Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 902506, 30 pages
doi:10.1155/2009/902506
Research Article

Quantum Energy Expectation in Periodic Time-Dependent Hamiltonians via Green Functions

César R. de Oliveira1,2 and Mariza S. Simsen3

1Departamento de Matemática, Universidade Federal de São Carlos, 13560-970 São Carlos, SP, Brazil
2Department of Mathematics, University of British Columbia, Vancouver, BC, CanadaV6T 1Z2
3Instituto de Ciências Exatas, Universidade Federal de Itajubá, 37500-000 Itajubá, MG, Brazil

Received 10 November 2008; Revised 29 January 2009; Accepted 21 February 2009

Academic Editor: Edson Denis Leonel

Copyright © 2009 César R. de Oliveira and Mariza S. Simsen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑈 𝐹 be the Floquet operator of a time periodic Hamiltonian 𝐻 ( 𝑡 ) . For each positive and discrete observable 𝐴 (which we call a probe energy), we derive a formula for the Laplace time average of its expectation value up to time 𝑇 in terms of its eigenvalues and Green functions at the circle of radius 𝑒 1 / 𝑇 . Some simple applications are provided which support its usefulness.

1. Introduction

Consider a general periodically driven quantum hamiltonian system 𝐻 ( 𝑡 ) = 𝐻 0 + 𝑉 ( 𝑡 ) ( 1 . 1 ) with period 𝜏 acting in a separable Hilbert space ℋ , and let 𝑈 𝐹 denote its Floquet operator, so that if 𝜉 is the initial state (at time zero) of the system, then 𝑈 𝑚 𝐹 𝜉 is this state at time 𝑚 𝜏 . Typically, the unperturbed hamiltonian 𝐻 0 is assumed to have purely point spectrum so that the same is true for 𝑒 − 𝑖 𝜏 𝐻 0 . What happens when 𝐻 0 is perturbed by 𝑉 ( 𝑡 ) ? A natural physical question is if the expectation values of the unperturbed energy 𝐻 0 remain bounded when 𝑉 ( 𝑡 ) ≠ 0 . This question is formulated based on many physical models, in particular on the Fermi accelerator in which a particle can acquire unbounded energy from collisions with a heavy periodically moving wall. Here quantum mechanics is considered and, more precisely, if s u p 𝑚 ∈ ℕ | | 𝑈 𝑚 𝐹 𝜉 , 𝐻 0 𝑈 𝑚 𝐹 𝜉 | | ( 1 . 2 ) is finite or not, where 𝜉 ∈ d o m 𝐻 0 ⊂ ℋ , the domain of 𝐻 0 .

Motivated by models with hamiltonians as above 𝐻 ( 𝑡 ) = 𝐻 0 + 𝑉 ( 𝑡 ) , one is suggested to probe quantum (in)stability through the behavior of an “abstract energy operator" which we call a probe operator and will be represented by a positive, unbounded, self-adjoint operator 𝐴 ∶ d o m 𝐴 ⊂ ℋ → ℋ and with discrete spectrum, 𝐴 𝜑 𝑛 = 𝜆 𝑛 𝜑 𝑛 , ( 1 . 3 ) 0 ≤ 𝜆 𝑛 < 𝜆 𝑛 + 1 , such that if 𝑈 𝑚 𝐹 𝜉 ∈ d o m 𝐴 for all 𝑚 ∈ ℕ , then, for each 𝑚 , the expectation value 𝐸 𝐴 𝜉 ( 𝑚 ) = ⟨ 𝑈 𝑚 𝐹 𝜉 , 𝐴 𝑈 𝑚 𝐹 𝜉 ⟩ is finite. It is convenient to write 𝐸 𝐴 𝜉 ( 𝑚 ) = + ∞ if 𝑈 𝑚 𝐹 𝜉 does not belong to d o m 𝐴 .

We say the system is A-dynamically stable when 𝐸 𝐴 𝜉 ( 𝑚 ) is a bounded function of time 𝑚 , and A-dynamically unstable otherwise (usually we say just (un)stable). If the function 𝐸 𝐴 𝜉 ( 𝑚 ) is not bounded, one can ask about its asymptotic behavior, that is, how does 𝐸 𝐴 𝜉 ( 𝑚 ) behave as 𝑚 goes to infinity? Usually this is a very difficult question and sometimes the temporal average of 𝐸 𝐴 𝜉 ( 𝑚 ) is considered, as we will do in this work.

Quantum systems governed by a time periodic hamiltonian have their dynamical stability often characterized in terms of the spectral properties of the corresponding Floquet operator. As in the autonomous case, the presence of continuous spectrum is a signature of unstable quantum systems; this is a rigorous consequence of the famous RAGE theorem [1], firstly proved for the autonomous case and then for time-periodic hamiltonians [2, 3]. At first sight a Floquet operator with purely point spectrum would imply stability, but one should be alerted by examples with purely point spectrum and dynamically unstable [4–6] in the autonomous case and, recently, also a time-periodic model with energy instability [7] was found.

Dynamical stability of time-dependent systems was studied, for example, in references [2, 8–19]. In [14] it was proved that the applicability of the KAM method gives a uniform bound at the expectation value of the energy for a class of time-periodic hamiltonians considered in [20].

