Quantum Evolution and Anticipation

by Hans-Rudolf Thomann

March 2010

Author address Im Altried 1h, 8051 Zürich, [email protected]

CONTENTS

0 Introduction ...... 4

1 Quantum Evolution Systematic ...... 5 1.1 Basic Terminology and Definitions ...... 5 Definition 1: Evolution Scenario ...... 5 Definition 2: Spectral measures and quantities ...... 6 Proposition 1: Spectral Measures ...... 7 Theorem 1: Embedded Orthogonal Evolution ...... 8 1.2 Characterization of Finite-dimensional Evolution ...... 8 1.2.1 Amplitudes, Characteristic Polynomial, Eigenvalues and Spectra ...... 8 1.2.2 Domain of Positivity ...... 9 1.2.3 Summary ...... 10 Theorem 2: Characterization of Finite-dimensional Evolution ...... 10 1.3 Characterization of Infinite-dimensional Evolution ...... 11 Theorem 3: Characterization of Infinite-dimensional Evolution ...... 11

2 Anticipation in Quantum Evolution ...... 12 2.1 Anticipation and Retrospection ...... 12 Definition 3: Anticipation Amplitudes and Probabilities ...... 12 2.2 Anticipation Amplitudes and the Spectral Difference ...... 12 Definition 4: Spectral Difference ...... 13 Lemma 1: Well-behaved measures ...... 13 Proposition 2: Representation of Anticipation Amplitudes ...... 13 2.3 Assessment Methodology ...... 14 Definition 5: ...... 14 2.4 Anticipation Strength ...... 14 Theorem 4: Anticipation Strength ...... 14 Lemma 2 ...... 15

3 Conclusions ...... 16

4 Bibliography ...... 17

Appendix A: Spectral-analytic Lemmata ...... 18 A.1 Inversion of the Discrete Fourier Transform 퐝훎^퐧 퐧 ∈ ℤ ...... 18 Lemma A.1: Inversion of the Discrete Fourier Transform 퐝훎^퐧 퐧 ∈ ℤ ...... 18 Examples ...... 20 A.2 Representation of the Dirac -Functional ...... 20 Lemma A.2: Representation of the Dirac -Functional ...... 21 Corollary A.2 ...... 21 A.3 Time-averages of 풅흁^풕ퟐ ...... 22

Quantum Evolution and Anticipation

Abstract A state is said to exhibit orthogonal evolution of order 퐿 and step size 푇 iff it evolves into or- 푇 thogonal states at times 0 < 푛 푇 ≤ 퐿. Measurement at time yields states from this ortho- 2 gonal set with certain probabilities. This effect, which we name anticipation, has been de- scribed in a previous paper in a special setting.

Here we extend our analyzes to general type quantum evolutions and spectral measures and prove that quantum evolutions possessing an embedded orthogonal evolution are characterized by positive joint spectral measure.

Furthermore, we categorize quantum evolution, assess anticipation strength and provide a framework of analytic tools and results, thus preparing for further investigation and experi- mental verification of anticipation in concrete physical situations such as the H-atom, which we have found to exhibit anticipation.

Quantum Evolution and Anticipation

0 Introduction

This paper generalizes and extends the findings of (Thomann, 2008), where we studied the special case of orthogonal evolutions of maximal order with pure point and absolutely continuous measures, to arbitrary types of spectral measures and evolutions.

푖퐻 − 푡 Under a Hamiltonian 퐻, a quantum state 푞 evolves into an orbit 푞푡 = 푈푡 푞, where 푈푡 = 푒 ℏ is defined on the closure of 퐷 퐻 . By spectral theory (Reed, 1980), the spectral measure 휇푞 휆 of 푞 and 퐻 is uniquely defined by 푞, 푓 퐻 푞 = ∫ 푓 퐻 푑휇푞 휆 for any analytic function 푓.

For fixed step size 푇 > 0, 푞0 = 푞 and 푛 ∈ ℤ, an evolution of order 퐿 ≥ 0 is given by the prima- 푛 푛 ry sequence 푞푛 = 푈푇 푞0 and the dual sequence, 푟푛 = 푈푇 푟0, such that 푟푘 , 푞푙 = 훿푘푙 푘, 푙 ∈ ℤ , 푘 − 푙 ≤ 퐿 (see definition 1). The amplitudes 푟푡, 푞0 are determined by the spectral measure 휌푑휇푞 (see definition 2). Positive evolutions are defined as those with positive 휌.

Theorem 1 states that every positive evolution contains an embedded orthogonal evolution, driving the component 푠0 of 푞0 of size 휁 through 퐿 + 1 mutually orthogonal states. Theorems 2 and 3 prove the existence of (not necessarily positive) evolutions of any order for point, singular-continuous and absolutely continuous spectrum, respectively.

Quantum-Mechanical anticipation (definition 3) is defined for the embedded orthogonal evolution in terms of the anticipation amplitudes 훼푛 = 푠푛 , 푈푇 2푠0 , and the anticipation probabili- 2 ties 푝푛 = 훼푛 , expressing the result of a measurement of 푠0 at time 푇 2, where 푛 ≤ 퐿. Quantum-mechanical retrospection is the time-reverse of anticipation. Measurements antici- pate future (or recall past) states 푠푛 with probability 푝푛 , which is the reason for the naming. Theorem 4 assesses the anticipation strength in various situations.

This paper provides analyses and tools facilitating further research about anticipation. In chapter 1 we introduce the basic terminology and definitions, characterize finite-dimensional and infinite-dimensional evolutions and their spectral measures, and explore the topology of the manifold of positive evolutions. Chapter 2 explains and defines anticipation, provides a representation of anticipation amplitudes and analyzes anticipation strength.

To cover not only point and absolutely continuous but also singular-continuous measures, we use some formalism from spectral theory an provide mathematical theorems in appendix A. Theorem A.1 establishes explicit formulae for the Fourier coefficients of continuous tions 퐹 supported on 퐾 whose second distributional derivatives yield a given measure 휈, in- verting the discrete Fourier transform 푑휈^ 푛 . Theorem A.2 proves a very general representation of Dirac's 훿 functional.

In a future paper we will present results from numerical simulations of the point spectrum of the H-atom as well as equidistant and random point spectra, demonstrating that anticipation and retrospection occur in common situations. The quantum anticipation simulator software running und Microsoft Windows is already downloadable from the author's homepage http://www.thomannconsulting.ch/public/aboutus/aboutus-en.htm.

Quantum Evolution and Anticipation

1 Quantum Evolution Systematic In this chapter we introduce the basic definitions and terminology, cast quantum evolutions into a set of categories and derive the characteristic spectral properties of each category.

1.1 Basic Terminology and Definitions Let 퐻 be an essentially self-adjoint Hamiltonian. The set of states we are considering is the Hilbert space ℋ which equals the closure of 퐷 퐻 . Throughout the paper all states will be assumed to have unit norm. New technical terms are introduced in italic type.

For any state 푞 ∈ ℋ the finite spectral measure 휇푞 휆 is uniquely defined by the inner product 푞, 푓 퐻 푞 = ∫ 푓 퐻 푑휇푞 휆 for any analytic function 푓 (see e.g. (Reed, 1980)) This relation 2 induces a unique unitary mapping ℑ푞 : ℋ → ℒ 푑휇푞 such that the image of 푞 is the constant function 1. We will sometimes identify states with their images under this mapping. 푖퐻 − 푡 The evolution operator 푈푡 = 푒 ℏ is a unitary isomorphism of ℋ, under which 푞 evolves into an orbit 푞푡 = 푈푡 푞. The amplitude 푞0, 푞푇 equals the Fourier transform ^ −푖휆푇 ℏ 푑휇푞 푇 = ∫ 푒 푑휇푞 휆 . By the unitarity of 푈푡 , 푞0, 푞푇 = 푞푡, 푞푡+푇 is independent from 푡.

Our subject is the evolution of states at equidistant time steps. To analyze them we define evolution scenarios comprising a primary and an associated dual sequence of states.

