Lipmann-Schwinger Equations for a System of Multiband Energy-Eigenvalue Spectrum

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1 New Horizos i Mathematical Physics, Vol. 2, No. 1, March Lipma-Schwiger Equatios for a System of Multibad Eergy-Eigevalue Spectrum M.A. Grado-Caffaro ad M. Grado-Caffaro M.A. Grado-Caffaro ad M. Grado-Caffaro- Scietific Cosultats, C/ Julio Palacios 11, 9-B, 2829-Madrid Spai); ma.grado-caffaro@sapiezastudies.com Abstract. For the first time, we establish the Lippma-Schwiger equatios for a system of multibad eergy-eigevalue spectrum so that the ivolved time-idepedet Schrödiger wavefuctios become matrix elemets. As a matter of fact, a supremum-semiorm of the total wavefuctio matrix is estimated i terms of the same semiorm for the partial wavefuctio matrix ad for two ey matrices of operators. I additio, a associated matrix-tesor formalism is preseted. Keywords: Lippma-Schwiger equatios; multibad eigevalue spectrum; supremum-semiorm; matrices of operators; matrix formalism. 1 Itroductio Treatig by usig elegat mathematical-physics methods, quatum systems of multibad eergyeigevalue spectra is certaily iterestig because these systems play a importat role i several areas of Physics. Let us regard, for istace, quatum systems of two-bad eergy eigevalue spectra [1,2]. A typical example of these systems ca be foud i magetism of amorphous solids [1-3]. I particular, we refer to determie the magetic susceptibility of amorphous solids [1-3]. I this cotext, a Waier-type represetatio was employed i ref.[1] while a geeralizatio of this represetatio, by itroducig a matrix formalism, was doe i ref.[2]. Really, the physics of codesed matter exhibits a umber of examples of quatum systems with multibad eigevalue spectra. Cosider, for istace, a sigificat two-bad eigevalue spectrum cosistig of spi-up ad spi-dow electroic bads as, for example, i metamagetic systems [4,5] ad aophysics [6-1]) ad a typical system with three-bad spectrum as, for istace, a semicoductor with the coductio bad, the valece bad, ad the forbidde bad bad gap). Certaily, systems with two or three eigevalue bads are relatively frequet i Physics. Therefore, it is atural to thi o geeralizig for may bads so that elaboratig aalytical models to characterize multibad systems is desirable. O the other had, the so-called Lippma-Schwiger model is suitable to discuss relevat issues i Nuclear Physics as, for example, eutro-ucleus scatterig [11] ad i Solid State Physics as, for istace, problems related to optical potetial [12]. It is well-ow see, for istace, ref.[11]) that the Lippma-Schwiger approach cosists of a system of equatios oe equatio for each quatum state) relative to the time-idepedet o-relativistic or relativistic Schrödiger equatio with a potetial viewed as a perturbatio. Here, we will cosider the static o-relativistic Schrödiger equatio with a time-idepedet exteral potetial. Withi this picture, electro-atom scatterig or electro-molecule scatterig may be cosidered by employig the Lippma-Schwiger equatios i relatio to a multibad eergy-eigevalue spectrum. I fact, amog other thigs, we will establish a matrix formalism to describe the aforemetioed scatterig. 2 Theory We cosider the ψ + Lippma-Schwiger equatios, which read see, for example, refs.[11,12]): ˆ ) 1 ψ = ϕ + E + iε H ψ 1) Copyright 218 Isaac Scietific Publishig