For hamiltonians 𝐻 ( 𝑡 ) = 𝐻 0 + 𝑉 ( 𝑡 ) , not necessarily periodic, with 𝐻 0 a positive self-adjoint operator whose spectrum consists of separated bands { 𝜎 𝑗 } ∞ 𝑗 = 1 such that 𝜎 𝑗 ⊂ [ 𝜆 𝑗 , Λ 𝑗 ] , upper bounds of the type ⟨ 𝑈 ( 𝑡 , 0 ) 𝜓 , 𝐻 0 𝑈 ( 𝑡 , 0 ) 𝜓 ⟩ ≤ c t e 𝑡 ( 1 + 𝛼 ) / 𝑛 𝛼 ( 1 . 4 ) were obtained in [10] if the gaps 𝜆 𝑗 + 1 − Λ 𝑗 grow like 𝑗 𝛼 , with 𝛼 > 0 , and if 𝑉 ( 𝑡 ) is strongly 𝐶 𝑛 with 𝑛 ≥ [ ( 1 + 𝛼 ) / 2 𝛼 ] + 1 . The proof is based on adiabatic techniques that require smooth time dependence and therefore do not apply to kicked systems. In [11, 13] upper bounds complementary to those of [10] described above are obtained.

In [2, 8, 9, 15] stability results are obtained through topological properties of the orbits 𝜉 ( 𝑡 ) = 𝑈 ( 𝑡 , 0 ) 𝜉 for 𝜉 ∈ ℋ , while in [16–19] lower bounds for averages of the type 1 𝑇 𝑇 𝑚 = 1 𝑈 𝑚 𝐹 𝜉 , 𝐻 0 𝑈 𝑚 𝐹 𝜉 ≥ 𝐶 𝑇 𝛾 ( 1 . 5 ) are obtained for periodic hamiltonians 𝐻 ( 𝑡 ) = 𝐻 0 + 𝑉 ( 𝑡 ) through dimensional properties of the spectral measure 𝜇 𝜉 associated with 𝑈 𝐹 and 𝜉 (the exponent 𝛾 depends on the measure 𝜇 𝜉 ).

In this work we study (in)stability of periodic time-dependent systems. As for tight-binding models (see [21] and references therein) we consider the Laplace-like average of ⟨ 𝑈 𝑚 𝐹 𝜉 , 𝐴 𝑈 𝑚 𝐹 𝜉 ⟩ , that is, 2 𝑇 ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ⟨ 𝑈 𝑚 𝐹 𝜉 , 𝐴 𝑈 𝑚 𝐹 𝜉 ⟩ , ( 1 . 6 ) where 𝐴 is a probe energy, 𝜉 is an element of d o m 𝐴 , and 𝑈 𝐹 is the Floquet operator. The main technical reason for working with this expression for the time average is that it can be written in terms of (see Theorem 2.3) the eigenvalues of 𝐴 , that is, 𝐴 𝜑 𝑗 = 𝜆 𝑗 𝜑 𝑗 , and the matrix elements ⟨ 𝜑 𝑗 , 𝑅 𝑧 ( 𝑈 𝐹 ) 𝜉 ⟩ of the resolvent operator 𝑅 𝑧 ( 𝑈 𝐹 ) = ( 𝑈 𝐹 − 𝑧 𝟏 ) − 1 (with 𝑧 = 𝑒 − 𝑖 𝐸 𝑒 1 / 𝑇 ) with respect to the orthonormal basis { 𝜑 𝑗 } of the Hilbert space (here 1 denotes the identity operator). Lemma 2.2 relates the long run of Laplace-like average with the usual Cesàro average. In Section 2 we shall prove this abstract results and present some applications in Section 3.

Since our main results are for temporal Laplace averages of expectation values of probe energies (see Section 2), in practice we will think of (in)stability in terms of (un)boundedness of such averages. Note that unbounded Laplace averages imply unboundedness of expectation values of probe energies themselves.

2. Average Energy and Green Functions

Consider a time-dependent hamiltonian 𝐻 ( 𝑡 ) with 𝐻 ( 𝑡 + 𝜏 ) = 𝐻 ( 𝑡 ) for all 𝑡 ∈ ℝ , acting in the separable Hilbert space ℋ . Suppose the existence of the propagators 𝑈 ( 𝑡 , 𝑠 ) , so that the Floquet operator 𝑈 𝐹 = 𝑈 ( 𝜏 , 0 ) is at our disposal. Let 𝐴 be a probe energy and 𝜆 𝑗 , 𝜑 𝑗 as in the introduction. Also, { 𝜑 𝑗 } ∞ 𝑗 = 1 is an orthonormal basis of ℋ .

The main interest is in the study of the expectation values, herein defined by 𝐸 𝐴 𝜉 𝑈 ( 𝑚 ) ∶ = 𝑚 𝐹 𝜉 , 𝐴 𝑈 𝑚 𝐹 𝜉 , i f 𝑈 𝑚 𝐹 𝜉 ∈ d o m 𝐴 , + ∞ , i f 𝑈 𝑚 𝐹 𝜉 ∈ ℋ ⧵ d o m 𝐴 , ( 2 . 1 ) as function of time 𝑚 ∈ ℕ . Another quantity of interest is the time dependence of the moment of this probe energy, which takes values in [ 0 , + ∞ ] and is defined by 𝑀 𝐴 𝜉 ( 𝑚 ) ∶ = ∞ 𝑗 = 1 𝜆 𝑗 | | ⟨ 𝜑 𝑗 , 𝑈 𝑚 𝐹 | | 𝜉 ⟩ 2 . ( 2 . 2 ) Our first remark is the equivalence of both concepts (under certain circumstances).