Definition 1: Evolution Scenario

An evolution scenario (or short: evolution) is characterized by the following elements:

a) A Hamiltonian 퐻. b) The step time 푇 > 0. c) The primary base state 푞0. 푛 d) The doubly-infinite primary sequence 푞푛 = 푈푇 푞0 푛 ∈ ℤ . e) The primary amplitudes 훽푛 = 푞0, 푞푛 satisfying 훽−푛 = 훽 푛 . f) The cyclic space 퐶 = Span 푞푛 푛 ∈ ℤ , which is invariant under 푈푇. g) The dimension 푑 = dim 퐶 , 0 < 푑 ≤ ∞. h) The integer order 퐿, 푑 > 퐿 ≥ 0. i) The subspace 퐶퐿 = Span 푞푛 0 ≤ 푛 ≤ 퐿 of 퐶. j) The dual base state 푟0 ∈ 퐶퐿 of unit norm meeting 푟0, 푞푛 = 훿푛 푛 ∈ ℤ , 0 ≤ 푛 ≤ 퐿

and 푟푘 , 푞푙 = 푟푙 , 푞푘 푘, 푙 ∈ ℤ . 푛 k) The doubly-infinite dual sequence 푟푛 = 푈푇 푟0 푛 ∈ ℤ in 퐶.

An evolution scenario is said to exhibit orthogonal evolution of order 퐿 iff 푞푛 = 푟푛. ∎

Evolution scenarios exist for any 푈푇, 푞0 and 0 < 퐿 ≤ 푑 − 1 2 < ∞, as we will prove in theorems 2 and 3. For 퐿 = ∞ see theorem 2e).

Provided its existence 푟0 is that vector in the orthogonal complement of the hyperplane Span 푞푛 0 < 푛 ≤ 퐿 relative to 퐶퐿 uniquely defined by the requirement that 푟0, 푞0 = 1, and ∞ > 푟0 2 ≥ 1.

Quantum Evolution and Anticipation

As, by the unitarity of 푈푡 , 푟0, 푞푛 = 푟푘 , 푞푘+푛 is independent from 푘, our construction ensures that 푟푛+푘 , 푞푘 = 훿푛 푛, 푘 ∈ ℤ , 0 ≤ 푛 ≤ 퐿 . Thus, within the order, the dual sequence is orthogonal to the primary sequence.

In case of orthogonal evolution, the primary and dual sequence coincide, thus 푟0, 푞0 = 2 푞0 2 = 1.

State evolution is completely determined by spectral measures and quantities, which we introduce now.

Definition 2: Spectral measures and quantities

푛 Until the end of this chapter ∙ 푛 denotes the ℒ 푑휇푞 norm. The Lebesgue norm is ∙ 푛,푑휆 .

a) The spectral measure of the primary and dual sequence are denoted by 휇푞 휆 and 휇푟 휆 , respectively. 2 휌 b) With 휌 = ℑ푞 푟0 ∈ ℒ 푑휇푞 the joint spectral measure 휇s 휆 is 푑휇푠 = 푑휇푞 . 휌 1 c) Positive evolution is characterized by 휌 being a non-negative function. 휅 0 ≤ 휅 < 휋 d) Shift 휅 = maps the interval 0,2 휋 to 퐾 = −휋,휋 . 휅 − 2휋 휋 ≤ 휅 < 2휋 e) For any finite measure 휇푥 and real 푚, the reduction modulo 푚 is 푑휇푥,푚 휅 ≝ 휅=휆 mod 푚 푑휇푥 휆 . f) For 푥 = 푞, 푟, 푠, the reduced measure 휈푥 is defined by 푑휈푥 Shift 휅 = 푑휇푥,2휋 휅 . g) 1′ = sign 휌. 휌 −푖휆푛푇 ℏ ′ h) ℑ푞 푠푛 = 푒 , ℑ푞 푠푛 = 1′ 휌 1 휌 . 휌 1 i) 휁 = 푠0, 푞0 . ∎

2 휇푥 ℝ = 푥0 2,푑휆 = 1 for 푥 = 푞, 푠, and the supports of these measures are equal up to a set of zero measure.

2 −푖휆푛푇 ℏ −푖휆푛푇 ℏ Obviously 푑휇푟 = 휌 푑휇푞 , ℑ푞 푞푛 = 푒 , ℑ푞 푟푛 = 휌푒 and −푖휆푛푇 ℏ ^ 푞0, 푟푛 = ∫ 휌푒 푑휇푞 = 휌푑휇푞 푛푇 ℏ .

The definition of 휇s requires 휌 1,푑휇푞 < ∞, which indeed follows from the Schwartz inequality and 1 2 = 휇푞 ℝ = 1: 휌 1 = 휌 ∙ 1 1 ≤ 휌 2 1 2 = 휌 2 = 푟0 2,푑휆 < ∞.

1 2 In case of positive evolution, 휌 1 = ∫ 휌 푑휇푞 = 푟0, 푞0 = 1, 푑휇푠 = 휌푑휇푞 and ℑ푞 푠0 = 휌 , otherwise 휌 1 > 1. 휌 휁 = = 푠 is a positive quantity. 0 < 휁 ≤ 1, as it is the inner product of two unit 휌 0 1 1 1 1 2 vectors. In case of positive evolution 휁 = ∫ 휌 푑휇푞 .

′ The definition of 1 requires that 휌 be real-valued. This holds because 푞0, 푟푛 + 푟−푛 = 2휌 cos 휆푛푇 ℏ, 푞0, 푟−푛 − 푟푛 = 2 ∫ 휌 sin 휆푛푇 ℏ 푑휇푞 and 푟푛 ± 푟−푛 , 푞0 = 0 0 < 푛 < 퐿 . As ∫ 휌 푑휇푞 = 1 > 0, 휌 is the unique solution of a system of purely real (integral) equations.

Quantum Evolution and Anticipation

휌 ′ −푖휆푛푇 ℏ ′ 푑휇푠 = 푑휇푞 is a positive measure, 푞0, 푟푛 = 휌 1 ∫ 1 푒 푑휇푠 = 푠0, 푠푛 . In case of 휌 1 ′ ′ positive evolution 1 = 1 a.e., 휌 1 = 1, 푠푛 = 푠푛 and 푠푛 has orthogonal evolution.

−푖휆푛푇 ℏ 휌 is periodic with period 2휋 ℏ 푇, as it has a representation 휌 = 푛 ≤퐿 푐푛 푒 for certain ′ coefficients 푐푛 , which satisfy 푐푛 = 푐 − 푛 because 휌 is real-valued. The same follows for 1 and ℑ푞 푠푛 .

2 The unitary mappings ℑ푥 : ℋ → ℒ 푑휇푥 , existing for 푥 = 푞, 푠, induce unitary mappings 2 휘푥 : 퐶 → ℒ 푑휈푥 between the cyclic space 퐶 and the Lebesgue spaces of the reduced spec- ′ −푖휆푛푇 ℏ tral measures. Due to the periodicity of 휌, 1 and ℑ푞 푠0 there holds 휘푞 푟푛 = 휌푒 in 휌 2 휌 −푖휆푛푇 ℏ 2 1,푑휈푞 ℒ 푑휈푞 , 푑휈푠 = 푑휈푞 and 휘푠 푠푛 = 푒 in ℒ 푑휈푠 , furthermore 휘푠 푞0 = and 휌 1 휌

휘푠 푟0 = 1′ 휌 1 휌 , as well as 휌 푛 = 휌 푛,푑휈푞 .

The reduced spectral measures as well determine the state evolution at integer times: ^ ^ 푞0, 푞푛 = 푑휈푞 푛푇 ℏ and 푟0, 푞푛 = 휌 1푑휈푠 푛푇 ℏ . The proof is given in the following proposition which summarizes the above findings.

Until stated otherwise, we assume below 푇 ℏ = 1 and omit this factor from all formulae.

Proposition 1: Spectral Measures

a) The measures 휇푥 푥 = 푞, 푠 are positive and have total measure 휇푥 ℝ = 1. 푑휇푟 = 2 −1 2 1 2 1 2 −1 2 휌 푑휇푞 , ℑ푠 푞0 = 휌 휌 1 and ℑ푠 푟0 = 1′ 휌 휌 1 . ′ 1 2 −1 2 b) 휌, 1 and ℑ푞 푠0 = 휌 휌 1 are real, periodic functions. c) The measures 휈푥 푥 = 푞, 푠 are positive and have total measure 휈푥 퐾 = 1. 푑휈푠 = −1 −1 2 1 2 1 2 −1 2 휌 1 휌 푑휈푞 , 휘푠 푞0 = 휌 휌 1 and 휘푠 푟0 = 1′ 휌 휌 1 . d) The state evolution at integer times is determined by the reduced measures: ^ ′ ^ 푞0, 푞푛 = 푑휈푞 푛 and 푠0, 푠푛 = 휌 1 1′푑휈푠 푛 . ′ e) In case of positive evolution 푠푛 has orthogonal evolution, 휌 1 = 1, ℑ푞 푠0 = ℑ푞 푠0 = 1 2 1 2 휌 and 휁 = ∫ 휌 푑휇푞 . f) 푞푛 has orthogonal evolution of order 퐿 ≥ 0 iff 휁 = 1 iff 휈푞 = 휈푟 = 휈푠.

