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A weak spectral condition for the controllability of the bilinear Schr\"odinger equation with application to the control of a rotating planar molecule

Ugo Boscain, Marco Caponigro, Thomas Chambrion, Mario Sigalotti

arXiv:1101.4313v4math.OCcs.SYeess.SYmath.AP

TL;DR

Exact controllability is unavailable in general for infinite-dimensional Schrödinger systems, while prior approximate-controllability results impose spectral restrictions such as nonresonant gaps and broad coupling requirements. The proof uses fine controllability of finite-dimensional Galerkin approximations, with a time-dependent transformation, convexification, and phase control before passing to the infinite-dimensional system. A non-resonant connectedness chain yields approximate controllability, and the planar-molecule system is not controllable when either control is fixed to zero.

  • Problem

    Exact controllability is unavailable in general for infinite-dimensional Schrödinger systems, while prior approximate-controllability results impose spectral restrictions such as nonresonant gaps and broad coupling requirements.

    Source: §1 · p. 2

  • Method

    The proof uses fine controllability of finite-dimensional Galerkin approximations, with a time-dependent transformation, convexification, and phase control before passing to the infinite-dimensional system.

    Source: §1 · p. 4

  • Results

    A non-resonant connectedness chain yields approximate controllability, and the planar-molecule system is not controllable when either control is fixed to zero.

    Source: §1 · p. 5, §2.2 · p. 7

  • Takeaways & Limitations

    Convexification of the transformed system enables approximate controllability with less restrictive non-resonance hypotheses.

    Source: §1 · p. 4

    For the planar-molecule application, fixing one control to zero prevents controllability because of parity-based uncoupling.

    Source: §8 · p. 26

Abstract

from arXiv · show

In this paper we prove an approximate controllability result for the bilinear Schr\"odinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schr\"odinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the $L^{1}$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.

1 Introduction

The paper establishes approximate controllability under a graph-based spectral condition and applies the result to a planar bipolar molecule controlled by two orthogonal fields.

1 Introduction

  • The framework covers Schrödinger systems beyond a Laplacian-plus-potential model, including dynamics where the uncontrolled operator has a more general form.The paper describes H0 and H1 as self-adjoint operators and the scalar control as an external field.Source: §1 · pp. 1–2
  • Earlier results impose stronger spectral or coupling assumptions, while exact controllability is unavailable in general for infinite-dimensional systems.The paper contrasts its hypotheses with, which requires rationally nonresonant gaps and coupling between all eigenvectors.Source: §1 · pp. 2–3
  • Approximate controllability follows when nonzero couplings contain a connected graph whose selected edges satisfy the paper’s spectral condition.The criterion allows a discrete spectrum that need not be simple and excludes direct coupling between degenerate energy levels.Source: §1 · p. 3
  • The result permits unbounded control operators, density-matrix and simultaneous controllability extensions, and does not require a simple spectrum.Sparse nonzero couplings can also reduce the number of spectral conditions that must be checked.Source: §1 · p. 3
  • The proof passes from finite-dimensional Galerkin systems to the infinite-dimensional equation, using convexification and a separate phase-control step.The method keeps higher-mode projections arbitrarily small along trajectories and yields estimates for the control’s L1 norm.Source: §1 · pp. 4–5
  • For the planar bipolar molecule, two orthogonal time-varying fields provide the controls; fixing either control to zero prevents controllability.The proposed steering argument uses controls along a direction determined by the symmetry of the initial state, with the ground state serving as an intermediate target.Source: §1 · pp. 4–5

2 Framework and main results

The section sets up an abstract framework for the Schrödinger control system and states approximate controllability under a non-resonant connectedness-chain condition. It also gives extensions to simultaneous control and density matrices, alongside an L1 control estimate.

2.1 Settings and notations

  • The framework uses skew-adjoint operators A and B and defines solutions through the propagator for piecewise-constant controls.It applies to possibly unbounded operators under the stated assumptions.Source: §2.1 · pp. 5–6

2.2 Main results

  • A non-resonant connectedness chain suffices for approximate controllability, without requiring a simple spectrum.The chain condition requires distinct gaps for its edges relative to other coupled levels.Source: §2.2 · p. 7
  • Proposition 2.8 provides an L1-norm estimate for controls steering one eigenvector into an ε-neighborhood of another along a chain edge.The proposition assumes a non-resonant connectedness chain and controls in [0, δ].Source: §2.2 · p. 7

2.3 Simultaneous controllability and controllability in the sense of density matrices

  • Theorem 2.11 gives approximate simultaneous controllability under the same non-resonant connectedness-chain condition.The result also yields approximate controllability for density matrices.Source: §2.3 · p. 8

3 Finite dimensional case

For finite-dimensional systems, the section gives a sufficient controllability condition based on a non-resonant connectedness chain. It relates the condition to Lie-algebra controllability and identifies the resulting reachable matrix groups.

