Results 1  10
of
13
Time asymptotics of the Schrödinger wave function in timeperiodic potentials
, 2004
"... We study the transition to the continuum of an initially bound quantum particle in R d, d = 1, 2, 3, subjected, for t ≥ 0, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete ..."
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Cited by 8 (5 self)
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We study the transition to the continuum of an initially bound quantum particle in R d, d = 1, 2, 3, subjected, for t ≥ 0, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow. We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows that complete ionization is generic and provides a convenient test for particular cases. When satisfied it implies absence of discrete spectrum and resonances of the associated Floquet operator. As an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property.
Transition to the continuum of a particle in timeperiodic potentials
 ADVANCES IN DIFFERENTIAL EQUATIONS AND MATHEMATICAL PHYSICS
, 2003
"... We present new results for the transition to the continuum of an initially bound quantum particle subject to a harmonic forcing. Using rigorous exponential asymptotics methods we obtain explicit expressions, as generalized Borel summable transseries, for the probability of localization in a specif ..."
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Cited by 6 (3 self)
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We present new results for the transition to the continuum of an initially bound quantum particle subject to a harmonic forcing. Using rigorous exponential asymptotics methods we obtain explicit expressions, as generalized Borel summable transseries, for the probability of localization in a specified spatial region at time t. The transition to the continuum occurs for general compactly supported potentials in one dimension and our results extend easily to higher dimensional systems with spherical symmetry. This of course implies the absence of discrete spectrum of the corresponding Floquet operator.
Local timedecay of solutions to Schrödinger equations with timeperiodic potentials
"... Let H(t) = #+V (t, x) be a timedependent Schrodinger operator on L ). We assume that V (t, x) is 2#periodic in time and decays su#ciently rapidly in space. Let U(t, 0) be the associated propagator. For u 0 belonging to the continuous spectral subspace of L ) for the Floquet operator U( ..."
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Cited by 5 (0 self)
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Let H(t) = #+V (t, x) be a timedependent Schrodinger operator on L ). We assume that V (t, x) is 2#periodic in time and decays su#ciently rapidly in space. Let U(t, 0) be the associated propagator. For u 0 belonging to the continuous spectral subspace of L ) for the Floquet operator U(2#, 0), we study the behavior of U(t, 0)u 0 as in the topology of xweighted spaces, in the form of asymptotic expansions. Generically the leading term is t 3/2 B 1 u 0 . Here B 1 is a finite rank operator mapping functions of x to functions of t and x, periodic in t. If n Z is an eigenvalue, or a threshold resonance of the corresponding Floquet Hamiltonian t + H(t), the leading behavior is t 1/2 B 0 u 0 . The point spectral subspace for U(2#, 0) is finite dimensional. If U(2#, 0)# j = e i2## # j , then U(t, 0)# j represents a quasiperiodic solution.
Ionization for Three Dimensional Timedependent Point Interactions
, 2004
"... We study the time evolution of a three dimensional quantum particle under the action of a timedependent point interaction fixed at the origin. We assume that the “strength ” of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients ..."
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Cited by 4 (2 self)
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We study the time evolution of a three dimensional quantum particle under the action of a timedependent point interaction fixed at the origin. We assume that the “strength ” of the interaction α(t) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of α(t), we prove that there is complete ionization as t → ∞, starting from a bound state at time t = 0. Moreover we prove also that, under the same conditions, all the states of the system are scattering states. Ref. SISSA/ISAS preprint 11/2004/FM 1
Metastable States in Parametrically Excited Multimode Hamiltonian Systems
, 2003
"... Consider a linear autonomous Hamiltonian system with m time periodic bound state solutions. In this paper we study their dynamics under time almost periodic perturbations which are small, localized and Hamiltonian. The analysis proceeds through a reduction of the original infinite dimensional dynam ..."
