Producción CientíficaThe exact Green function is constructed for a quantum system, with known Green
function, which is decorated by two delta function impurities. It is shown that when
two such impurities coincide they behave as a single singular potential with combined
amplitude. The results are extended to N impurities and higher dimensions.Ministerio de Economía, Industria y Competitividad (project ct MTM2014-57129-C2-1)Junta de Castilla y León (project VA057U

Glasser, M. Lawrence

Frontiers in Physics

English

Repositorio Documental de la Universidad de Valladolid

ORIGINAL RESEARCHpublished: 29 January 2019doi: 10.3389/fphy.2019.00007Frontiers in Physics | www.frontiersin.org 1 January 2019 | Volume 7 | Article 7Edited by:Luiz A. Manzoni,Concordia College, United StatesReviewed by:Fabio Rinaldi,Universitá degli Studi GuglielmoMarconi, ItalyFabiano Andrade,Universidade Estadual de PontaGrossa, Brazil*Correspondence:M. L. Glasserlaryg@tds.netSpecialty section:This article was submitted toMathematical Physics,a section of the journalFrontiers in PhysicsReceived: 01 October 2018Accepted: 11 January 2019Published: 29 January 2019Citation:Glasser ML (2019) A Note on theExact Green Function for a QuantumSystem Decorated by Two or MoreImpurities. Front. Phys. 7:7.doi: 10.3389/fphy.2019.00007A Note on the Exact Green Functionfor a Quantum System Decorated byTwo or More ImpuritiesM. L. Glasser 1,2*1Departamento de Física Teórica, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain, 2 Physics Department,Clarkson University, Potsdam, NY, United StatesThe exact Green function is constructed for a quantum system, with known Greenfunction, which is decorated by two delta function impurities. It is shown that whentwo such impurities coincide they behave as a single singular potential with combinedamplitude. The results are extended to N impurities and higher dimensions.Keywords: quantum green function, delta function potential, contact interaction, Schroedinger equation, quantumgraphPacs: 02.30Gp, 02.30Hq, 03.65Db1. INTRODUCTIONThe one dimensional harmonic oscillator or square well, for example, for which the energy-dependent Green function G0(x, x′;E) is known, have been taken for many years as solvablemodels for semi-conductor quantum wells [1]. Frequently delta function potentials are placedat various points to simulate defects or impurities. In the case of a single impurity potential,V(x) = λδ(x− a), the Green function for the composite system is known to be [2]G(x, x′;E) = G0(x, x′)+ λG0(x, a;E)G0(a, x′)1− λG0(a, a;E)(1)In this note a corresponding formula is derived for the case V(x) = λδ(x − a) + µδ(x − b) Onthe basis of the analogy of the algebraic structure the result is extended to N-impurities and for astandard interpretation of the Dirac delta function, to higher dimension.2. CALCULATIONWe first note that the same argument can be used for the time dependent-, as well as theenergy-dependent Green functions, so we shall omit the third argument and write simply G(x, x′).Beginning with the Dyson equation, noting that G0(x, y) = G0(y, x)G(x, x′) = G0(x, x′)+∫G0(x, y)V(y)G(y, x′)dy, (2)where the integration extends over the system domain, one has the set of equationsG(x, x′) = G0(x, x′)+ λG0(x, a)G(a, x′)+ µG0(x, b)G(b, x′) (3)G(a, x′) = G0(a, x′)+ λG0(a, a)G(a, x′)+ µG0(a, b)G(b, x′), (4)G(b, x′) = G0(b, x′)+ λG0(a, b)G(a, x′)+ µG0(b, b)G(b, x′). (5)Glasser Green Function for Two ImpuritiesThe linear Equations (4) and (5) are easily solved for G(a, x′) andG(b, x′) :G(a, x′) =G0(a, x′)+ µ[G0(b, x′)G0(a, b)− G0(a, x′)G0(b, b)]D(6)G(b, x′) =G0(b, x′)+ λ[G0(a, x′)G0(a, b)− G0(b, x′)G0(a, a)]D(7)withD = [1− λG0(a, a)][1− µG0(b, b)]− λµ[G0(a, b)]2. (8)By inserting (6) and (7) into (3) we obtain the desired expressionG(x, x′) = G0(x, x′)+1D{λG0(x, a)G0(a, x′)+ µG0(x, b)G0(b, x′)+λµ[G0(x, a)(G0(a, b)G0(a, x′)− G0(b, b)G0(b, x′))+G0(x, b)(G0(a, b)G0(a, x′)− G0(a, a)G0(a, x′))]}. (9)3. DISCUSSIONBy setting µ to 0 (9) reduces to (1), proving this expressionas well. The most salient feature of (9) is the denominator Dwhose zeros form the exact spectrum of the composite system.For example, when a and b coincide, D reduces to 1 − (λ +µ)G(a, a) and (9) reduces to (1) with λ replaced by the amplitudeλ + µ. I.e., the two impurities combine to form one withcombined amplitude. This generalizes the result of Fasssari andRinaldi [3], for two identical defects symmetrically placed withrespect to the center of a harmonic oscillator. An expressionsimilar to (9) has been derived recently by Horing (privatecommunication) for the case of a quantum dot in a magneticfield.Two further points can be made. Nothing in the derivationof (9) restricts it to the line. If we accept the standard definitionδ(Ex) = 5dj=1δ(xj), then (9), and its consequences, are valid ford-dimensional quantum systems. This has been proven function-theoretically for the three dimensional quantum dot with twosymmetrically placed identical impurities by Albeverio et al.