Proposition 2.1. If 𝑈 𝑚 𝐹 𝜉 ∈ d o m 𝐴 for all 𝑚 , then 𝐸 𝐴 𝜉 ( 𝑚 ) = 𝑀 𝐴 𝜉 ( 𝑚 ) , ∀ 𝑚 . ( 2 . 3 ) This holds, in particular, if d o m 𝐴 is invariant under the time evolution 𝑈 𝑚 𝐹 and 𝜉 ∈ d o m 𝐴 .

Proof. Since d o m 𝐴 ⊂ d o m 𝐴 1 / 2 [1], one has 𝑈 𝑚 𝐹 𝜉 ∈ d o m 𝐴 1 / 2 , for all 𝑚 , and so 𝑀 𝐴 𝜉 ( 𝑚 ) = ∞ 𝑗 = 1 | | 𝜆 𝑗 1 / 2 𝜑 𝑗 , 𝑈 𝑚 𝐹 𝜉 | | 2 = ∞ 𝑗 = 1 | | 𝐴 1 / 2 𝜑 𝑗 , 𝑈 𝑚 𝐹 𝜉 | | 2 = ∞ 𝑗 = 1 | | 𝜑 𝑗 , 𝐴 1 / 2 𝑈 𝑚 𝐹 𝜉 | | 2 = ‖ ‖ 𝐴 1 / 2 𝑈 𝑚 𝐹 𝜉 ‖ ‖ 2 = ⟨ 𝐴 1 / 2 𝑈 𝑚 𝐹 𝜉 , 𝐴 1 / 2 𝑈 𝑚 𝐹 𝜉 ⟩ = ⟨ 𝑈 𝑚 𝐹 𝜉 , 𝐴 𝑈 𝑚 𝐹 𝜉 ⟩ = 𝐸 𝐴 𝜉 ( 𝑚 ) , ( 2 . 4 ) which is the stated result.

We introduce the temporal Laplace average of 𝐸 𝐴 𝜉 (see also the appendix) by the following function of 𝑇 > 0 , which also takes values in [ 0 , + ∞ ] : 𝐿 𝐴 𝜉 ( 2 𝑇 ) ∶ = 𝑇 ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 𝐸 𝐴 𝜉 ( 𝑚 ) . ( 2 . 5 ) Under certain conditions, the next result shows that the upper 𝛽 + and lower 𝛽 − growth exponents of this average, that is, roughly they are the best exponents so that for large 𝑇 there exist 0 ≤ 𝑐 1 ≤ 𝑐 2 < ∞ with 𝑐 1 𝑇 𝛽 − ≤ 𝐿 𝐴 𝜉 ( 𝑇 ) ≤ 𝑐 2 𝑇 𝛽 + , ( 2 . 6 ) and the corresponding exponents for the temporal Cesàro average 𝐶 𝐴 𝜉 1 ( 𝑇 ) = 𝑇 𝑇 𝑚 = 0 𝐸 𝐴 𝜉 ( 𝑚 ) ( 2 . 7 ) are closely related; this follows at once by Lemma 2.2, which perhaps could be improved to get equality also between lower exponents. Note that, although not indicated, these exponents depend on the initial condition 𝜉 .

Lemma 2.2. If ( ℎ ( 𝑚 ) ) ∞ 𝑚 = 0 is a nonnegative sequence and ℎ ( 𝑚 ) ≤ 𝐶 𝑚 𝑛 for some 𝐶 > 0 and 𝑛 ≥ 0 , then 𝛽 + 𝑒 = 𝛽 + 𝑑 and 𝛽 − 𝑒 ≤ 𝛽 − 𝑑 , where 𝛽 + 𝑒 = l i m s u p 𝑇 → ∞ ∑ l o g 𝑇 𝑚 = 0 ℎ ( 𝑚 ) l o g 𝑇 , 𝛽 − 𝑒 = l i m i n f 𝑇 → ∞ ∑ l o g 𝑇 𝑚 = 0 ℎ ( 𝑚 ) , 𝛽 l o g 𝑇 + 𝑑 = l i m s u p 𝑇 → ∞ ∑ l o g ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) l o g 𝑇 , 𝛽 − 𝑑 = l i m i n f 𝑇 → ∞ ∑ l o g ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) . l o g 𝑇 ( 2 . 8 )