Proof: For parts a-c) and e-g) see the above derivations. Part d) follows from

2휋 푚+1 휋 −푖휆푛 −푖휆푛 −푖휅푛 푞0, 푞푛 = 푒 푑휇푞 휆 = 푒 푑휇푞 휆 = 푒 푑휇푞 휆 푚∈ℤ 휆=2휋푚 휅=−휋 휅=Shift 휆 mod 2휋 ^ = 푑휈푞 푛 ∎

From this proposition there follows a simple, but very important observation, which we state in our first theorem.

Quantum Evolution and Anticipation

Theorem 1: Embedded Orthogonal Evolution

Let 푞0 have positive evolution of order 퐿. Then there is a component of 푞0 which has orthogonal evolution of order 퐿. This component is 휁푠0 and has norm 0 < 휁 ≤ 1. Furthermore, Supp 휇푠 ⊂ Supp 휇푞 .

The evolution of 푠0 is the embedded orthogonal evolution of 푞0, and 휁 is its size.

Proof: By proposition 1, 푠0 has positive evolution. The inclusion of the supports is proper, as e.g. a spectral measure of dimension 푑 whose support contains an equidistant point spectrum of dimension 퐿 + 1 ≤ 푑 has an embedded orthogonal evolution of order 퐿. ∎

Until the end of this chapter we will, based on proposition 1d, confine to the reduced spectral measures.

1.2 Characterization of Finite-dimensional Evolution In the finite-dimensional case the cyclic space and the Lebesgue spaces have finite dimension. The spectrum is pure point, and the reduced spectral measure has support on 푑 points. This occurs if either the underlying state space ℋ is finite-dimensional, or if the support of 휇푞 has a regular structure such that the reduction yields only finitely many points. An example is the reduction of the spectrum of the Schrödinger operator −∆ + 푋2 modulo a rational number.

1.2.1 Amplitudes, Characteristic Polynomial, Eigenvalues and Spectra In this section we explore the relationships of amplitudes, characteristic polynomial, eigenvalues and spectra.

−푖휅푛 Let 휅푛 0 ≤ 푛 < 푑 be the ordered sequence of the reduced spectrum, 휔푛 = 푒 the eigenvalues of 푈푇 퐶, and 푤푛 = 휈푞 휅푛 the weights.

Finiteness of the dimension implies 푞푑 = 0≤푛<푑 푎푑−푛 푞푛 , or equivalently 푑 푛 푈푇 푞0 = 0≤푛<푑 푎푑−푛 푈푇 푞0. As 푞0 can be replaced by any other base vector of 퐶, we conclude 푑 푛 푈푇 퐶 = 0≤푛<푑 푎푑−푛 푈푇 퐶. The eigenvalues of 푈푇 퐶 therefore are roots of the characteristic polynomial

푑 푛 (1) 휔푘 = 0≤푛<푑 푎푑−푛 휔푘 0 < 푘 < 푑 .

Notice that the coefficients 푎푘 have only 푑 real degrees of freedom, as they are functions of 푛 the 휔푘 and thus the 휅푛 . Indeed, −1 푎푛 is the basic symmetric function of order 푛 of the 휔푘 , particularly 푎푑 = 1, 푎푛 = 푎푑 푎 푑−푛 and thus 푎푛 = 푎푑−푛 .

The weights 푤푛 are determined by the primary amplitudes from the equation system

푘 (2) 훽푘 = 0≤푛<푑 푤푛 휔푛 0 ≤ 푘 < 푑 .

Given 2푑 − 1 primary amplitudes, the coefficients of the characteristic polynomial are easily found from the linear system

Quantum Evolution and Anticipation

(3) 훽푑+푘 = 0≤푛<푑 푎푛 훽푛+푘 0 ≤ 푘 < 푑 ,

However, 훽0, … , 훽 푑 2 contain 푑 real degrees of freedom and are sufficient to determine 휈푞 , as the 푎푘 are as well solutions of the non-linear system obtained from (3) by the replacement 푑 푎 = 푎 푎 0 < 푛 ≤ and 훽 = 훽 . 푛 푑 푑−푛 2 −푛 푛

Once the characteristic polynomial is found, (1) yields the eigenvalues and (2) the weights.

For any 0 < 퐿 ≤ 푑 − 1 2 and any 휈푞 there are 푑 − 1 − 2퐿 linearly independent real solutions 휌푛 = 휌 휅푛 satisfying the system

(4.1) 훿푘 = 0≤푛<푑 휌푛 푤푛 cos 푘휅푛 0 ≤ 푘 ≤ 퐿

(4.2) 0 = 0≤푛<푑 휌푛 푤푛 sin 푘휅푛 0 < 푘 ≤ 퐿

^ Now 휈푟 and 휈푠 are obtained from proposition 1c). 휌 1 1′푑휈푠 푛 = 훿푛 푛 ≤ 퐿 as required by definition 1j). Thus 휈푞 and 퐿 ≤ 푑 − 1 2 determine the 푑 − 1 − 2퐿-dimensional linear space 퐷푑,퐿 of solutions of (4.1-2) and the corresponding evolution scenarios, where 휈푟 and 휈푠 obey (4.1-2) and satisfy definition 1j).

휈푠 and 휁 do not generally completely determine 휈푞 , as we will show in theorem 2.

For 퐿 > 푑 − 1 2 the system (4) is over-determined. (1) and (4.1) imply 푟0, 푞퐿+1 = 푟0, 푎푘 푞퐿+1−푘 = 푎퐿+1 and 푟0, 푞−퐿−1 = 푟퐿+1, 푞0 = 푎푘 푟퐿+1−푘 , 푞0 = 푎 퐿 + 1 . If 푎퐿+1 = 0, then definition 1j) is met and the evolution scenario has order 퐿 + 1. Induction and 푎푛 = 푎푑−푛 imply 푎퐿 = ⋯ = 푎푑−퐿 = 0. Thus, for special sets of eigenvalues, (4) has a solution, and evolution scenarios with 푑 − 1 2 < 퐿 < 푑 do exist.

푝 푖휗 1 th In case of essentially periodic motion, 푈 = 푒 for some 휗, and the 휔푛 are essentially 푝 unit roots. If 푝 = 푑 < ∞, then the eigenvalues form a complete set of unit roots, and 퐿 = 푑 − 1 −1 iff 푑휈푠 has equal weights 휍푛 ≝ 휈푠 휅푛 = 푝 , as this is the unique solution of (4). In turn, evolution of 퐿 = 푑 − 1 is only possible in the minimal periodic case 푝 = 푑, because equation −1 (4) implies 휍푛 = 푝 .

1.2.2 Domain of Positivity

Let 푑 < ∞ and 휍푛 ≝ 휈푠 휅푛 . The domain of positivity of dimension 푑 and order 퐿 is the set of eigenvalues 휅푛 admitting positive 푑-dimensional evolutions of order 퐿, 푃푑,퐿 = 0 ≤ 휅0 <

⋯ < 휅푑 ≠ 휅0 < 2휋 휍푛 > 0 0 ≤ 푛 < 푑 , where the 휍푛 's satisfy

푘 (4.4) 훿푘 = 0≤푛<푑 휍푛 휔푛 0 ≤ 푘 ≤ 퐿 .

푃푑,0 is the convex sub-set of the 푑-cube of edge length 2휋 defined by the constraint 0 ≤ 휅0 < ⋯ < 휅푑 ≠ 휅0 < 2휋. 푃푑,퐿 ⊂ 푃푑,퐿−1 by definition 1j).

If 0 ≤ 퐿 ≤ 푑 − 1 2 , then 푃푑,퐿 is an open manifold of dimension 푑, because the system (4) is always solvable and establishes in the interior of 푃푑,퐿 a continuous (even analytic) relation- ship between the 휍푛 's and the 휅푛 's. Due to continuity, in some neighborhood of any element of 푃푑,퐿 the 휍푛 's remain positive.