[4].A second observation is that D is simply the Cramerdeterminant for the pair of simultaneous linear Equations (4) and(5). In the case of impurity potentialV(x) =N∑j=1λjδ(x− aj) (10)there will be N such equations and the determinant is easilyevaluated. The general result isIf a quantum system having Green function G0(x, y) isdecorated with N delta function impurities λjδ(x − aj),j = 1, 2, · · ·N, then the new energy levels are theroots ofDN =N∏j=1Ajj −N∑j=2(−1)j∑1≤k1<···kj≤NAk1k2Ak2k3 · · ·Akjk1 = 0(11)where Alm = δlm − λlG0(al, am).Thus,D3 =3∏j=1[1− λjG0(aj, aj)]−∑i<jλiλjG0(ai, aj)G0(aj, ai)+λ1λ2λ3G0(a1, a2)G(a2, a3)G0(a3, a1), (12) :which reduces to the N = 1 and N = 2 cases appropriately andshows that any two coinciding impurities coalesce as indicatedabove.Note that if all the λs and a’s coincide thenDN = (1−λG0(a, a))N−N∑j=2(Nj)λjG0(a, a)j = (1−NλG0(a, a)).(13)Equation (11) might offer a new approach to Kronig-Penney-type systems for periodic or random unitcells.Finally, it should be pointed out that the work in this noteis paralleled in the theory of quantum graphs introduced byLinus Pauling about 1930 to describe electrons in moleculeswhich has developed into a sophisticated and important branchof quantum physics [5]. For relations of this discipline to thepresent work see the papers by Andrade et al. [6] and Andradeand Severini [7].AUTHOR CONTRIBUTIONSThe author confirms being the sole contributor of this work andhas approved it for publication.ACKNOWLEDGMENTSThe author thanks Prof. S. Fassari and Prof. L. M. Nietofor helpful comments and acknowledges the financialsupport of MINECO (Project MTM2014-57129-C2-1-P)and Junta de Castilla y Leon (VA057U16). The authorthanks a referee for pointing out the relevance to quantumgraphs.Frontiers in Physics | www.frontiersin.org 2 January 2019 | Volume 7 | Article 7Glasser Green Function for Two ImpuritiesREFERENCES1. Albeverio S, Gesztesy F, Hoegh-Krohn R, Holden H. Solvabe Models inQuantumMechanics. Providence, RI: AMS-Chelsea (2004).2. Glasser ML, Nieto LM. The energy level structures of a varietyof one-dimensional confining potentials and the effects of a localsingular perturbation. Can J Phys. (2015) 93:1–9. doi: 10.1139/cjp-2015-03563. Fassari S, Rinaldi F. On the spectrum of the Schroedinger Hamiltonian ofthe one-dimensional harmonic oscillator perturbed by two identical attractivepoint interactions. Rep Math Phys. (2012) 69: 353–70.4. Albeverio S, Fassari S, Rinaldi F. Spectral properties of a symmetric three-dimensional quantum dot with a pair of identical attractive impuritiessymmetrically situated around the origin. Nanosyst Phys Chem Math. (2016)7: 268–89. doi: 10.17586/222080542016722682895. Berkolaiko G, Kuchement P. Introduction to Quantum Graphs. Providence, RI:AMS Mathematical Surveys and Monographs 186 (2013).6. Andrade FM, Schmidt GM, Vicentini E, Cheng BK, da Luz MGE. Green’sFunction approach to quantum graphs: An overview. Phys Rep. (2016) 647:1.doi: 10.1016/j.physrep.2016.07.0017. Andrade FM, Severini S. Unitary equivalence between the Green’s Functionand Schroedinger approaches for quantum graphs. arXiv:1808.01154v1.Conflict of Interest Statement: The author declares that the research wasconducted in the absence of any commercial or financial relationships that couldbe construed as a potential conflict of interest.Copyright © 2019 Glasser. This is an open-access article distributed under theterms of the Creative Commons Attribution License (CC BY). The use, distributionor reproduction in other forums is permitted, provided the original author(s)and the copyright owner(s) are credited and that the original publication inthis journal is cited, in accordance with accepted academic practice. No use,distribution or reproduction is permitted which does not comply with theseterms.Frontiers in Physics | www.frontiersin.org 3 January 2019 | Volume 7 | Article 7

A note on the exact Green function for a quantum system decorated by two or more impurities

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A note on the exact Green function for a quantum system decorated by two or more impurities

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