Proof. Note that for 0 ≤ 𝑚 ≤ 𝑇 we have 𝑒 − 2 ≤ 𝑒 − 2 𝑚 / 𝑇 ≤ 1 , and so 𝑇 𝑚 = 0 ℎ ( 𝑚 ) ≤ 𝑇 𝑚 = 0 𝑒 2 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) ≤ 𝑒 2 ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) . ( 2 . 9 ) Hence 𝛽 ± 𝑒 ≤ 𝛽 ± 𝑑 .
On the other hand, for each 𝜖 > 0 , denoting by ⌈ 𝑥 ⌉ the smallest integer larger or equal to 𝑥 , one has ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) = 𝑇 1 + 𝜖 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) + ∞ 𝑇 𝑚 = 1 + 𝜖 + 1 𝑒 − 2 𝑚 / 𝑇 ≤ ℎ ( 𝑚 ) 𝑇 1 + 𝜖 𝑚 = 0 ℎ ( 𝑚 ) + 𝐶 ∞ 𝑇 𝑚 = 1 + 𝜖 + 1 𝑒 − 2 𝑚 / 𝑇 𝑚 𝑛 . ( 2 . 1 0 ) Now, for 𝑇 large enough 𝑛 𝑇 / 2 < 𝑇 1 + 𝜖 ≤ ⌈ 𝑇 1 + 𝜖 ⌉ . Thus ∞ 𝑇 𝑚 = 1 + 𝜖 + 1 𝑒 − 2 𝑚 / 𝑇 𝑚 𝑛 ≤ ∞ 𝑇 1 + 𝜖 𝑒 − 2 𝑡 / 𝑇 𝑡 𝑛 𝑑 𝑡 . ( 2 . 1 1 ) Therefore, for each 𝜖 > 0 and 𝑇 large enough ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) ≤ 𝑇 1 + 𝜖 𝑚 = 0 ℎ ( 𝑚 ) + 𝐶 ∞ 𝑇 1 + 𝜖 𝑒 − 2 𝑡 / 𝑇 𝑡 𝑛 ≤ 𝑑 𝑡 𝑇 1 + 𝜖 𝑚 = 0 ℎ ( 𝑚 ) + 𝐶 𝑒 − 2 𝑇 𝜖 𝑇 𝑛 . ( 2 . 1 2 ) Since 𝑒 − 2 𝑇 𝜖 𝑇 𝑛 → 0 as 𝑇 → ∞ , it follows that 𝛽 + 𝑑 = l i m s u p 𝑇 → ∞ ∑ l o g ∞ 𝑚 = 0 𝑒 − 2 𝑚 / 𝑇 ℎ ( 𝑚 ) l o g 𝑇 ≤ l i m s u p 𝑇 → ∞ ∑ l o g 𝑇 1 + 𝜖 𝑚 = 0 ℎ ( 𝑚 ) l o g 𝑇 = l i m s u p 𝑇 → ∞ ∑ l o g 𝑇 1 + 𝜖 𝑚 = 0 ℎ ( 𝑚 ) 𝑇 l o g 1 + 𝜖 𝑇 l o g 1 + 𝜖 l o g 𝑇 ≤ l i m s u p 𝑇 → ∞ ∑ l o g 𝑇 1 + 𝜖 𝑚 = 0 ℎ ( 𝑚 ) 𝑇 l o g 1 + 𝜖 l o g ( 𝑇 + 1 ) 1 + 𝜖 l o g 𝑇 = ( 1 + 𝜖 ) l i m s u p 𝑇 → ∞ ∑ l o g 𝑇 1 + 𝜖 𝑚 = 0 ℎ ( 𝑚 ) 𝑇 l o g 1 + 𝜖 ≤ ( 1 + 𝜖 ) 𝛽 + 𝑒 . ( 2 . 1 3 ) As 𝜖 > 0 was arbitrary, 𝛽 + 𝑑 ≤ 𝛽 + 𝑒 .

Recall that the Green functions 𝐺 𝜉 𝑧 ( 𝑗 ) associated with the operators 𝐴 , 𝑈 𝐹 at 𝜉 ∈ ℋ and 𝑧 ∈ ℂ , | 𝑧 | ≠ 1 , are defined by the matrices elements of the resolvent operator 𝑅 𝑧 ( 𝑈 𝐹 ) = ( 𝑈 𝐹 − 𝑧 1 ) − 1 along the orthonormal basis { 𝜑 𝑗 } ∞ 𝑗 = 1 , that is, 𝐺 𝜉 𝑧 ( 𝑗 ) ∶ = ⟨ 𝜑 𝑗 , 𝑅 𝑧 𝑈 𝐹 𝜉 ⟩ . ( 2 . 1 4 ) Note that 𝐺 𝜉 𝑧 ( 𝑗 ) is always well defined since for | 𝑧 | ≠ 1 that resolvent operator is bounded. Theorem 2.3 is the main reason for considering the temporal averages 𝐿 𝐴 𝜉 ( 𝑇 ) . It presents a formula that translates the Laplace average of wavepackets at time 𝑇 into an integral of the Green functions over “energies” in the circle of radius 𝑒 1 / 𝑇 in the complex plane (centered at the origin). As 𝑇 grows, the integration region approaches the unit circle where the spectrum of 𝑈 𝐹 lives and 𝑅 𝑧 ( 𝑈 𝐹 ) takes singular values, so that (hopefully) 𝐴 -(in)stability can be quantitatively detected.