Quantum Evolution and Anticipation

If 푑 > 퐿 > 푑 − 1 2 , then 푃푑,퐿 is an open manifold of dimension 2 푑 − 퐿 − 1, because in this case, as we have seen in the preceding section, some 푎푛 = 0, and each of these complex equations constrains two real degrees of freedom.

The boundary of 푃푑,퐿 consists on the one hand of the manifold 푀푑,퐿 of eigenvalues admitting at least one non-negative but no positive solution of order 퐿, and on the other hand of degenerate elements with one or more degenerate pairs 휅푛 = 휅푚 . By continuity, the 휍푛 's either converge to positive limits as elements of 푃푑,퐿 are degenerating, or the degenerate elements are lying on 푀푑,퐿.

In the former case we identify the degenerate element with an element of 푃푒,퐿, where 퐿 < 푒 < 푑, as 푃푒,퐿 = ∅ for 푒 ≤ 퐿. Each element of 푃푒,퐿 corresponds to 푒 푑 − 푒 degenerate elements of 푃푑,퐿.

Limit points of 푃푑,퐿 (degenerate and non-degenerate) with some zero 휍푛 's are lying on 푀푑,퐿. Each element of 푀푑,퐿 can be constructed as an element of 푃푒,퐿 combined with 푑 − 푒 points in 0,2휋 of zero weight, where 퐿 < 푒 < 푑.

Let 푁푑,퐿 = 푃푑,0 for 퐿 ≤ 푑 − 1 2 , and the sub-set of 푃푑,0 satisfying the special constraints mentioned above in case of 퐿 > 푑 − 1 2 . Then 푃푑,퐿 is open in the topology of 푁푑,퐿, and 푃푑,퐿 ∪ 푀푑,퐿 ⊂ 푁푑,퐿. 푃푑,퐿 ⊂ 푃푑,퐿−1, as 푃푑,퐿 satisfies all constraints that 푃푑,퐿−1 does. 푀푑,퐿 partitions 푃푑,퐿−1 ∩ 푁푑,퐿 into 푃푑,퐿 and 푃푑,퐿−1 ∩ 푁푑,퐿 − 푃푑,퐿.

1.2.3 Summary The following theorem summarizes in parts a-c and e-f the results of the preceding two sec- tions and completes them in parts d and g. It does however not reproduce the full details and all finesses of the above discussion.

Theorem 2: Characterization of Finite-dimensional Evolution If 푑 < ∞, then the reduced spectrum is pure point and

a) 휈푞 is completely determined by 훽푘 0 ≤ 푘 ≤ 푑 2 . b) Evolution scenarios exist for any 휈푞 and 0 < 퐿 ≤ 푑 − 1 2 , as well as for 푑 − 1 2 < 퐿 ≤ 푑 provided that 푎푛 = 0 푑 − 1 2 < 푛 < 퐿 . c) The solutions of the system (4.1-2) form the 푑 − 1 − 2퐿-dimensional linear space 퐷푑,퐿. Each solution completely determines 휈푟 , 휈푠 and 휁. d) If 푑 > 2, then for any 휈s and 0 < 휁 < 1 there are several solutions 휈푞 , 휈푟 . −1 e) 퐿 = 푑 − 1 iff the motion is essentially periodic with 푝 = 푑 iff 휍푛 = 푝 . f) The domain of positivity, 푃푑,퐿, is an open manifold of dimension 푑 푑 > 2퐿 ≥ 0 and 2 푑 − 퐿 − 1 2퐿 ≥ 푑 > 퐿 > 0 , respectively, nested into 푃푑,퐿−1. Its boundary is con- structible from 푃푒,퐿, where 퐿 < 푒 < 푑. g) Necessary and sufficient for any measure 휈푞 to admit non-negative solutions of order 퐿 = 1 is that sup Supp 휈푞 − inf Supp 휈푞 > 휋.

Proof: For parts a-c and e-f) see the derivations above. Part d) We prove the proposition for 1 < 푑 ≤ ∞ and arbitrary spectrum. Let the support of the reduced measures be partitioned into two sets 퐴, 퐵 with non-zero measure 푎, 푏 with respect

Quantum Evolution and Anticipation

to 푑휈푠, respectively. Then there is exactly one solution such that ℑ푠 푞0 has constant value 0 < 푥 ≤ 1, 푦 ≥ 1 on 퐴, 퐵, respectively, because proposition 1 implies 푎푥 + 푏푦 = 휁 and 푎푥2 + 푏푦2 = 1, which yields a quadratic equation with non-negative discriminant, such that there is a unique solution meeting the inequalities. The claim follows from the fact that in the case of pure point spectrum there are 2푑 − 2 partitions satisfying the premises. Part g) One verifies this criterion easily in case of point spectrum by noticing that it guaran- tees opposite sign of the minimum and maximum terms. The extension to continuous spectrum is straight-forward. ∎

1.3 Characterization of Infinite-dimensional Evolution In the infinite-dimensional case the cyclic space and the Lebesgue spaces have infinite dimension. The spectrum is a mixture of absolutely continuous, singular-continuous and discontinuous spectrum. Infinite-dimensional reduced point spectrum has isolated or even dense points of accumulation. An example for the latter is the reduction of the spectrum of 푋2 modulo an irrational number. However, the formulae of theorem 2 below hold even in this case.

Relationships similar to those in the finite-dimensional case hold between the various quantities, with slight differences.

Theorem 3: Characterization of Infinite-dimensional Evolution If 푑 = ∞, the reduced spectrum is a mixture of absolutely continuous, singular-continuous and discontinuous spectrum. The motion is aperiodic.

2 a) 휈푞 is completely determined by 훽푛 푛 ∈ ℤ, 푛 ≥ 0 , 푑휈푞 = 퐷 퐹 − 훽0훿휋 − 훾0퐷훿휋 for the continuous function 퐹 and constant 훾0 given by Lemma A.1 below. b) If 휈푞 has dense support in 퐼 = 휅0, 휅∞ , then for any 0 < 퐿 < ∞ there is a solution 휌. c) For any real measure 휈s, 0 < 휁 < 1 and 퐿 < ∞ there are infinitely many solutions 휈푞 . d) 휁 = 1 implies orthogonal evolution 휈푞 = 휈푟 = 휈푠 of length 퐿 ≤ ∞. 1 e) 퐿 = ∞ implies absolutely continuous spectrum 푑휈 휅 = 푑휅 on 퐾, while 0 < 휁 ≤ 1. 푠 2휋 f) Let 휈푞 have dense support in 휅푛 − 휀, 휅푛 + 휀 , where the 푑-tuple of 휅푛 's is an ele- −1 ment of 푃푑,퐿. Then positive evolution occurs for sufficiently small 0 < 휀 ≪ 퐿 .

Proof: Part a) From 훽−푛 = 훽푛 and Lemma A.1 below.

−푖푛휅 Part b) Partition the support of 휈푞 into 2퐿 + 1 intervals 퐼푚 , define 푐푛푚 = ∫ 푒 푑휈푞 휅 and 퐼푚 2 solve 푚 휌푚 푐푛푚 = 훿푛 , 휌푚 = 1. Then 휌 휅 = 휌푚 휅 ∈ 퐼푚 is a solution.

Part c) The proof of theorem 1b) applies and yields the claim. Part d) By proposition 1f. Part e) By Lemma A.1, the given solution (see example A.1) is unique.

Part f) Partition the support into 푁 intervals and determine 휌 as in part b). As 푁 → ∞, the solutions approximate those of the finite-dimensional problem, and the openness of 푃푑,퐿 guar- antees positivity. ∎ All rights reserved 11/22

Quantum Evolution and Anticipation

2 Anticipation in Quantum Evolution In this chapter we formally define anticipation for positive evolutions, establish a representation of the anticipation amplitudes in terms of the spectral difference and set up the statistical model for the assessment of anticipation in the remaining chapters.

