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arxiv: 2405.18094 · v3 · submitted 2024-05-28 · 🧮 math.NA · cs.NA

A space-time variational formulation for the many-body electronic Schr{\"o}dinger evolution equation

Mi-Song Dupuy (LJLL (UMR\_7598)) , Virginie Ehrlacher (MATHERIALS , CERMICS UMR 9032) , Cl\'ement Guillot (MATHERIALS This is my paper

Pith reviewed 2026-05-24 01:22 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords time-dependent Schrödinger equationspace-time variational formulationleast-squares minimizationGalerkin discretizationmany-body quantum dynamicsdynamical low-rank approximationvariational principle
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The pith

The many-body time-dependent Schrödinger equation solution is the minimizer of a global space-time quadratic functional.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the solution of the time-dependent Schrödinger equation equals the minimizer of a quadratic problem posed over the full space-time domain. This reformulation uses a least-squares approach chosen so that its Euler-Lagrange equations recover the original evolution even when the potential contains Coulomb singularities. A reader would care because the variational statement opens the door to Galerkin schemes that discretize space and time together and to low-rank approximations whose existence can be proved globally in time rather than step by step.

Core claim

We prove that the solution of the time-dependent Schrödinger equation can be expressed as the solution of a global space-time quadratic minimization problem that is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation. The present analysis can be applied to the electronic many-body time-dependent Schrödinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities.

What carries the argument

Global space-time quadratic minimization problem obtained from a least-square formulation whose minimizer recovers the TDSE solution.

Load-bearing premise

There exists an appropriate least-square formulation whose minimizer exactly recovers the many-body TDSE solution even in the presence of Coulomb singularities.

What would settle it

A direct computation on the hydrogen atom showing that the candidate quadratic functional is not minimized by the known exact TDSE solution.

read the original abstract

We prove in this paper that the solution of the time-dependent Schr{\"o}dinger equation can be expressed as the solution of a global space-time quadratic minimization problem that is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation. The present analysis can be applied to the electronic many-body time-dependent Schr{\"o}dinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities. We motivate the interest of the present approach with two goals: first, the design of Galerkin space-time discretization methods; second, the definition of dynamical low-rank approximations following a variational principle different from the classical Dirac-Frenkel principle, and for which it is possible to prove the global-in-time existence of solutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves that the solution of the time-dependent many-body electronic Schrödinger equation (with Coulomb singularities) is the unique global minimizer of a space-time quadratic least-squares functional. The Euler-Lagrange equation of this functional recovers the TDSE together with the initial condition. The formulation is designed to support Galerkin space-time discretizations and a new variational principle for dynamical low-rank approximations that guarantees global-in-time existence.

Significance. If the central result holds, the work supplies a self-contained variational framework for space-time methods on the TDSE that is distinct from the Dirac-Frenkel principle and directly accommodates the Coulomb singularities of the many-body problem. The explicit construction of the functional, the function space, and the proof that the minimizer satisfies the TDSE are strengths that could enable new discretization schemes and low-rank approximations with rigorous existence guarantees.

minor comments (3)
  1. [§2] §2: the precise definition of the space-time function space (including the handling of the initial condition in the norm) should be stated before the minimization problem is introduced, to make the well-posedness argument easier to follow.
  2. The proof that the Euler-Lagrange equation recovers both the TDSE and the initial condition is given, but a short remark on why the Coulomb singularity does not destroy the density of the test functions would help readers outside the immediate field.
  3. Figure 1 (schematic of the space-time cylinder) would benefit from an explicit indication of the initial-time slice and the lateral boundary conditions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its potential for space-time Galerkin methods and alternative low-rank approximations, and the recommendation of minor revision. No specific major comments appear in the provided report, so we have no individual points requiring rebuttal or revision at this time.

Circularity Check

0 steps flagged

Self-contained mathematical proof; no reduction to inputs or self-citations

full rationale

The paper presents an explicit construction of a space-time least-squares functional whose Euler-Lagrange equation recovers the TDSE (including initial condition) for the many-body electronic case with Coulomb singularities. The argument is a direct variational equivalence proof internal to the manuscript, with no fitted parameters, no renaming of empirical patterns, and no load-bearing self-citations whose validity would depend on the present result. The derivation chain therefore does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the paper invokes standard existence of TDSE solutions and the suitability of a least-squares reformulation, but supplies no further free parameters or invented entities.

axioms (1)
  • domain assumption The time-dependent many-body Schrödinger equation possesses solutions that can be recovered exactly by minimization of an appropriate quadratic least-squares functional.
    The central claim rests on this equivalence holding for Coulomb-singular potentials.

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