Quantum-mechanical Signal Processing And

- Only title to specifically address the use of quantum mechanical methods in spectral analysis and signal processing.- Chapter summaries for ease of reference, and an extensive reference list for further reading.- Established author, well known in both atomic/molecular and medical physics communities, with recent book in similar area.

Preface

ix

Acknowledgments

xii

Acronyms used in the text

xiii

Introduction

1

(19)

Auto-correlation functions

20

(3)

The time-independent Schrodinger equation

23

(4)

The time-dependent Schrodinger equation

27

(4)

Equivalence: auto-correlation functions and time signals

31

(7)

Orthonormality and completeness of expansion functions

38

(6)

Difference equations and the harmonic inversion

44

(12)

The Schrodinger eigenproblem in global spectral analysis

56

(4)

Dimensionality reduction in the frequency domain

60

(7)

Dimensionality reduction in the time domain

67

(6)

The basic features of the Pade approximant (PA)

73

(8)

Gaussian quadratures and the Pade approximant

81

(5)

Pade-Schur approximant (PSA) with no spurious roots

86

(16)

Spectral analysis and systems of inhomogeneous linear equations

102

(2)

Exact iterative solution of a system of linear equations

104

(16)

Relaxation methods and sequence accelerations

120

(2)

Quantification: harmonic inversion in the time domain

122

(7)

Enhanced convergence of sequences by nonlinear transforms

129

(4)

The Shanks transform as the exact filter for harmonic inversion

133

(17)

The base-transient concept in signal processing

150

(2)

Extraction of the exact number of transients from time signals

152

(8)

Explicit filtering of one transient cn = d1un1

152

(2)

Explicit filtering of two transients cn = d1un1 + d2un2

154

(2)

Explicit filtering of three transients cn = d1un1 + d2un2 + d3un3

156

(2)

Explicit filtering of K transients cn = d1un1 +d2un2 +d3un3 + ... + dkunk

158

(2)

The Lanczos recursive algorithm for state vectors ψn

160

(8)

The Lanczos orthogonal polynomials Pn(u) and Qn(u)

168

(6)

Recursions for derivatives of the Lanczos polynomials

174

(2)

The secular equation and the characteristic polynomial

176

(4)

Power series representations for two Lanczos polynomials

180

(4)

The Wronskian for the Lanczos polynomials

184

(2)

Finding accurate zeros of high-degree polynomials

186

(3)

Recursions for sums involving Lanczos polynomials

189

(5)

The Lanczos finite-dimensional linear vector space LM

194

(3)

Completeness proof for the Lanczos polynomials

197

(6)

Duality: the states ψn and polynomials Qn(u)

203

(2)

An analytical solution for the overlap determinant det Sn

205

(2)

The explicit Lanczos algorithm

207

(3)

The explicit Lanczos wave packet propagation

210

(3)

The Pade-Lanczos approximant (PLA)

213

(5)

Inversion of the Schrodinger matrix u1 - U by the PLA

218

(4)

The exact spectrum via the Green function series

222

(5)

Uniqueness of the amplitudes {dk}

227

(3)

Partial fractions in the PLA only from the denominator polynomial

230

(2)

The Lanczos continued fractions (LCF)

232

(3)

Equations for eigenvalues uk via continued fractions

235

(2)

The exact analytical expression for the general Pade spectrum

237

(6)

The exact analytical result for any continued fraction coefficients

243

(15)

Calculation of coefficients a1 and a2

244

(1)

Calculation of coefficient a3

245

(1)

Calculation of coefficient a4

246

(1)

Calculation of coefficient a5

247

(2)

Calculation of coefficient a6

249

(2)

Calculation of coefficient a7

251

(7)

Stieltjes' formula for any continued fraction coefficients

258

(3)

The exact analytical expressions for any Lanczos couplings {α n, β n}

261

(3)

The Lanczos continued fractions and contracted continued fractions

264

(2)

Exact retrieval of time signals from the Pade polynomial coefficients

266

(7)

Auto-correlation functions at large times

273

(5)

The power moment problem in celestial mechanics

278

(8)

Mass positivity and the power moment problem

286

(4)

The power moment problem and Gaussian numerical quadratures

290

(2)

The power moment problem of the generalized Stieltjes type

292

(2)

Matrix elements (um | Qn (u)) and the Stieltjes integral

294

(4)

The modified moment problem

298

(6)

The mixed moment problem

304

(6)

Proof for zero-valued amplitudes of spurious resonances

310

(7)

Mapping from monomials un to polynomials Qn, (u)

317

(2)

Mapping of state vectors: Schrodinger Lanczos

319

(4)

The Gram-Schmidt versus the Lanczos orthogonalization

323

(5)

Analytical inversion of the Schrodinger overlap matrix

328

(6)

A variational principle for a quadratic Hankel form

334

(3)

Eigenvalues without eigenproblems or characteristic equations

337

(4)

Tridiagonal inhomogeneous systems of linear equations

341

(9)

Delayed time series

350

(6)

Delayed Green function

356

(2)

The quotient-difference (QD) recursive algorithm

358

(4)

The product-difference (PD) recursive algorithm

362

(6)

Delayed Lanczos continued fractions

368

(7)

Delayed Pade-Lanczos approximant

375

(4)

Delayed Pade approximant convergent outside the unit circle

379

(5)

Delayed Pade approximant convergent inside the unit circle

384

(6)

Illustrations

390

(8)

The uncertainty principle and resolution improvement

398

(2)

Prediction/extrapolation for resolution improvement

400

(2)

Main advantages of the Pade-based spectral analysis

402

(9)

Conclusions

411

(5)

Linear mappings among vector spaces

416

(7)

Non-classical polynomials in the expansion methods

423

(13)

Classical polynomials in the expansion methods

436

(3)

References

439

(15)

Index

454

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Quantum-mechanical Signal Processing And Spectral Analysis

Summary

Table of Contents

Supplemental Materials