2.1 Anticipation and Retrospection

Definition 3: Anticipation Amplitudes and Probabilities

Consider a positive evolution of order 퐿. Then 푠0 has orthogonal evolution of order 퐿, and for 0 ≤ 푛 ≤ 퐿 and 0 < 휁 ≤ 1

1 1 휋 −푖 푛+ 휅 th a) 훼 = 휁 푈 푠 , 푈 푠 = 휁푑휇 푛 + = 휁 ∫ 푒 2 푑휇 휅 is the 푛 anticipation 푛 푛푇 0 −푇 2 0 푠 2 −휋 푠 amplitude 2 th b) 푝푛 = 훼푛 is the 푛 anticipation probability. ∎

Let 퐴 be an observable defined on Span 푠푛 0 ≤ 푛 ≤ 퐿 (whose dimension lies in the range 퐿 + 1, 푑 ), such that 푠푛 0 ≤ 푛 ≤ 퐿 is an eigenfunction of 퐴. Then 푝푛 is the probability to ob- 푇 푇 tain 푠 , when measuring 퐴 at time − . Thus 푝 is the probability to observe at time − that 푛 2 푛 2 component of the embedded orthogonal evolution appearing in the sequence of discrete evo- 푇 lution steps not before the 푛th step: With probability 푝 , measurement at time − anticipates 푛 2 the state at time 푛푇, and projects the target into state 푠푛 .

This is the reason to name the effect anticipation. Due to anticipation, measurement interac- tions can speed-up processes and de-stabilize states. Let's define "computation" as "con- trolled state preparation by quantum evolution". Then due to anticipation measurements at 푇 time − may provide a rapid advance of computation results. 2

푇 If 퐵 is an observable with eigenfunctions 푠 −퐿 ≤ 푛 ≤ 0 , then measurement of 퐵 at time + 푛 2 will project the state back to a previous state. This effect, which we name retrospection, can slow-down processes and stabilize states. Retrospection amplitudes and probabilities can be defined analogous to definition 3 above, and all of the following analyzes of anticipation hold as well for retrospection.

In case of non-positive evolution measurements are less likely to project states forward or backwards on their trajectory, as there is no embedded orthogonal evolution. Therefore we confine our definition and analysis of anticipation and retrospection to positive evolutions. The occurrence of positive evolutions will be investigated in a future paper.

2.2 Anticipation Amplitudes and the Spectral Difference

In chapter 1 we have seen that the evolution is determined by the reduced spectra. The anticipation amplitudes being defined at half-step time thus can be represented by the reduced spectra of the evolution with step size 1 2. In definition a) below the modular factor selects from the real axis the odd and even numbered intervals of length 2휋, respectively.

Recall from (Last, 1996) that the measure 푔 is absolutely continuous with respect to the measure 푕, iff 푑푔 = 푓푑푕, for some Borel function 푓.

Quantum Evolution and Anticipation

Definition 4: Spectral Difference

a) 푑푥푠,푗 Shift 휅 = 푑휇푠,4휋 휅 + 2휋푗 푗 = 0,1 0 ≤ 휅 ≤ 2휋, 푗 = 0,1 . b) 푑푥푠 = 푑푥푠,0 − 푑푥푠,1 is the spectral difference measure. c) 휇 is well-behaved iff 푑푥푠,푖 = 푔푖 푑푕 푎. 푒. for some measure 푕 and Borel functions 푔푖. 푔 −푔 d) Let 휇 be well-behaved. Then 푦 = 0 1 is the spectral difference function. 푔0+푔1 See definition 2d-e) for the notation used in a). ∎

In the sequel, we require 휇 to be well-behaved, where 푕 is admitted to be a spectral measure of arbitrarily general type. The following lemma shows that this is a weak condition.

Lemma 1: Well-behaved measures

a) If 휇 is pure point, then it is well-behaved. b) If 휇 is absolutely continuous, then it is well-behaved. c) If the support of 푑푥푠 can be partitioned into measurable sets, such that on each set the restrictions of 푑푥푠,0 and 푑푥푠,1 to this set are normally related, then 휇 is well- behaved. Two measures, 푓 and 푔, supported on a set 푆, are said to be normally related if either 푓 is absolutely continuous w.r.t. 푔, or vice versa, or 푆 has zero measure under one of them.

Proof:

a) 푑푥푠,0 and 푑푥푠,1 are pure point and meet the premises of c). b) 푑푥푠,0 and 푑푥푠,1 are absolutely continuous and meet the premises of c). c) Consider a set 푆 and the restriction of the measures to this set. If 푑푥푠,0 is absolutely 푑푥 푠,0 continuous w.r.t. 푑푥푠,1, then set 푑푕 = 푑푥푠,1, 푔1 = 1, 푔0 = . If 푑푥푠,0 is singular w.r.t. 푑푥 푠,1 푑푥푠,1, then set 푑푕 = 푑푥푠,0, 푔0 = 1, 푔1 = 0. The reversed cases are analogous. ∎

In the well-behaved case the following representations hold:

Proposition 2: Representation of Anticipation Amplitudes

a) 푑푥푠,0 + 푑푥푠,1 = 푑휈푠, 푑푥푠 = 푦푑휈푠. 1 1 휋 −푖 푛+ 휅 b) 훼 = 푑푥^ 푛 + = ∫ 푒 2 푑푥 휅 . 푛 푠 2 −휋 푠 2 c) 푦 is a real function in ℒ 푑휈푠 , 푦 ≤ 1. 1 d) 훼 = 휁 푦푑휈 ^ 푛 + . 푛 푠 2

Proof:

a) By definition 2 and 4) and the well-behavior of 휇. b) 휋 1 1 −푖 푛+ 휅 −푖 푛+ 2휋+휅 훼푛 = 휁 푒 2 푑푥푠,0 + 푒 2 푑푥푠,1 −휋

Quantum Evolution and Anticipation

휋 1 −푖 푛+ 휅 = 휁 푒 2 푑푥푠,0 − 푑푥푠,1 . −휋 c) By definition 4d). d) Straight-forward from part a-b) and definition 2. ∎

2.3 Assessment Methodology To assess anticipation strength, we consider laboratory and field situations where the primary base state is sampled from a certain distribution and its evolution at a certain step time is observed. This distribution induces a distribution of 푑휇푞 defined on a probability space of spectral measures. In our analyses we focus on distributions conditional under 푑푥푞,0 + 푑푥푞,1 = 푑휈푞 , from which any other distributions are obtained as products with the distribution of 푑휈푞 . In this conditional model only the spectral difference measure 푑푥푠 is stochastic, whe- reas 푑휈푞 is pre-scribed.

In contrast to (Thomann, 2008), where we asserted, that technologies for the preparation of orthogonal evolutions of a given step size were likely to exhibit orthogonal anticipation of sig- nificant strength, we do not make here any specific assumptions on the distribution of the 2 spectral difference, but will make reference to its mean 푦1 and variance 푦2 = 퐸 푦 − 푦1 under the given distribution, which by the boundedness of 푦 always exist.

The following definitions are used below:

Definition 5:

a) 푃푁 = 0≤푛<푁 푝푛 is the probability to measure one of the states 푞푛푇 0 ≤ 푛 ≤ 푁 at time 푇, 2 푟 푟 b) 푛 푁 ≝ 0≤푛<푁 푛 푝푛 . The special case 푟 = 1 is the anticipation look-ahead. ∎

2.4 Anticipation Strength In this chapter we determine the anticipation strength for model and general type states and measures with 퐿 ≤ ∞. We focus on positive evolutions with well-behaved measures and their embedded orthogonal evolutions. These results generalize those of (Thomann, 2008).