Theorem 2.3. Assume that 𝑈 𝑚 𝐹 𝜉 ∈ d o m 𝐴 for all 𝑚 ≥ 0 . Then 𝐿 𝐴 𝜉 ( 1 𝑇 ) = 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜆 𝑗 0 2 𝜋 | | 𝐺 𝜉 𝑧 | | ( 𝑗 ) 2 𝑑 𝐸 , 𝑧 = 𝑒 − 𝑖 𝐸 + 1 / 𝑇 . ( 2 . 1 5 )

Before the proof of this theorem, we underline that this formula, that is, the expression on the right-hand side of (2.15), is a sum of positive terms and so it is well defined for all 𝜉 ∈ ℋ if we let it take values in [ 0 , + ∞ ] ; hence, in principle it can happen that this formula is finite even for vectors 𝑈 𝑚 𝐹 𝜉 not in the domain of 𝐴 , where 𝐿 𝐴 𝜉 ( 𝑇 ) = + ∞ . The general case, that is, ∀ 𝜉 ∈ ℋ , can then be gathered in the following inequality: 𝐿 𝐴 𝜉 ( 1 𝑇 ) ≥ 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜆 𝑗 0 2 𝜋 | | 𝐺 𝜉 𝑧 | | ( 𝑗 ) 2 𝑑 𝐸 , 𝑧 = 𝑒 − 𝑖 𝐸 + 1 / 𝑇 , ( 2 . 1 6 ) so that lower bound estimates for this formula always imply lower bound estimates for the Laplace average.

Proof of Theorem 2.3. First note that, by hypothesis, 𝑈 𝑚 𝐹 𝜉 ∈ d o m 𝐴 1 / 2 for each 𝑚 ∈ ℕ . Denote by 𝜇 𝑗 the spectral measure of 𝑈 𝐹 associated with the pair ( 𝜑 𝑗 , 𝜉 ) and by ℱ the Fourier transform ℱ ∶ L 2 [ 0 , 2 𝜋 ] → 𝑙 2 ( ℤ ) . By the spectral theorem for unitary operators ⟨ 𝜑 𝑗 , 𝑈 𝐹 𝜉 ⟩ = 0 2 𝜋 𝑒 − 𝑖 𝐸 ′ 𝑑 𝜇 𝑗 𝐸 . ( 2 . 1 7 ) For each 𝑗 let 𝑎 ( 𝑗 ) = ( 𝑎 ( 𝑗 ) ( 𝑚 ) ) 𝑚 ∈ ℤ be the sequence 𝑎 ( 𝑗 ) ( ⎧ ⎪ ⎨ ⎪ ⎩ 𝑒 𝑚 ) = 0 , i f 𝑚 < 0 , − 𝑚 / 𝑇 0 2 𝜋 𝑒 − 𝑖 𝐸 ′ 𝑚 𝑑 𝜇 𝑗 𝐸 , i f 𝑚 ≥ 0 . ( 2 . 1 8 ) Since 𝑎 ( 𝑗 ) ∈ 𝑙 1 ( ℤ ) ∩ 𝑙 2 ( ℤ ) and ℱ is a unitary operator, it follows that ‖ 𝑎 ( 𝑗 ) ‖ 𝑙 2 ( ℤ ) = ‖ ℱ − 1 𝑎 ( 𝑗 ) ‖ L 2 [ 0 , 2 𝜋 ] and also ℱ − 1 𝑎 ( 𝑗 ) ( 1 𝐸 ) = √ 2 𝜋 ∞ 𝑚 = − ∞ 𝑒 𝑖 𝐸 𝑚 𝑎 ( 𝑗 ) ( = 1 𝑚 ) √ 2 𝜋 ∞ 𝑚 = 0 𝑒 𝑖 𝐸 𝑚 𝑒 − 𝑚 / 𝑇 0 2 𝜋 𝑒 − 𝑖 𝐸 ′ 𝑚 𝑑 𝜇 𝑗 𝐸 = 1 √ 2 𝜋 0 2 𝜋 ∞ 𝑚 = 0 𝑒 𝑖 𝑚 ( 𝐸 − 𝐸 ′ ) + 𝑖 / 𝑇 𝑑 𝜇 𝑗 𝐸 = 1 √ 2 𝜋 0 2 𝜋 1 1 − 𝑒 𝑖 ( 𝐸 − 𝐸 ′ + 𝑖 / 𝑇 ) 𝑑 𝜇 𝑗 𝐸 = 1 √ 2 𝜋 0 2 𝜋 𝑑 𝜇 𝑗 𝐸 𝑒 𝑖 ( 𝐸 + 𝑖 / 𝑇 ) 𝑒 − 𝑖 ( 𝐸 + 𝑖 / 𝑇 ) − 𝑒 − 𝑖 𝐸 ′ 1 = − √ 2 𝜋 𝑒 𝑖 ( 𝐸 + 𝑖 / 𝑇 ) 0 2 𝜋 𝑑 𝜇 𝑗 𝐸 𝑒 − 𝑖 𝐸 ′ − 𝑒 − 𝑖 ( 𝐸 + 𝑖 / 𝑇 ) 1 = − √ 2 𝜋 𝑒 𝑖 ( 𝐸 + 𝑖 / 𝑇 ) ⟨ 𝜑 𝑗 , 𝑅 𝑧 𝑈 𝐹 1 𝜉 ⟩ = − √ 2 𝜋 𝑒 𝑖 𝐸 𝑒 − 1 / 𝑇 𝐺 𝜉 𝑧 ( 𝑗 ) , ( 2 . 1 9 ) with 𝑧 = 𝑒 − 𝑖 𝐸 + 1 / T . Therefore | | ℱ − 1 𝑎 ( 𝑗 ) | | 2 1 ( 𝐸 ) = 2 𝜋 𝑒 − 2 / 𝑇 | | 𝐺 𝜉 𝑧 | | ( 𝑗 ) 2 , ( 2 . 2 0 ) and so ‖ ‖ ℱ − 1 𝑎 ( 𝑗 ) ‖ ‖ 2 L 2 [ 0 , 2 𝜋 ] = 1 2 𝜋 𝑒 − 2 / 𝑇 0 2 𝜋 | | 𝐺 𝜉 𝑧 | | ( 𝑗 ) 2 𝑑 𝐸 . ( 2 . 2 1 ) From such relation it follows that 𝐿 𝐴 𝜉 ( 𝑇 ) = ∞ 𝑚 = 0 2 𝑇 𝑒 − 2 𝑚 / 𝑇 𝑀 𝐴 𝜉 ( = 𝑚 ) ∞ 𝑗 = 1 𝜆 𝑗 ∞ 𝑚 = 0 2 𝑇 𝑒 − 2 𝑚 / 𝑇 | | 𝜑 𝑗 , 𝑈 𝑚 𝐹 𝜉 | | 2 = ∞ 𝑗 = 1 𝜆 𝑗 2 𝑇 ∞ 𝑚 = 0 | | | | 𝑒 − 𝑚 / 𝑇 0 2 𝜋 𝑒 − 𝑖 𝐸 ′ 𝑚 𝑑 𝜇 𝑗 𝐸 | | | | 2 = ∞ 𝑗 = 1 𝜆 𝑗 2 𝑇 ‖ ‖ 𝑎 ( 𝑗 ) ‖ ‖ 2 𝑙 2 ( ℤ ) = ∞ 𝑗 = 1 𝜆 𝑗 2 𝑇 ‖ ‖ ℱ − 1 𝑎 ( 𝑗 ) ‖ ‖ 2 L 2 [ 0 , 2 𝜋 ] = 1 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜆 𝑗 0 2 𝜋 | | 𝐺 𝜉 𝑧 | | ( 𝑗 ) 2 𝑑 𝐸 , ( 2 . 2 2 ) which is exactly the stated result.