3 Finite dimensional case

  • The standard SU(n) criterion is that A and B generate a Lie algebra containing su(n), while the section proposes an easier sufficient condition.The authors describe their condition as slightly improving earlier sufficient conditions.Source: §3 · p. 9
  • A non-resonant connectedness chain with distinct eigenvalues on its edges ensures controllability on the unit sphere and SU(n), when U contains at least two points.If tr A or tr B is nonzero, controllability also holds on U(n).Source: §3 · p. 9
  • The proof obtains su(n) from the connectedness structure, then uses trajectories on U(n) to connect points on the unit sphere.The generated algebra is u(n) when either A or B has nonzero trace.Source: §3 · pp. 9–10

4 Convexification procedure

The proof develops transformations and finite-dimensional control constructions to establish the main controllability result. These include a time reparametrization, a convexification argument, an auxiliary system, and a reordered eigenbasis with connected chains at every finite order.

4 Convexification procedure

  • The proof sections develop the convexification procedure to establish Theorem 2.11 in the infinite-dimensional case.Source: §4 · p. 10

4.1 Time-reparametrization

  • Time reparametrization transfers trajectories of the original system to a transformed system, with transformed time equal to the control's L1 norm.Controllability for controls in (0, δ] is equivalent to controllability in the transformed system with controls in [1/δ, ∞).Source: §4.1 · p. 10

4.2 Convexification

  • The convexification argument constructs widely separated times whose coordinate averages cancel selected coordinates while the first coordinate remains bounded away from zero.The construction places the resulting average in the convex hull of the trajectory.Source: §4.2 · pp. 11–12

4.3 An auxiliary system

  • The auxiliary system is linear in x and preserves the unit sphere because its control matrices are skew-adjoint with zero trace.The system can also be lifted to SU(n).Source: §4.3 · p. 13

4.4 Existence of a n-connectedness chain

  • A recursively constructed reordering of the eigenbasis makes the connectedness chain an n-connectedness chain for every n.The proposition uses a bijection of the eigenvector indices.Source: §4.4 · p. 13

5 Modulus tracking

The proof builds infinite-dimensional modulus tracking from tracking results for finite-dimensional systems, passing from SU(n) trajectories through Galerkin approximations to the original system. It also derives L1 control estimates for eigenvector transitions.

5 Modulus tracking

  • Under the stated assumptions, a piecewise-constant control can approximate a prescribed unitary curve on any finite span of eigenvectors, up to time reparametrization.The proposition requires (A) and a non-resonant connectedness chain.Source: §5 · p. 13

5.1 Tracking: definitions and general facts

  • A system tracks a curve up to time reparametrization when its controlled trajectory stays within any chosen tolerance after an increasing change of time.Without time reparametrization, the change of time must be the identity.Source: §5.1 · p. 14

5.2 Tracking in (Σn)

  • If the connectedness chain is non-resonant and n-connected for every n, each finite system (Σn) can track any curve in SU(n).The tracking criterion uses symmetric available vector fields and a Lie algebra spanning the tangent space.Source: §5.2 · p. 14, §5.1 · p. 14

5.3 Tracking trajectories of (Σn) in (ΘN)

  • For N ≥ n, system (ΘN) can track a trajectory of (Σn) without time reparametrization when projected onto its first n components.The construction uses piecewise-constant controls valued in [1/δ, +∞).Source: §5.3 · p. 15

5.4 Tracking trajectories of (Σn) in the original system

  • The finite-dimensional tracking result extends to the original system, whose propagator approximates the target trajectory on the first n eigenstate components.The proof chooses a sufficiently large Galerkin dimension using the square-summability of coupling coefficients.Source: §5.4 · p. 17

5.5 Proof of modulus tracking

  • Finite SU(n) curves approximate the relevant matrix coefficients of a unitary curve uniformly, enabling the proof to assemble modulus tracking for the infinite-dimensional system.The construction first approximates the target columns in a finite-dimensional span and completes them to a basis.Source: §5.5 · p. 18

5.6 Estimates of the L1 norm of the control

  • The L1 estimates focus on controls that implement eigenvector permutations; each chain transposition exchanges a pair up to phase while leaving the other eigenstates invariant.The stated conclusion uses the transition-time result and the estimate ν > 2/5.Source: §5.6 · pp. 19–20

6 Phase tuning

The section uses phase tuning to turn approximate steering up to phase into approximate controllability, including simultaneous control when suitable spectral orbits are dense in a torus. It establishes that density condition for the spectral cases considered.