Theorem 4: Anticipation Strength 푛+푚 푁 Define 휅 = 0 ≤ 푛 < 퐿, 0 ≤ 푚 ≤ 푀, 0 < 푀 ≤ 푁 . 푛,푚 퐿

a) Let 휇푞 be well-behaved and 푞0 have positive evolution of order 퐿 and size 휁. Then 1. 2 2 2 푃퐿 ≤ 휁 푦 2,푑휈푠 ≤ 휁 ≤ 1 2 2 2. 퐸 푃 ≤ 휁 푦 + 퐿 푦 2 , where 푑휈 is the pure point compo- 퐿 1 2,푑휈푠 2 1,푑휈푠,푝푝 푠,푝푝 2 nent of 푑휈푠 and 푑휈푠,푝푝 its square. 푐 b) Let 푦 be such that 푥 휅 , 휅 = for some 0 < 퐿, 0 < 푐 ≤ 1, 0 < 푀 = 휀푁 and 푠 푛,푚 푛,푚+1 퐿푀 0 < 휀 < 1. Then for 퐿 → ∞ and 푁 → ∞

Quantum Evolution and Anticipation

1 2 sin ε휋 푛− 퐿 2 푐 2 4푐 1. 푝푛 ≈ 1 1 with maximum value ≈ at 푛 ∈ 0,1, 퐿 − sin 휋 푛− 퐿 ε휋 푛− 휋2 2 2 ε휋 퐿 1, 퐿 , minimum value ≈ 4ε−2π−2퐿−2푐2 sin2 → 퐿−2푐2 ε → 0 at 푛 = , and 2 2 퐿 푝 ∈ 푂 푛−2 푛 → . 푛 2 2. 푛1 = ln 퐿 + 푂 1 , 푛푟 = 푂 퐿푟−1 푟 > 1 . 2 −1 3. 푃퐿 = 푐 1 + O 퐿 . 푐 −1 푛 c) Let 푦 be such that 푥 휅 , 휅 = for some 0 < 퐿, 0 < 푐 ≤ 1, 0 < 푀 = 휀푁 푠 푛푚 푛,푚+1 퐿푀 and 0 < 휀 < 1. Then for 퐿 → ∞ and 푁 → ∞ 1 2 sin ε휋 푛− 퐿 푐 2 −2 2 1. 푝푛 ≈ 1 1 with minimum value ≈ 퐿 푐 at 푛 ∈ cos 휋 푛− 퐿 ε휋 푛− 2 2 ε휋 4푐 2 퐿 0,1, 퐿 − 1, 퐿 , maximum value ≈ 16ε−2π−4푐2 sin2 → ε → 0 at 푛 = , 2 휋2 2 퐿 −2 퐿 and 푝 ∈ 푂 − 푛 푛 → . 푛 2 2 2. 푛푟 = 푂 퐿푟 푟 > 1 . ε휋 3. 푃 = 4ε−2π−2푐2 sin2 1 + O 퐿−1 → 푐2 휀 → 0, 퐿 → ∞ . 퐿 2

Proof: Part a.1) By theorem 2, 푠0 is the embedded orthogonal evolution. As 푠0, … , 푠퐿 are mutually orthogonal, 2 , where equals the projection of on . 푃퐿 = 푧 2,푑휈푠 푧 푦 Span 푠0, … , 푠퐿 Part a.2) By (Last, 1996) 휈푠 can be decomposed in a continuous and a pure point part, which are mutually singular. Therefore the second integral in the decomposition 1 −푖 푛+ 휅−휅′ 2 ′ ′ 2 only 퐸 푃퐿 = ∬휅≠휅′ 푒 푦1 휅 푦1 휅 푑휈푠 휅 푑휈푠 휅 + ∫ 퐸 푦 휅 푑휈푠 휅 푑휈푠 휅 2 2 2 depends on 푑휈푠,푝푝 . The result follows from the identity 푦2 = 퐸 푦 − 푦1 . Notice that, if the evolution is of order 퐿, then 푑휈푠,푝푝 is either zero or at least of dimen- 2 2 sion 퐿 + 1, thus 퐿 + 1 ∫ 푑휈푠,푝푝 ≤ ∫ 푑휈푠,푝푝 . Part b.1) As 푁 → ∞ the anticipation amplitudes approximate 1 1 1 휀 2 −2휋푖 푛− 푚+휅 퐿 푑휅 휀 2 −2휋푖 푛− 휅 퐿 푑휅 −2휋푖 푛− 푚 퐿 훼 = 퐿−1 ∫ 푐푒 2 = 푐 ∫ 푒 2 퐿−1 푒 2 . 푛 푚=0 −휀 2 2휋 −휀 2 2휋 푚=0 The maxima, minima and asymptotics are elementary. Part b.2) From b.1) by the Euler summation formula. Part b.3) By Lemma 2 below. Part c.1) In the same way as part b.1). 퐿 푟 Part c.2) From c.1) and c.3), as 푛푟 ≥ 푃 . 4 퐿 Part c.3) By Lemma 1 below. ∎

Notice in parts b-c that 푥푠 approximates an evolution of order 퐿, but this 퐿 is unrelated to the order of the evolution under 휇푞 . The following lemma is obtained by direct summation, adapt- ing theorem 4.9a of (P.Henrici, 1988) to periodic functions.

Lemma 2 1 푝−1 푝−2 sin−2 휋 푛 − 푝 = 1. 푛=0 2

Quantum Evolution and Anticipation

3 Conclusions We have classified and analyzed quantum evolutions with arbitrary spectrum and shown in theorem 1 that every positive evolution (see definition 2.c) possesses an embedded orthogonal evolution. Our findings on eigenvalues, spectra and domains of positivity have been summarized in theorem 2 for the finite-dimensional and in theorem 3 for the infinit- dimensional case. Anticipation and retrospection have been introduced (see definition 3) for positive evolutions, and their strength has been assessed in theorem 4 in the general setting of well-behaved measures (see definition 4.c).

Our analyzes show that anticipation and retrospection occur in a wide class of quantum evolutions, generalizing and extending (Thomann, 2008) where only equally distributed measures were considered.

An interesting open question is how systems exhibiting anticipation and retrospection interact with other systems, particularly with the environment. In the presence of einselection (Zurek, 2003), interesting phenomena may occur. E.g., if decoherence sets in randomly in the inter- 퐿−1 val 0, 퐿푇 , then theorem 4.a.1. implies 푛 ≤ 휁2 푦 2 . Given suitable timing, a Zeno- 퐿 2 2,푑휈푠 like effect may occur, but due to anticipation and retrospection with rapid evolution forward or even backward in time.

While the domain of positivity has been analyzed in our study, the requirement of positive evolution in the definition of anticipation and retrospection raises the demand for the demon- stration of the presence of positive evolution in physical model systems. This will be the subject of a future paper, presenting numerical simulations showing that anticipation and retrospection occur in a variety of spectra including that of the H-atom. The quantum anticipation simulator software running und Microsoft Windows is already downloadable from the author's homepage http://www.thomannconsulting.ch/public/aboutus/aboutus-en.htm.

Quantum Evolution and Anticipation

4 Bibliography

Aitken, A. (1956). Determinants and Matrices. Oliver and Boyd. Killip, R. ,. (2001). Dynamical Upper Bounds on Wavepacket Spreading. arXiv:math.SP/0112078v2. Kiselev, A. L. (1999). Solutions, Spectrum and Dynamics for Schrödinger Operators on Infinite Domains. math.SP/9906021. Last, Y. (1996). Quantum Dynamics and Decompositions of Singular-Continuous Spectra. J. Funct. Anal. 142, pp.406-445. Nielsen, O. (1997). An introduction to integration and measure theory. New York: Wiley. P.Henrici. (1988). Applied Computational and Complex Analysis", Vol.I . Wiley. Reed, M. S. (1980). Methods of Mathematical Physics, Vol.1-4. Academic Press. Rudin, W. (1973). Functional Analysis. Mc Graw Hill. Simon, B. (1990). Absence of Ballistic Motion. Commun. Math. Phys. 134, pp.209-212. Strichartz, R. (1990). Fourier asymptotics of fractal measures. J.Funct.Anal. 89 , pp.154-187. Thomann, H. (2006). Instant Computing – a new computing paradigm. arXiv:cs/0610114v3. Thomann, H. (2008). Orthogonal Evolution and Anticipation. arXiv:0810.1183v1. Thomann, H. (July 2009). Quantum anticipation simulator. http://www.thomannconsulting.ch ("About us" page) . Zurek, W. (July 2003). Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys., Vol 75 .

Quantum Evolution and Anticipation

Appendix A: Spectral-analytic Lemmata This appendix contains some useful lemmas from spectral analysis which are referenced in the main body of this paper. Our proofs are original, and we do not know of any references in the literature for them.

A.1 Inversion of the Discrete Fourier Transform 퐝훎^ 퐧 퐧 ∈ ℤ Integrating any finite measure 휇 yields a continuous function 푓. As the reduced measure 푑휈 has support in the compact set 퐾, it depends only on 퐹 = 푓 퐾, which in turn can be represented by a Fourier series. Explicit formulae for the coefficients of this series in terms of the discrete Fourier transform 푑휈^ 푛 are given below.