Theorem 2.3 clearly remains true if the eigenvalues 𝜆 𝑗 of 𝐴 have finite multiplicity. In this case, for each 𝜆 𝑗 consider the corresponding orthonormal eigenvectors 𝜑 𝑗 1 , … , 𝜑 𝑗 𝑘 , and one obtains 𝐿 𝐴 𝜉 1 ( 𝑇 ) = 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜆 𝑗 𝑘 𝑛 = 1 0 2 𝜋 | | ⟨ 𝜑 𝑗 𝑛 , 𝑅 𝑧 ( 𝑈 𝐹 | | ) 𝜉 ⟩ 2 𝑑 𝐸 , ( 2 . 2 3 ) with 𝑧 as before.

In case the initial condition is 𝜉 = 𝜑 1 , put 𝜂 ( 𝑧 ) ∶ = 𝑅 𝑧 ( 𝑈 𝐹 ) 𝜑 1 . Thus, ( 𝑈 𝐹 − 𝑧 ) 𝜂 ( 𝑧 ) = 𝜑 1 and so 𝑈 𝐹 𝜂 ( 𝑧 ) = 𝑧 𝜂 ( 𝑧 ) + 𝜑 1 . Hence 𝜑 𝑗 , 𝑈 𝐹 𝜂 ( 𝑧 ) = 𝑧 ⟨ 𝜑 𝑗 , 𝜂 ( 𝑧 ) ⟩ + 𝛿 𝑗 , 1 ( 2 . 2 4 ) and by denoting 𝐺 𝑧 ( 𝑗 ) ∶ = 𝐺 𝜑 1 𝑧 ( 𝑗 ) , ( 2 . 2 5 ) one concludes.

Lemma 2.4. 𝐺 𝑧 ⎧ ⎪ ⎨ ⎪ ⎩ 1 ( 𝑗 ) = 𝑧 ⟨ 𝜑 1 , 𝑈 𝐹 𝜂 ( 𝑧 ) 1 ⟩ − 1 , i f 𝑗 = 1 , 𝑧 𝜑 𝑗 , 𝑈 𝐹 𝜂 ( 𝑧 ) , i f 𝑗 > 1 . ( 2 . 2 6 )

In Section 3 we discuss some Floquet operators that are known in literature and analyze their Green functions through the equation 𝑈 𝐹 𝜂 − 𝑧 1 ( 𝑧 ) = 𝜑 1 . ( 2 . 2 7 )

3. Applications

This section is devoted to some applications of the formula obtained in Theorem 2.3. In general it is not trivial to get expressions and/or bounds for the Green functions of Floquet operators, and so one of the main goals of the applications that follow is to illustrate how to approach the method we have just proposed.

