A_0,\psi) of the coupled Maxwell–Schrödinger system in R3\mathbb R^3 in the Coulomb gauge div⁡A=0\operatorname{div}\mathbf A=0, with time-independent external potentials, admits the long-time limits

(A(x,t),A0(x,t),ψ(x,t))∼(A±(x),A0,±(x),e−iω±tψ±(x)),t→±∞,(\mathbf A(x,t),A_0(x,t),\psi(x,t))\sim(\mathbf A_\pm(x),A_{0,\pm}(x),e^{-i\omega_\pm t}\psi_\pm(x)),\qquad t\to\pm\infty,

where (A±,e−iω±tψ±)(\mathbf A_\pm,e^{-i\omega_\pm t}\psi_\pm) are stationary orbits, with convergence in the L2L^2 norm (square-integrable functions) and the H1H^1 norm (the Sobolev norm, which also controls the first spatial derivatives) on every bounded region of R3\mathbb R^3; the Coulomb gauge leaves a residual freedom χ(t)\chi(t) which shifts A0,±A_{0,\pm} and ω±\omega_\pm by constants and maps stationary orbits to stationary orbits, so the limiting orbit is determined up to this freedom; or exhibit a finite-energy solution for which this asymptotics fails.

(ii) Translation-invariant regime (Eq. (15.3)): with Aext=A0ext=0\mathbf A^{ext}=A_0^{ext}=0, show that every finite-energy solution decomposes at t→±∞t\to\pm\infty into a finite sum of moving solitons plus dispersive radiation, i.e., as in the source Eq. (15.3),

A(x,t)∼∑kA±k(x−v±kt)+A±(x,t),ψ(x,t)∼∑kψ±k(x−v±kt) eiΦ±k(x,t)+ψ±(x,t),t→±∞,\mathbf A(x,t)\sim\sum_k\mathbf A^k_\pm(x-v^k_\pm t)+\mathbf A_\pm(x,t),\qquad \psi(x,t)\sim\sum_k\psi^k_\pm(x-v^k_\pm t)\,e^{i\Phi^k_\pm(x,t)}+\psi_\pm(x,t),\qquad t\to\pm\infty,

where the soliton profiles carry explicit space–time phases Φ±k(x,t)\Phi^k_\pm(x,t) (a bare translated profile ψk(x−vt)\psi^k(x-vt) is not by itself a solution of this Galilean-covariant system) and A±(x,t),ψ±(x,t)\mathbf A_\pm(x,t),\psi_\pm(x,t) are the dispersive radiation; convergence is meant again in local L2L^2/H1H^1 seminorms, in the comoving frame for the soliton terms; or disprove this decomposition.

The same question for the Maxwell–Pauli system (the spin-12\frac12 variant, where the Schrödinger equation is replaced by the Pauli equation with the Pauli matrices coupling spin to the field) is equally open and constitutes part of the same conjecture; it is noted here as context — items (i)–(ii) concern the Maxwell–Schrödinger system. A resolution must handle all finite-energy solutions, not a perturbative subclass, and both time directions t→±∞t\to\pm\infty.

References

  1. A. Komech, 'Lectures on Quantum Mechanics for mathematicians', arXiv:1907.05786 (2019). https://arxiv.org/abs/1907.05786
  2. A. Komech, 'On quantum jumps and attractors of the Maxwell–Schrödinger equations', Annales mathématiques du Québec 46 (2022) 139–159, DOI 10.1007/s40316-021-00179-1; arXiv:1907.04297. https://doi.org/10.1007/s40316-021-00179-1
  3. A. Comech, A. Komech, E. Kopylova, 'Attractors of Hamiltonian Nonlinear Partial Differential Equations', in: Partial Differential Equations and Functional Analysis (Trends in Mathematics), Birkhäuser/Springer, Cham (2023) 197–244, DOI 10.1007/978-3-031-33681-2_22; arXiv:2212.14152. https://doi.org/10.1007/978-3-031-33681-2_22
  4. A. Komech, E. Kopylova, 'Attractors of nonlinear Hamiltonian partial differential equations', Russian Mathematical Surveys 75:1 (2020) 1–87, DOI 10.1070/rm9900. https://doi.org/10.1070/rm9900
  5. Y. Guo, K. Nakamitsu, W. Strauss, 'Global finite-energy solutions of the Maxwell–Schrödinger system', Communications in Mathematical Physics 170 (1995) 181–196, DOI 10.1007/bf02099444. https://doi.org/10.1007/bf02099444
  6. M. Nakamura, T. Wada, 'Global existence and uniqueness of solutions to the Maxwell–Schrödinger equations', Communications in Mathematical Physics 276 (2007) 315–339, DOI 10.1007/s00220-007-0337-9. https://doi.org/10.1007/s00220-007-0337-9
  7. I. Bejenaru, D. Tataru, 'Global wellposedness in the energy space for the Maxwell–Schrödinger system', Communications in Mathematical Physics 288 (2009) 145–198, DOI 10.1007/s00220-009-0765-9. https://doi.org/10.1007/s00220-009-0765-9
  8. P. Antonelli, P. Marcati, R. Scandone, 'Global well-posedness for the non-linear Maxwell-Schrödinger system', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5) 23 (2022), no. 3, 1293–1324, DOI 10.2422/2036-2145.202010_033; arXiv:1910.05297. https://doi.org/10.2422/2036-2145.202010_033
  9. G. M. Coclite, V. Georgiev, 'Solitary waves for Maxwell-Schrödinger equations', Electronic Journal of Differential Equations, Vol. 2004 (2004), No. 94, 1–31; arXiv:math/0303142. https://eudml.org/doc/124709
  10. R. Chapling, 'Bound States of the One-Dimensional Maxwell–Schrödinger Equations', arXiv:1608.02637 (2016). https://arxiv.org/abs/1608.02637
  11. E. Kopylova, A. Komech, 'On global attraction to solitons for 3D Maxwell-Lorentz equations', arXiv:2308.03611 (2023). https://arxiv.org/abs/2308.03611
  12. A. Komech, 'On absorbing set for 3D Maxwell–Schrödinger damped driven equations in bounded region', arXiv:2104.10723 (2021). https://arxiv.org/abs/2104.10723
  13. N. Leopold, P. Pickl, 'Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian', SIAM Journal on Mathematical Analysis 52:5 (2020) 4900–4936, DOI 10.1137/19m1307639; arXiv:1609.01545. https://doi.org/10.1137/19m1307639