Lemma A.1: Inversion of the Discrete Fourier Transform 퐝훎^ 퐧 퐧 ∈ ℤ

Let 휈 be a finite complex measure supported in the half-open interval 퐾 = −휋,휋 , and ^ ^ −푖휅푡 훽푛 = 푑휈 푛 푛 ∈ ℤ be given, where 푑휈 푡 = ∫ 푒 푑휈 휆 . Then

a) There is an absolutely continuous function 퐹 supported on 퐾 such that 2 푑휈 = 퐷 퐹 − 훽0훿휋 − 훾0퐷훿휋 , 휕 where the derivative 퐷 = 2휋푖 is taken in the distributional sense. 휕휅

휅2 훽 b) 2휋 2퐹 = 퐵 휅 + 푐 − 2휋푖훾 휅 − 훽 휅 ≤ 휋 , where 퐵 휅 = 푛 푒−푖푛휅 , 0 0 2 푛≠0 푛2 휕 휋 훾 = 퐵 −휋 + 훽 휋 and 푐 = −퐵 −휋 − 푖훾 휋 + 훽 . 0 휕휅 0 0 0 4

c) For any test function 휙 훽 2휋푖휈 ∙ 휙 = 푛 푒−푖푛휅 ∙ 휙 + 푖훽 휅 − 2휋훾 ∙ 휙 푛 0 0 푛≠0

−푖푛휅 푑휈 ∙ 휙 = 훽푛 푒 ∙ 휙

Proof:

All references qualified by an "R" in the proof below address (Rudin, 1973). Notice that, other than (Rudin, 1973), we are scaling 퐷 by the factor 2휋, to let 퐾 have unit Lebesgue measure.

W.l.o.g. 훽푛 ≤ 1. Define 휈 휅 = 휈 −∞,휅 . Then 휈 −휋 = 0 and 휈 휋 = 훽0. Let 훾푛 = 휋 푑휅 ∫ 휅푛 휈 휅 . −휋 2휋

First notice that Λ휙 = ∫ 휙푑휈, defined for test functions 휙, is a functional of zero order in the sense of the note following theorem R6.8, as Λ휙 ≤ max휅∈퐾 휙 . By theorem R6.27 and ex- ercise R6.17 there is a continuous function 푓 ∈ 퐶 ℝ such that 푑휈 = 퐷2푓 in the distributional sense, i.e. 퐷Λ휙 = −Λ퐷휙 (see R6.12). To adjust for the factor 2휋푖, 퐷푓 = −푖 2휋 −1휈. W.l.o.g. 푓 휅 = 0 휅 ≤ −휋 .

Quantum Evolution and Anticipation

We represent 퐹 = 푓 퐾 as a function on ℝ by defining 퐹 = 푓 1 − 퐻휋 , using the Heaviside 휋 푑휅 function 퐻 = 휅 ≥ 휋 . With 퐷퐻 = 2휋푖훿 , 푔훿 = 푔 푥 훿 and 푓 휋 = ∫ −푖 휈 휅 = 휋 푥 푥 푥 푥 −휋 2휋 2 −1 −푖 2휋 훾0 we find

(1.1) 퐹 = 푓 1 − 퐻휋 −1 −1 (1.2) 퐷퐹 = −푖 2휋 휈 1 − 퐻휋 − 2휋푖푓훿휋 = −푖 2휋 휈 1 − 퐻휋 − 훾0훿휋 휕 (1.3) 퐷2퐹 = 1 − 퐻 푑휈 − 휈훿 − 훾 퐷훿 = 푑휈 − 훽 + 2휋푖훾 훿 휋 휋 0 휋 0 0 휕휅 휋

The products 푓 1 − 퐻휋 and 휈 1 − 퐻휋 are functionals, have been differentiated using the product rule, which is not generally applicable (see R6.14 and R6.15). However, integration 휋 by parts of the r.h.s. of ′ ′ yields the r.h.s. of 퐷푓 1 − 퐻휋 ∙ 휙 = −퐷푓 1 − 퐻휋 ∙ 휙 = − ∫−휋 푓휙 ∞ ∞ 1.2, and −1 ′ ′ ′ . 퐷 −푖 2휋 휈 1 − 퐻휋 ∙ 휙 = −휈 1 − 퐻휋 ∙ 휙 = − ∫−∞ 휈휙 + ∫휋 휈휙 = 푑휈 − 휈훿휋 ∙ 휙

This proves part a).

Now by theorem 7.15, the Fourier transform 퐷2퐹 ^ 푡 = 2휋 2푡2퐹^, by example 7.16.3, ex- ^ −1 −푖휋푡 ^ −푖휋푡 ample 7.16.5 and theorem 7.2a) 훿휋 = 2휋 푒 and 퐷훿휋 = 푡푒 , thus

푑휈 ^− 훽 +2휋훾 푡 푒−푖휋푡 (2) 퐹^ = 0 0 . 2휋 2푡 2

As 퐹 is continuous on a compact set (the closure of 퐾), the Fourier series

−푖푛휅 (3) 퐹 = 푏푛 푒 exists, and its Fourier transform

sin 휋 푡−푛 (4) 퐹^ 푡 = 푏 . 푛 휋 푡−푛

Combined with (6) this yields

훽 − −1 푛 훽 +2휋푛훾 (5) 푏 = 퐹^ 푛 = 푛 0 0 . 푛 2휋 2푛2

The singularity at the origin is removable; 푑휈^ is analytic, as 푑휈 has support in 퐾 and −1 ^ ^ 푡 푑휈 푡 = 휈 푡 = 훾0 푡 = 0 . Therefore 푏0 is well defined.

We insert (5) into (3) and sum the terms on the r.h.s. of (5) up separately into the absolutely convergent series

훽 (6) 퐵 휅 = 푛 푒−푖푛휅 푛≠0 푛2

휕 −1 푛 −1 푛 1+푒−푖휅 and the two sums considered next. As 푖 푒−푖푛휅 = 푒−푖푛휅 = ln = 휕휅 푛≠0 푛2 푛≠0 푛 1+푒푖휅 휅2 ln 푒−푖휅 = −푖휅 휅 < 휋 , the absolutely convergent series on the l.h.s. equals 푐 − for some 2 constant 푐. Redefining 푐 = 푐 + 푏0 we obtain

Quantum Evolution and Anticipation

−푖푛휅 (7.3) 푑휈 ∙ 휙 = 푛≠0 훽푛 푒 ∙ 휙 + 훽0 ∙ 휙 휕 (7.4) 2휋푖훾 = 퐵 −휋 + 훽 휋 0 휕휅 0 휋2 (7.5) 푐 = −퐵 −휋 − 2휋2푖훾 + 훽 0 0 2 where (7.1) is defined for 휅 ≤ 휋 due to the continuity of 퐹, (7.2) and (7.3) for test functions 휙 with support in 퐾.

(7.1) follows from the norm convergence of 퐵, (7.2) and (7.3) from 6.17, as 퐵 ∙ 휙 = 훽 푛 푒−푖푛휅 ∙ 휙 exists for every 휙 due to the norm convergence of 퐵 and boundedness of 푛≠0 푛2 푒−푖푛휅 .

This proves part b-c).

(7.4) and (7.5) are determined from the boundary condition 휈 −휋 = 푓 −휋 = 0. 퐹 is absolutely continuous if 휈 is continuous, but may not even be one-sided differentiable in the singular-continuous spectrum and at points of accumulation of the discontinuous spectrum. In that case the r.h.s. of (7.4) cannot be evaluated. 훾0 may be approximated from (7.2) using a sequence of test functions converging to 퐻−휋 − 퐻휋 . 푑휈 is always well defined by (7.3). ∎

The following examples demonstrate the power of lemma 1.

Examples

1) Infinite orthogonal evolution 훽푛 = 훿푛 . 1 2휅 휅2 1 휅 퐵 = 0, 퐹 = −1 − − , 휈 = 1 + , 푑휈 = 1. 8 휋 휋2 2 휋 2) Periodic evolution 훽푛 = 훿푛 mod 푝. For 0 ≤ 휆 = 휅 + 휋 − 2휋푚푝−1 < 2휋푝−1, 0 ≤ 푚 < 푝: 휕 −푖 1−푒−푖푝휅 퐵 = 푒−푖푛푝휅 = 푖푝−1 ln = 푖푝−1 ln 푒−푖 휋+푝휅 = 푖푝−1 ln 푒−푖 휋+푝휆 = 휆 + 휕휅 푛≠0 푛푝 1−푒+푖푝휅 푝−1휋. Inserting this into (11.2) yields 휈 = 푚푝−1, as the 휆 term is cancelled out and the −푖푛푝휅 boundary condition at – 휋 is met. (11.3) yields 퐵 = 푛≠0 푒 which for 푝휅 ≠ 푒−푖푝휅 푒푖푝휅 0 mod 2휋 equals + = −1, i.e. 푑휈 = 퐵 + 1 = 0, but at the singularities 1−푒−푖푝휅 1−푒푖푝휅 −1 equals (by lemma A.2.c below) the 훿 functional, so 푑휈 = 0≤푚<푝 푝 훿2휋푚 푝−1 . 푖푛휅푚 3) Finite or infinite point spectrum 훽푛 = 훼푚 푒 , 훼푚 = 1. −푖푛 휅−휅푚 By (7.3) and example 2, 푑휈 = 훽0 + 푛≠0 푚 훼푚 푒 = 훽0 + 푚 훼푚 훿휅푚 − 1 =

푚 훼푚 훿휅푚 .