3.1. Time-Independent Hamiltonians

As a first example and illustration of the formula proposed in Theorem 2.3, we consider the special case of autonomous hamiltonians. In this case 𝐻 ( 𝑡 ) = 𝐻 0 for all 𝑡 , and we assume that 𝐻 0 is a positive, unbounded, self-adjoint operator and with simple discrete spectrum, 𝐻 0 𝜑 𝑗 = 𝜒 𝑗 𝜑 𝑗 , so that { 𝜑 𝑗 } ∞ 𝑗 = 1 is an orthonormal basis of ℋ and 0 ≤ 𝜒 1 < 𝜒 2 < 𝜒 3 < ⋯ with 𝜒 𝑗 → ∞ . For 𝑞 > 0 we can consider 𝐻 𝑞 0 as our abstract energy operator 𝐴 , so that its eigenvalues are 𝜆 𝑗 = 𝜒 𝑞 𝑗 (since 𝐴 and 𝐻 0 have the same eigenfunctions, we are justified in using the notation 𝜑 𝑗 for the eigenfunctions of 𝐻 0 ). We take 𝑈 𝐹 = 𝑒 − 𝑖 𝐻 0 (time 𝑡 = 1 ) and for 𝜉 ∈ ℋ 𝐺 𝜉 𝑧 ( 𝜑 𝑗 ) = 𝑗 , 𝑅 𝑧 𝐻 0 𝜉 = 𝑅 𝑧 𝐻 0 𝜑 𝑗 = , 𝜉 ⟨ 𝜑 𝑗 , 𝜉 ⟩ 𝑒 − 𝑖 𝜒 𝑗 − 𝑧 . ( 3 . 1 ) Since d o m 𝐻 𝑞 0 is invariant under the time evolution 𝑒 − 𝑖 𝑡 𝐻 0 , then for 𝑧 = 𝑒 − 𝑖 𝐸 𝑒 1 / 𝑇 and 𝜉 ∈ d o m 𝐻 𝑞 0 we have 𝐿 𝑞 𝜉 ( 𝑇 ) ∶ = 𝐿 𝐻 𝑞 0 𝜉 = 1 ( 𝑇 ) 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜒 𝑞 𝑗 0 2 𝜋 | | 𝐺 𝜉 𝑧 | | ( 𝑗 ) 2 = 1 𝑑 𝐸 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜒 𝑞 𝑗 0 2 𝜋 | | ⟨ 𝜑 𝑗 | | , 𝜉 ⟩ 2 | 𝑒 − 𝑖 𝜒 𝑗 − 𝑧 | 2 = 1 𝑑 𝐸 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜒 𝑞 𝑗 | | ⟨ 𝜑 𝑗 | | , 𝜉 ⟩ 2 0 2 𝜋 𝑑 𝐸 | 𝑒 − 𝑖 𝜒 𝑗 − 𝑧 | 2 . ( 3 . 2 )

Thus we need to calculate the integral 𝐼 𝑗 ∫ ∶ = 0 2 𝜋 ( 𝑑 𝐸 / | 𝑒 − 𝑖 𝜒 𝑗 − 𝑧 | 2 ) . Let 𝛾 be the closed path in ℂ given by 𝛾 ( 𝐸 ) = 𝑒 𝑖 𝐸 with 0 ≤ 𝐸 ≤ 2 𝜋 , 𝛼 𝑗 = 𝑒 1 / 𝑇 𝑒 𝑖 𝜒 𝑗 and 𝛽 𝑗 = 𝑒 − 1 / 𝑇 𝑒 𝑖 𝜒 𝑗 , then 𝐼 𝑗 = 0 2 𝜋 𝑑 𝐸 𝑒 − 𝑖 𝜒 𝑗 𝑒 − 𝑧 𝑖 𝜒 𝑗 − 𝑧 = 0 2 𝜋 𝑑 𝐸 𝑒 − 𝑖 𝜒 𝑗 − 𝑒 − 𝑖 𝐸 𝑒 1 / 𝑇 𝑒 𝑖 𝜒 𝑗 − 𝑒 𝑖 𝐸 𝑒 1 / 𝑇 = 0 2 𝜋 𝑑 𝐸 𝑒 2 / 𝑇 𝑒 − 1 / 𝑇 𝑒 − 𝑖 𝜒 𝑗 − 𝑒 − 𝑖 𝐸 𝑒 − 1 / 𝑇 𝑒 𝑖 𝜒 𝑗 − 𝑒 𝑖 𝐸 1 = − 𝑒 2 / 𝑇 0 2 𝜋 𝑑 𝐸 𝑒 − 𝑖 𝐸 𝑒 − 1 / 𝑇 𝑒 − 𝑖 𝜒 𝑗 𝑒 𝑖 𝐸 − 𝛼 𝑗 𝑒 𝑖 𝐸 − 𝛽 𝑗 1 = − 𝑒 1 / 𝑇 𝑒 − 𝑖 𝜒 𝑗 1 𝑖 0 2 𝜋 𝑖 𝑒 𝑖 𝐸 𝑑 𝐸 𝑒 𝑖 𝐸 − 𝛼 𝑗 𝑒 𝑖 𝐸 − 𝛽 𝑗 = 𝑖 𝑒 1 / 𝑇 𝑒 − 𝑖 𝜒 𝑗 𝛾 𝑑 𝑤 𝑤 − 𝛼 𝑗 𝑤 − 𝛽 𝑗 . ( 3 . 3 ) As | 𝛼 𝑗 | > 1 and | 𝛽 𝑗 | < 1 , 𝛽 𝑗 is the unique pole in the interior of 𝛾 . Thus, by using residues, 𝐼 𝑗 = 𝑖 𝑒 1 / 𝑇 𝑒 − 𝑖 𝜒 𝑗 1 2 𝜋 𝑖 𝛽 𝑗 − 𝛼 𝑗 = 2 𝜋 𝑒 2 / 𝑇 − 1 ( 3 . 4 ) and 𝐼 𝑗 is independent of 𝜒 𝑗 .

