1-d Quantum harmonic oscillator with time quasi-periodic quadratic perturbation: Reducibility and growth of Sobolev norms

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Abstract

For a family of 1-d quantum harmonic oscillators with a perturbation which is C2 parametrized by EIR and quadratic on x and ix with coefficients quasi-periodically depending on time t, we show the reducibility (i.e., conjugation to time-independent) for a.e. E. As an application of reducibility, we describe the behaviors of solutions in Sobolev space:

  • Boundedness w.r.t. t is always true for “most” EI.

  • For “generic” time-dependent perturbation, polynomial growth and exponential growth to infinity w.r.t. t occur for E in a “small” part of I.

Concrete examples are given for which the growths of Sobolev norm do occur.

Résumé

Pour une famille des oscillateurs harmoniques quantiques unidimensionnels avec une perturbation qui est parametrée par EIR d'une manière C2 et qui est quadratique sur x et ix avec des coefficients qui dépendent du temps t d'une manière quasi-periodique, on montre la réductibilité (c'est-à-dire la conjugaison à l'indépendant du temps) pour presque tout E. Comme une application de la réductibilité, on décrit les comportements des solutions dans l'espace de Sobolev :

  • La bornitude par rapport à t est toujours vraie pour la ≪ plupart ≫ de EI.

  • Pour la perturbation ≪ générique ≫ qui dépend du temps, la croissance polynomiale et la croissance exponentielle à l'infini par rapport à t ont lieu pour E dans une ≪ petite ≫ partie de I.

Des exemples concrets sont donnés pour lesquels les croissances de la norme de Sobolev vraiment ont lieu.

Section snippets

Introduction and main results

Consider the one-dimensional Schrödinger equationitu=ν(E)2H0u+W(E,ωt,x,ix)u,xR, where, we assume that

  • the frequencies ωRd, d1, satisfy the Diophantine condition (denoted by ωDCd(γ,τ) for γ>0, τ>d1):infjZ|n,ωj|>γ|n|τ,nZd{0},

  • the parameter EI, an interval R, and νC2(I,R) satisfies|ν(E)|l1,|ν(E)|l2,EI, for some l1,l2>0,

  • H0 is the one-dimensional quantum harmonic oscillator, i.e.(H0u)(x):=(x2u)(x)+x2u(x),uL2(R),

  • W(E,θ,x,ξ) is a quadratic form of (x,ξ):W(E,θ,x,ξ)=12(a(E,θ)x2

Classical Hamiltonian and quantum Hamiltonian

To give some preliminary knowledge, let us recall the definition of Weyl quantization, which relates the classical and quantum mechanics, and its properties. The conclusions listed in this section can also be found in [7].

The Weyl quantization is the operator OpW:ffW for any symbol f=f(x,ξ), with x,ξRn, where fW is the Weyl operator of f:(fWu)(x)=1(2π)ny,ξRnf(x+y2,ξ)u(y)dydξ,uL2(Rn). In particular, if f is a polynomial of degree at most 2 in (x,ξ), then fW is a polynomial of degree at

An abstract theorem on reducibility

Consider the 1-d time-dependent equationitu=LW(ωt,x,ix)u,xR, where LW(ωt,x,ix) is a linear differential operator, ωTd, d1, and the symbol L(θ,x,ξ) is a quadratic form of (x,ξ) with coefficients analytically depending on θTd. More precisely, we assume thatL(θ,x,ξ)=12(a(θ)x2+b(θ)xξ+b(θ)ξx+c(θ)ξ2), with coefficients a,b,cCω(Td,R).

Through Weyl quantization, the reducibility for the time-dependent PDE can be related to the reducibility for the sl(2,R)-linear system (ω,A()):X=A(ωt)X,ACω

Conjugation between classical hamiltonians

Given two quadratic classical Hamiltonianshj(ωt,x,ξ)=12(aj(ωt)x2+2bj(ωt)xξ+cj(ωt)ξ2),j=1,2, which can be presented ashj(ωt,x,ξ)=12(xξ)JAj(ωt)(xξ),j=1,2 with J:=(0110) and Aj()=(bj()cj()aj()bj())Cω(Td,sl(2,R)). The corresponding equations of motion are given byx=hjξ,ξ=hjx,j=1,2, which are the linear systems (ω,Aj):(x(t)ξ(t))=Aj(ωt)(x(t)ξ(t)).

Proposition 4

If the linear system (ω,A1()) is conjugated to (ω,A2()) by a time quasi-periodic SL(2,R)-transformation, i.e.,ddteZ(ωt)=A1(ωt)eZ(ωt)eZ(

Proof of Theorem 1 and 2

In view of Theorem 6, to show the reducibility of Eq. (1), it is sufficient to show the reducibility of the corresponding sl(2,R)-linear system.

For EI, the symbol of the quantum Hamiltonian (1) ishE(ωt,x,ξ)=ν(E)2(ξ2+x2)+W(E,ωt,x,ξ) which corresponds the quasi-periodic linear system (ω,A0+F0)(xξ)=[(0ν(E)ν(E)0)+(b(E,ωt)c(E,ωt)a(E,ωt)b(E,ωt))](xξ), where, for every EI,A0(E):=(0ν(E)ν(E)0)sl(2,R),F0(E,):=(b(E,)c(E,)a(E,)b(E,))Crω(Td,sl(2,R)) with |EmF0|r<ε0, m=0,1,2, sufficiently

Proof of Theorem 35

In this section, we show that the measure of the subset Oε0 is positive for the equations (3) – (5), which implies the growths of Sobolev norm.

Acknowledgements

The authors would like to thank Prof. D. Bambusi and Prof. J. You for their interests and fruitful discussions which helped to improve the proof. They also appreciate the anonymous referees for helpful remarks and suggestions in modifying this manuscript. Z. Zhao would like to thank the support of Visiting Scholars of Shanghai Jiaotong University (SJTU) and Visiting Scholars of Chern Institute of Mathematics (CIM) during his visits.

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    Z. Liang was partially supported by NSFC grant (11371097, 11571249) and Natural Science Foundation of Shanghai (19ZR1402400).

    2

    The work of Z. Zhao was partially supported by the French government through the National Research Agency (ANR) grant ANR-15-CE40-0001-03 for the project BeKAM and through the UCA JEDI Investments in the Future project managed by ANR with the reference number ANR-15-IDEX-01.

    3

    Q. Zhou was partially supported by support by NSFC grant (11671192, 11771077), the Science Fund for Distinguished Young Scholars of Tianjin (No. 19JCJQJC61300) and Nankai Zhide Foundation.

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