6.1 Phase tuning for the control of a single wave function

  • For a single state, steering toward a common eigenvector, waiting with zero control, and reversing a second steering path aligns the phases to approximate the target.The construction applies the tracking result to the forward and time-reversed systems.Source: §6.1 · p. 21

6.2 Phase tuning for simultaneous control

  • Simultaneous approximate controllability follows when, for every number of states and tolerance, a joint spectral orbit is dense in the corresponding phase torus.The argument uses a torus of dimension r to tune the phases of r states.Source: §6.2 · p. 21
  • When A has infinitely many distinct eigenvalues, the paper establishes the existence of distinct indices whose joint orbit is η-dense in the associated torus.This density property supplies the spectral condition used for simultaneous control.Source: §6.2 · pp. 21–22

6.2.2 The spectrum of A is unbounded

  • For an unbounded spectrum, choosing successive eigenvalues with magnitudes much larger than the previous ones yields the required torus-density approximation.The argument uses recurrence to extend density from a lower-dimensional torus.Source: §6.2.2 · p. 22

6.2.3 The spectrum of A is bounded and dimQ

  • For any previously chosen indices, one can select the next so the joint period τ(k1, . . . , km) ≫τ(k1, . . . , km−1); if the existing set is η-dense, adding the next torus preserves η-density in the expanded torus.Source: §6.2.3 · p. 23

6.2.4 The spectrum of A is bounded and 1 < dimQ

  • For every η > 0, the cited argument chooses indices ¯k1, …, ¯kr and shows their orbit {(eitλ¯k1, …, eitλ¯kr) | t ∈ R} is rmη-dense in Tr.Source: §6.2.4 · p. 24

7 Example: Infinite potential well

The paper applies its controllability results to a particle in an infinite potential well. Because resonances prevent direct application, it perturbs the drift and then obtains simultaneous controllability and L1 estimates.

7 Example: Infinite potential well

  • For a particle confined to (−1/2, 1/2), the paper extends earlier controllability results to simultaneous control and estimates the L1 norm of transfer controls.Earlier work cited here includes exact controllability on dense subsets and approximate controllability using Lyapunov techniques.Source: §7 · p. 24
  • Numerous resonances prevent direct application of the general result to the well model, so the authors shift the drift to Aη = A + ηB.The shifted model uses controls in [−η, δ − η].Source: §7 · p. 25
  • The shifted spectrum yields a non-resonant connectedness chain, allowing simultaneous approximate controllability and L1 estimates to be applied to the well.The chain consists of neighboring eigenstate pairs for a suitable 0 < η < ε.Source: §7 · p. 25

8 Orientation of a bipolar molecule in the plane by means of two external fields

The paper proves approximate controllability for the planar molecule using two orthogonal fields. Its proof controls even or odd components within fixed reflection-symmetry subspaces, then iteratively brings any unit-norm state close to the constant ground state.

8 Orientation of a bipolar molecule in the plane by means of two external fields

  • The two fields are necessary: setting either one to zero leaves parity-based uncoupled wave functions.The cos(θ) potential does not couple odd and even functions, while sin(θ) does not couple functions of the same parity.Source: §8 · p. 26
  • For a fixed α, piecewise-constant controls can approximately steer either parity component to any target of the same norm.The even and odd components are treated separately, with their L2 norms preserved.Source: §8 · p. 26
  • The proof applies Theorem 2.6 to a subsystem formed by restricting the wave functions and admissible controls.For initial states even with respect to an axis, the control vector varies along that axis's direction (cos α, sin α).Source: §8 · pp. 26–27
  • With a fixed control direction, the circle Laplacian and the coupling operator preserve the corresponding even and odd subspaces.The coupling operator is multiplication by −i cos(θ−α).Source: §8 · p. 27
  • The reduced system has adjacent-mode couplings with frequency gaps λk+1−λk=−2k−1, giving a non-resonant connectedness chain.This permits application of Theorem 2.6 to steer within the selected symmetry subspace.Source: §8 · p. 27
  • Proposition 8.2 establishes that system (8.1) is approximately controllable, with the proof reducing this to steering every wave function of norm one arbitrarily close to a constant function.Source: §8 · p. 27
  • Repeatedly choosing a reflection axis reduces the residual component's norm until the state is ε-close to 1/√(2π).Each iteration improves the constant component by an amount bounded below independently of the residual, so only finitely many iterations are needed.Source: §8 · pp. 27–28

A Appendix: relations between controllability notions

The appendix relates three stated controllability conditions, including a result that condition (iii) implies density-matrix controllability under an additional assumption. Its proof uses time reversibility and approximations on finite subsets of states.

A Appendix: relations between controllability notions

  • The appendix states finite-state approximation criteria quantified over every r, ε, and target unitary, with condition (iii) allowing phases for the first r states.The displayed fragments identify these as conditions (i) and (iii), respectively.Source: A Appendix: relations between… · p. 28
  • Under Proposition A.1's hypotheses and purely imaginary matrix elements ⟨φj,Bφk⟩, condition (iii) implies condition (ii).Condition (ii) is identified as controllability in the sense of density matrices.Source: A Appendix: relations between… · pp. 28, 30
  • The proof invokes time reversibility: reversing a solution in time gives a solution under the time-reversed control.This property is stated under the condition on B in Proposition A.2.Source: A Appendix: relations between… · p. 30
  • For density matrices, the proof truncates the state expansion so the omitted tail has total weight below ε, then approximates the action on the retained states.It uses finite collections of eigenstates and controls to approximate the corresponding unitary action.Source: A Appendix: relations between… · p. 30
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