Therefore by (3.2) it follows that 𝐿 𝑞 𝜉 ( 1 𝑇 ) = 𝜋 𝑒 − 2 / 𝑇 1 𝑇 ∞ 𝑗 = 1 𝜒 𝑞 𝑗 | | ⟨ 𝜑 𝑗 | | , 𝜉 ⟩ 2 2 𝜋 𝑒 2 / 𝑇 = 2 − 1 𝑒 − 2 / 𝑇 1 𝑇 1 𝑒 2 / 𝑇 − 1 ∞ 𝑗 = 1 𝜒 𝑞 𝑗 | | ⟨ 𝜑 𝑗 | | , 𝜉 ⟩ 2 = 2 1 − 𝑒 − 2 / 𝑇 1 𝑇 ‖ ‖ 𝐻 0 𝑞 / 2 𝜉 ‖ ‖ 2 . ( 3 . 5 ) Since ( 1 − 𝑒 − 2 / 𝑇 ) = 2 / 𝑇 + 𝒪 ( 1 / 𝑇 2 ) , for large 𝑇 it is found that 𝐿 𝑞 𝜉 ‖ ‖ 𝐻 ( 𝑇 ) ≈ 0 𝑞 / 2 𝜉 ‖ ‖ 2 , ( 3 . 6 ) with (for 𝜉 ∈ d o m 𝐻 𝑞 0 ) l i m 𝑇 → ∞ 𝐿 𝑞 𝜉 ( 𝑇 ) = ⟨ 𝜉 , 𝐻 𝑞 0 𝜉 ⟩ . ( 3 . 7 ) Then we conclude that the function 𝑒 ℕ ∋ 𝑚 ⟼ − 𝑖 𝐻 0 𝑚 𝜉 , 𝐻 𝑞 0 𝑒 − 𝑖 𝐻 0 𝑚 𝜉 ( 3 . 8 ) is bounded for 𝜉 ∈ d o m 𝐻 𝑞 0 , which is (of course) an expected result (see Proposition 2.1).

3.2. Lower-Bounded Green Functions

As a first theoretical application we get dynamical instability from some lower bounds of the Green functions. See [21] for a similar result in the one-dimensional discrete Schrödinger operators context; there, a relation to transfer matrices allows interesting applications to nontrivial models, what is not available in the unitary setting yet (and it constitutes of an important open problem). As before, 𝜆 𝑗 denote the increasing sequence of positive eigenvalues of the abstract energy operator 𝐴 , the ones we use to probe (in)stability.

Let [ ⋅ ] denotes the integer part of a real number, and | ⋅ | indicates Lebesgue measure.

Theorem 3.1. Suppose that there exist 𝐾 > 0 and 𝛼 > 0 such that for each 2 𝑁 > 0 large enough there exists a nonempty Borel set 𝐽 ( 𝑁 ) ⊂ 𝑆 1 such that | | 𝐺 𝜉 𝑧 | | ≥ 𝐾 ( 𝑗 ) 𝑁 𝛼 , 𝑁 ≤ 𝑗 ≤ 2 𝑁 ( 3 . 9 ) holds for all 𝑧 = 𝑒 − 𝑖 𝐸 + 1 / 𝑇 with 𝐸 ∈ 𝐽 𝑇 ( 𝑁 ) = { 𝐸 ∈ 𝑆 1 ∶ ∃ 𝐸 ∈ 𝐽 ( 𝑁 ) ; | 𝐸 − 𝐸 | ≤ 1 / T } (the ( 1 / T ) -neighborhood of 𝐽 ( 𝑁 ) ). Let 𝛿 > 0 , then for 𝑇 large enough such that 𝑁 ( 𝑇 ) = [ 𝑇 𝛿 ] , one has 𝐿 𝐴 𝜉 ( 𝑇 ) ≥ c t e 𝜆 [ 𝑇 𝛿 ] 𝑇 𝛿 ( 1 − 2 𝛼 ) − 2 . ( 3 . 1 0 ) Moreover, if 𝜆 j ≥ c t e 𝑗 𝛾 , 𝛾 ≥ 0 , then 𝐿 𝐴 𝜉 ( 𝑇 ) ≥ c t e 𝑇 𝛿 ( 𝛾 − 2 𝛼 + 1 ) − 2 . ( 3 . 1 1 )

Proof. By the formula in Theorem 2.3, or its more general version (2.16), 𝐿 𝐴 𝜉 ( 1 𝑇 ) ≥ 𝜋 𝑒