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Factorization-Algebraization-Path Integration and Dynamical Groups

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Symmetries in Science II

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Abstract

Schrödingerl proposed an elegant method of factorization of a quantum-mechanical second-order linear differential equation into a product of two first-order differential operators often referred to as ladder operators. These ladder operators when acting on respective eigenfunctions create new eigenfunctions with a quantum number raised or lowered by one unit. Schrödinger’s method was further systematically studied for a class of second-order linear differential equations in particular by Infeld and Hull2 who have shown that a second-order differential equation which may be brought into the form:

EquationSource% MathType!MTEF!2!1!+- % feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8x1dy % 2aa0baaSqaaiaab2gaaeaacaGGIaaaaOGaaiikaGqaaiaa+HhacaGG % PaGaey4kaSIaai4waiab-f8aYnaaBaaaleaacaqGTbaabeaakiaacI % cacaqG4bGaaiykaiabgUcaRiab-T7aSjab-1faDjab-v9aMnaaBaaa % leaacaqGTbaabeaakiaacIcacaqG4bGaaiykaiabg2da9iaaicdaca % GG7aGaaeiiaiaab2gacaqGGaGaeyicI4SaaeiiaiaabMeacaqGobaa % aa!5465!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\phi _{\rm{m}}^(x) + [{\rho _{\rm{m}}}({\rm{x}}) + \lambda ]{\phi _{\rm{m}}}({\rm{x}}) = 0;{\rm{ m }} \in {\rm{ IN}}$$
(0.1)

may be factorized into products of two first-order ladder operators EquationSource% MathType!MTEF!2!1!+- % feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqamaaDa % aaleaacaqGTbaabaGaeyySaelaaaaa!39C4!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\rm{A}}_{\rm{m}}^ \pm $$ such that

EquationSource% MathType!MTEF!2!1!+- % feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGbb % Waa0baaSqaaiaab2gaaeaacqGHRaWkaaGccaqGbbWaa0baaSqaaiaa % b2gaaeaacqGHsislaaaccaGccqWFvpGzdaWgaaWcbaGaaeyBaaqaba % GccaGGOaGaaeiEaiaacMcacqGH9aqpcaGGOaGae83UdWMae8NeI0Ia % e8xSde2aaSbaaSqaaiaab2gaaeqaaOGaaiykaiab-v9aMnaaBaaale % aacaqGTbaabeaakiaacIcacaqG4bGaaiykaiaacYcaaeaacaqGbbWa % a0baaSqaaiaab2gaaeaacqGHsislaaGccaqGbbWaa0baaSqaaiaab2 % gaaeaacqGHRaWkaaGccqWFvpGzdaWgaaWcbaGaaeyBaaqabaGccaGG % OaGaaeiEaiaacMcacqGH9aqpcaGGOaGae83UdWMae8NeI0Iae8xSde % 2aaSbaaSqaaiaab2gacaqGRaGaaeymaaqabaGccaGGPaGae8x1dy2a % aSbaaSqaaiaab2gaaeqaaOGaaiikaiaabIhacaGGPaGaaiOlaaaaaa!6870!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{array}{l} {\rm{A}}_{\rm{m}}^ + {\rm{A}}_{\rm{m}}^ - {\phi _{\rm{m}}}({\rm{x}}) = (\lambda - {\alpha _{\rm{m}}}){\phi _{\rm{m}}}({\rm{x}}),\\ {\rm{A}}_{\rm{m}}^ - {\rm{A}}_{\rm{m}}^ + {\phi _{\rm{m}}}({\rm{x}}) = (\lambda - {\alpha _{{\rm{m + 1}}}}){\phi _{\rm{m}}}({\rm{x}}). \end{array}$$
(0.2)

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Authors and Affiliations

  1. Department of Physics, State University of New York, Albany, New York, 12222, USA

    Akira Inomata

  2. Department of Mathematics, University of Texas, San Antonio, Texas, 78285, USA

    Raj Wilson

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  1. Akira Inomata
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  2. Raj Wilson
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Editors and Affiliations

  1. Southern Illinois University, Carbondale, Illinois, USA

    Bruno Gruber  & Romuald Lenczewski  & 

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Inomata, A., Wilson, R. (1986). Factorization-Algebraization-Path Integration and Dynamical Groups. In: Gruber, B., Lenczewski, R. (eds) Symmetries in Science II. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1472-2_21

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  • DOI: https://doi.org/10.1007/978-1-4757-1472-2_21

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