Algebraic Structures and Operator Calculus

Feinsilver, Philip; Schott, René

doi:10.1007/978-0-585-28003-5

Cited by 29 publications

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“…[49][50][51][52]66,67 Clearly, the closure property under commutation is a requirement to have a Lie algebra set, and c ijk are called structural constants. Then the set M is called Lie algebra with basis set of operators S j .…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…15,[43][44][45][46][47][48][49][50][51][52][53]68,74 It is sometimes helpful to combine two or three disentangling techniques together in case of complicated exponential operator. There are many schemes in the literature that have dealt with exponential operator disentanglement.…”

Section: A Exponential Operator Disentanglementmentioning

confidence: 99%

“…͑For the very same reason, analytically treating anharmonic electron-phonon coupling has been a longstanding problem.͒ Although exponential operator disentanglement through only Zassenhaus formula was employed in Ref. As a brief note on Lie algebra, 43,[48][49][50][51][52][53] it is a set of operators that satisfy a set of algebraic axioms, one of which, and most fundamental, is closure under commutation. Spin-boson coupling Hamiltonian was not exploited in the treatment of dissipative systems, 15 in an effort to do spectroscopy, to keep the mathematics and physics at a simple level and more tractable.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic approach to electronic spectroscopy and dynamics

Toutounji

2008

The Journal of Chemical Physics

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Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear absorption spectra. This new methodology should easily pave the way to calculating the four-point correlation function, F(tau(1),tau(2),tau(3),tau(4)), of which the optical nonlinear response function may be procured, as evaluating F(tau(1),tau(2),tau(3),tau(4)) is only evaluating the optical linear dipole moment correlation function iteratively over different time intervals, which should allow calculating various optical nonlinear temporal/spectral signals.

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Section: Mathematical Backgroundmentioning

confidence: 99%

Section: A Exponential Operator Disentanglementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic approach to electronic spectroscopy and dynamics

Toutounji

2008

The Journal of Chemical Physics

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“…Bernoulli systems in one variable are explained in [5], with higher-dimensional Bernoulli systems appearing in [6], where the basic methods of this work appear initially. As a good resource, the Berezin approach was applied to the Schrödinger algebra in [4].…”

Section: Introductionmentioning

confidence: 99%

“…We will use the notation O to denote a real orthogonal matrix. (5) Given N ≥ 0, B is defined as the multi-indexed matrix having as its only non-zero entries…”

Section: Introductionmentioning

confidence: 99%

Krawtchouk-Griffiths systems II: as Bernoulli systems

Feinsilver¹

2016

COSA

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We call Krawtchouk-Griffiths systems, KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. Here we present a Fock space construction with raising and lowering operators. The operators of "multiplication by X" are found in terms of boson operators and corresponding recurrence relations presented. The Riccati partial differential equations for the differentiation operators, Berezin transform and associated partial differential equations are found. These features provide the specifications for a Bernoulli system as a quantization formulation of multivariate Krawtchouk polynomials.

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Operator Calculus Approach to Solving Analytic Systems

Feinsilver

Schott

2006

Artificial Intelligence and Symbolic Computation

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Abstract. Solving analytic systems using inversion can be implemented in a variety of ways. One method is to use Lagrange inversion and variations. Here we present a different approach, based on dual vector fields.For a function analytic in a neighborhood of the origin in the complex plane, we associate a vector field and its dual, an operator version of Fourier transform. The construction extends naturally to functions of several variables.We illustrate with various examples and present an efficient algorithm readily implemented as a symbolic procedure in Maple while suitable as well for numerical computations using languages such as C or Java.

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Algebraic Structures and Operator Calculus

Cited by 29 publications

References 0 publications

Algebraic approach to electronic spectroscopy and dynamics

Algebraic approach to electronic spectroscopy and dynamics

Krawtchouk-Griffiths systems II: as Bernoulli systems

Operator Calculus Approach to Solving Analytic Systems

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