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Cited by 3 (2 self)
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Consider a linear autonomous Hamiltonian system with m time periodic bound state solutions. In this paper we study their dynamics under time almost periodic perturbations which are small, localized and Hamiltonian. The analysis proceeds through a reduction of the original infinite dimensional dynamical system to the dynamics of two coupled subsystems: a dominant mdimensional system of ordinary differential equations (normal form), governing the projections onto the bound states and an infinite dimensional dispersive wave equation. The present work generalizes previous work of the authors, where the case of a single bound state is considered. Here, the interaction picture is considerably more complicated and requires deeper analysis, due to a multiplicity of bound states and the very general nature of the perturbation’s time dependence. Parametric forcing induces coupling of bound states to continuum radiation modes, of bound states directly to bound states, as well as coupling among bound states, which is mediated by continuum modes. Our analysis elucidates these interactions and we prove the metastability (long life time) and eventual decay of bound states for a large class of systems. The key hypotheses for the analysis are: appropriate local energy decay estimates for the unperturbed evolution operator, restricted to the continuous spectral part of the Hamiltonian, and a matrix Fermi Golden rule condition, which ensures coupling of bound states to continuum modes. Problems of the type considered arise in many areas of application including ionization physics, quantum molecular theory and the propagation of light in optical fibers in the presence of defects.
Nonperturbative Analysis of a Model Quantum System Under Time Periodic Forcing
, 2001
"... We analyze the time evolution of a onedimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation #(t). We show that for generic #(t), which includes the sum of any finite number of harmonics, the system, st ..."
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Cited by 3 (1 self)
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We analyze the time evolution of a onedimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation #(t). We show that for generic #(t), which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as t ##irrespective of the magnitude or frequency of #(t). For the case #(t)=r sin(#t) we find an explicit representation of the probability of ionization. There are however exceptional, very nongeneric #(t), that do not lead to full ionization. These include rather simple explicit periodic #(t) for which the system evolves to a nontrivial localized stationary state related to eigenfunctions of the Floquet operator.
Stucchio C., Ionization in a 1dimensional dipole model
 Rev. Math. Phys
"... We study the evolution of a one dimensional model atom with δfunction binding potential, subjected to a dipole radiation field E(t)x with E(t) a 2π/ωperiodic realvalued function. Starting with ψ(x,t = 0) an initially localized state and E(t) a trigonometric polynomial, complete ionization occurs; ..."
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Cited by 3 (0 self)
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We study the evolution of a one dimensional model atom with δfunction binding potential, subjected to a dipole radiation field E(t)x with E(t) a 2π/ωperiodic realvalued function. Starting with ψ(x,t = 0) an initially localized state and E(t) a trigonometric polynomial, complete ionization occurs; the probability of finding the electron in any fixed region goes to zero. For ψ(x,0) compactly supported and general periodic fields, we construct a resonance expansion. Each resonance is given explicitly as a Gamow vector, and is 2π/ω periodic in time and behaves like the exponentially growing Green’s function near x = ±∞. The remainder is given by an asymptotic power series in t −1/2 with coefficients varying with x. 1
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, 2005
"... Abstract. We study the dispersive properties of the linear Schrödinger equation with a timedependent potential V (t, x). We show that an appropriate integrability condition in space and time on V, i.e. the boundedness of a suitable L r tL s x norm, is sufficient to prove the full set of Strichartz e ..."
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Abstract. We study the dispersive properties of the linear Schrödinger equation with a timedependent potential V (t, x). We show that an appropriate integrability condition in space and time on V, i.e. the boundedness of a suitable L r tL s x norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials V ∈ L r t Ls x. 1.
Energy Transfer via Point Interaction Control
, 2005
"... We consider the problem of energymass transfer from scattering to bound states for a one body quantum system uder the action of a time dependent point interaction. Under suitable assumptions on the initial state of the particle, we prove a result of local controllability of this process 1 ..."
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We consider the problem of energymass transfer from scattering to bound states for a one body quantum system uder the action of a time dependent point interaction. Under suitable assumptions on the initial state of the particle, we prove a result of local controllability of this process 1
Point Interaction Controls for the Energy Transfer in 3D Quantum Systems.
, 2007
"... We consider the problem of energymass transfer from scattering to bound states for a one body quantum system subject to the action of a time dependent point interaction in 3D. Under suitable assumptions on the initial state of the particle, we prove a result of local controllability of this proces ..."
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We consider the problem of energymass transfer from scattering to bound states for a one body quantum system subject to the action of a time dependent point interaction in 3D. Under suitable assumptions on the initial state of the particle, we prove a result of local controllability of this process. Our proof exploits the finite time asymptotic analysis of fractional integral equations and the rank theorem for maps defined on Banach spaces. 1