A.2 Representation of the Dirac -Functional This lemma establishes a representation of the Dirac 훿-functional for a wide range of measures, including Lebesgue and pure point measure. It complements theorem 6.32 from (Rudin, 1973).

Quantum Evolution and Anticipation

Recall from (Last, 1996) that a positive measure 휇 is 훼-Hölder-continuous at 푥, if 휇 푥−훿 2,푥+훿 2 lim⁡sup < ∞ for some 0 ≤ 훼 ≤ 1, 훼-Hölder-continuous in a set 퐴 if 훼- 훿→0 훿훼 Hölder-continuous for all 푥 ∈ 퐴 and uniformly 훼-Hölder-continuous (푈훼퐻) in 퐴, if 휇 푥−훿 2,푥+훿 2 lim⁡sup < 퐶 < ∞ for all 푥 ∈ 퐴. 훿→0 훿훼

Lemma A.2: Representation of the Dirac -Functional Let 휇 be a finite, positive measure 훼-Hölder-continuous at the origin and define the functional 훿 by 훿 휙 = 휙 0 for any test function 휙. For any test function Ψ supported in 퐼 = 1 1 훼 − , + define Ψ푁 휅 = Ψ 푁휅 . Then for 0 ≤ 푐 = lim⁡sup ∫ 푁 Ψ푁 휅 푑휇 휅 < ∞ 2 2 푁→∞

훼 (a) lim sup 푁 Ψ푁푑휇 = 푐 훿 , in the sense that, for any test function 휙: 푁→∞ 훼 lim⁡sup ∫ 푁 Ψ푁 휅 휙 휅 푑휇 휅 = 푐 휙 0 . 푁→∞ (b) If 훼 = 1 and 푑휇 휅 = 푓푑휅, where 푓 is continuous at the origin, then the limit lim⁡푁Ψ푁푑휇 = c훿 exists and 푐 = 푓 0 ∫ Ψ 푑휅. 푁→∞ (c) If 훼 = 0, then lim Ψ푁푑휇 = 푐훿 exists and 푐 = Ψ 0 휇 0 . 푁→∞

Proof: Part a) 1 1 As Ψ has support 퐼 = − , + , the integration domain is 퐼 . Therefore 푁 푁 2푁 2푁 푁 훼 훼 lim⁡sup 푁 ∫ Ψ푁 휅 휙 휅 푑휇 휅 = lim⁡sup 휙 0 푁 ∫ Ψ푁 휅 푑휇 휅 = 푐 휙 0 , where 푁→∞ 푁→∞ 훼 훼 훼 푐 = lim⁡sup ∫ 푁 Ψ푁 휅 푑휇 휅 ≤ lim⁡sup푁 ∫ Ψ푁 휅 푑휇 휅 ≤ Ψ ∞,푑휅 lim⁡sup 푁 휇 퐼 < ∞. 푁→∞ 푁→∞ 푁→∞ The last inequality follows from the premises.

Part b) holds as by definition 푑휇 휅 = 푓 휅 푑휅 ≈ 푓 0 푑휅 휅 ∈ 퐼푁 .

Part c) is a consequence of the definitions and , where the ∫ Ψ푁푑휇 = ∫휅≠0 Ψ푁푑휇 + Ψ 0 휇 0 first integral vanishes in the limit due to the contraction of I푁. ∎

Corollary A.2 Let 휇 and 퐼 be as in lemma A.2, 휇 supported in 퐼. Then

훼−1 2휋푖푛휅 (a) There is a constant 0 ≤ 푐 ≤ ∞ such that lim⁡sup푁→∞ 푁 푛 <푁 2 푒 푑휇 휅 = sin 휋푁휅 푐 훿 , in the sense that lim⁡sup푁훼−1 ∫ 휙 휅 푑휇 휅 = 푐 휙 0 for any test func- 푁→∞ sin 휋휅 tion 휙 supported in 퐼. (b) If 훼 = 1 and 푑휇 휅 = 푓푑휅, where 푓 is continuous at the origin, then the limit 훼−1 2휋푖푛휅 lim푁→∞ 푁 푛 <푁 2 푒 푑휇 휅 = c훿 exists and 푐 = 푓 0 . (c) If 훼 = 0 and the origin is an isolated point of the support of 휇, then 훼−1 2휋푖푛휅 lim푁→∞ 푁 푛 <푁 2 푒 푑휇 휅 = c훿 exists and 푐 = 휇 0 . (d) If 휇 is supported in ℝ and 훼-Hölder-continuous at 푛 ∈ ℤ then 훼−1 2휋푖푛휅 lim⁡sup푁→∞ 푁 푛 <푁 2 푒 푑휇 휅 = 푛∈ℤ 푐푛 훿휅−푛 .

Quantum Evolution and Anticipation

Proof: Part a) 1 1 2 If ≥ 휅 ≥ 휀 > 0, then sin 휋휅 ≥ sin 휋휀 and 푁훼−1 ∫ sin 휋푁휅 푑휇 휅 ∈ 푂 푁훼−1 for 훼 < 1, and 2 휀 1 2 cos 휋푁휅 for 훼 = 1: ∫ sin 휋푁휅 푑휅 = − ∈ 푂 푁−1 . Thus this part of the integral vanishes. 휀 휋푁

sin 휋푁휅 sin 휋푁휅 If 0 ≤ 휅 < 휀, then sin 휋휅 = 휋휅 + 푂 휀2 . Therefore 푁훼−1 ≈ 푁훼 ≈ 푁훼 . As the sin 휋휅 휋푁휅 sin 휋휅 integral vanishes for 휅 > 휀, the proof of lemma A.2.a extends to . However, 푐 may be 휋휅 infinite (see remark below).

sin 휋휅 2 Part (b) follows from ∫ 푑휅 = Si ∞ = 1. 휋휅 휋

sin 푁휋휅 Part (c) follows from → 1 휅 → 0 and 푁−1 1 = 1 + 푂 푁−1 . 푁휋휅 푛 <푁 2

Part (d) is an immediate consequence of (a) and the periodicity of sin 휋푁휅 . sin 휋휅 ∎

Even uniform 훼-Hölder-continuity of 휇 in a neighborhood of the origin is not sufficient for finiteness of the upper bound in corollary A.2.a above, as it does not imply more than 푁−1 푁−1 휀 sin 휋푁휅 1 휇 − ,+ ∫ 푁훼 푑휇 휅 ≤ ∙ 2 2 ∈ 푂 ln 푁 . On the other hand, the Cantor −휀 푁휋휅 푛 <푁휀 푛 푁−훼 measure and many other self-similar measures, as well as the conditions of corollary A.2.b-c, yield finite bounds. Thus this corollary is widely applicable.

A.3 Time-averages of 풅흁^ 풕 ퟐ 2 훼−1 푇 2 2 (Last, 1996) analyzes the time average 1−훼 and proves 푑휇^ 푡 푇 = 푇 ∫−푇 2 푑휇^ 푡 푑푡 in theorem 2.5 the following result first established by (Strichartz, 1990):

Let 휇 be a finite 푈훼퐻 measure and 푓 ∈ ℒ2 푑휇 . Then there is a constant 퐶 only depending on 2 2 휇 such that 푓푑휇 ^ 푡 푇1−훼 < 퐶 푓 2,푑휇 .

Notice however, that these are asymptotic results. In evolutions of order 퐿 theorem 4.a.1 im- 푇 2 plies 2 for , while the l.h.s. diverges for and by Stri- ∫−푇 2 푑휇^ 푡 푑푡 ≤ 1 푇 = 퐿 훼 < 1 푇 → ∞ chartz' theorem.