2 2 New Horizos i Mathematical Physics, Vol. 2, No. 1, March 218 where ψ deotes elastic coheret) total wavefuctio = 1,2,... ), ϕ is partial wavefuctio, Ĥ idicates field-free Hamiltoia operator, E is the th eergy eigevalue, V ˆ is the ivolved, timeidepedet, potetial-eergy operator, ad i is the imagiary uit. We have the time-idepedet Schrödiger equatios, amely, H ˆ ϕ = E ϕ ad Hˆ + ) ψ = E ψ. Now we are iterested i extrapolate formula 1) to a quatum system of multibad eergyeigevalue spectrum. I this case, the correspodig time-idepedet Schrödiger equatios are: Hˆ ϕ = E ϕ 2) ˆ ˆ ) H + V ψ = E ψ 3) where m is the bad idex ad labels the boud) states i each bad; we have that m = 1,2,..., N where N is the umber of bads) ad = 1, 2,... We may also cosider the total Hamiltoia operator amely Hˆ Hˆ + such that H ˆ ψ = E ψ see eq.3)). O the other had, let us assume that the groud state of the system is a state such that all the states i the m = 1 bad are occupied. Therefore, the Lippma-Schwiger equatios for the system of multibad spectrum i questio are: which implies: ˆ ) 1 ψ = ϕ + E + iε H ψ 4) ψ ϕ ε ψ 1 r) = r) + E + i Hˆ ) r) r),,, Note that i eq.5) it is idicated the explicit depedece o the spatial coordiates of the ivolved wavefuctios which, of course, are complex-valued fuctios) ad the exteral potetial. Also i relatio to eq.5), for the sae of brevity of otatio, we omit the rage of values of m ad ito the summatio symbols. It is clear that, i both formulae 4) ad 5), the total wavefuctio ad the partial oe are matrix elemets so that we may spea of the matrices, amely, ψ ) ad Φ ϕ ), where, for the sae of brevity of otatio, depedece o spatial coordiates ad Dirac s otatio of et vectors is ot show explicitly. I order to elaborate later a matrix-tesor formalism, we are iterested i cosiderig square matrices so that we defie ψ = ϕ = for m = N + 1, N + 2,... Now, istead of utilizig the usual orm of the wavefuctio Hilbert space, this orm beig uity ormalized wavefuctios), the we defie the followig supremum-semiorms as figures of merit of the quatum multibad states i questio: sup ψ 6) Φ r, sup ϕ r, where R is the ivolved spatial regio which is assumed to be compact. Note that the existece of the right-had side of 6) ad 7) is maifest because the fuctios ψ r ) ad ϕ r ) are twofold differetiable o a compact set so they are bouded i this set so the sum idicated i 6) ad 7) is also bouded i R. Notice that, i effect, expressios 6) ad 7) defie semiorms sice, although the direct assertio of the first axiom of a orm is fulfilled if or Φ, the right-had side of 6) or 7) is zero), the the reciprocal assertio is ot fulfilled, that is, if the right-had side of 6) or 7) is zero, the the lefthad side ca be differet from zero. Taig supremum at both sides of 5) ad usig relatios 6) ad 7), it follows: Φ + Γ ˆ 8) 5) 7) Copyright 218 Isaac Scietific Publishig

3 New Horizos i Mathematical Physics, Vol. 2, No. 1, March where ˆΓ is a matrix whose elemets are liear operators such that ˆ γˆ ) ˆ E ) 1 iε H Γ ad γ + which are bouded. Observe that ˆΓ ad ˆ ˆ V are semiorms, which ca be viewed as stregths of the respective operators ˆΓ ad V ˆ. Oe has: Γ ˆ Γˆ sup 9) sup Similarly, we may regard ad Φ as stregths of ad Φ, respectively. O the other had, if we adopt the followig orms rather tha the semiorms 6) ad 7): sup ψ 11) Φ r, sup ϕ r, as well as for the operators ˆΓ ad V ˆ, the it is easy to see that iequality 8) holds where, of course, ow the symbol meas orm. O the other had, relatioship 5) ca be expressed as follows: ψ = ϕ + tr Γˆ ) 1) 12) 13),, where tr deotes trace, the tesorial product is ot the stadard tesor product i the Kröecer sese used i quatum mechaics but the dyadic-tesor product of matrices [2,13]. This product comes from multiplyig dyadically two vectors [2,13]. As i precedig otatio, for the sae of brevity, i relatio 13) we abado Dirac s otatio ad depedece o spatial coordiates is ot show explicitly. Taig supremum at both sides of 13) ad employig formulas 6) ad 7), oe gets: Φ + sup tr Γˆ V ˆ 14) By combiig iequalities 8) ad 14), we fid: ˆ ˆ ) ) sup tr ΓV = c Γˆ where c is a umber such that < c 1 or c>1. The we euciate: Theorem 1.- There exists a real umber amely c < c 1 or c > 1 ) such that equality 15) is satisfied. O the other had, from formula 8) it follows: Φ 1 Γ ˆ 16) Expressio 16) gives a lower boud for the ratio of the stregth of the partial wavefuctio matrix to the stregth of the total wavefuctio matrix. The above boud depeds sigificatly o the semiorms of ˆΓ ad V ˆ. Now we have: Defiitio 1.- The so-called trasitio operator amely ˆT is such that Tˆ ϕ = ψ. Defiitio 2.- We defie the scatterig operator amely Ŝ such that S ˆ ϕ = ψ. I additio, oe euciates: Propositio 1.- It is verified: Φ 1 + Γ ˆ T ˆ ) 17) where, of course, the semiorm of ˆT is the same as for ˆΓ ad V ˆ. Proof: By usig Defiitio 1, taig supremum at both sides of 5) ad cosiderig formulae 6) ad 7), the relatio 17) is easily foud let us regard expressio 8)) 15) Copyright 218 Isaac Scietific Publishig

4 4 New Horizos i Mathematical Physics, Vol. 2, No. 1, March 218 Theorem 2.- There exists a real umber amely c such that < c 1 or c > 1 ad such that: ˆ ˆ) ) sup tr ΓT Φ = c Γˆ Tˆ Φ Proof: Taig supremum at both sides of 13), by usig relatios 6) ad 7) as well as Defiitio 1, the oe gets: ˆT ˆ ) ) Φ + sup tr Γ Φ regard formula 14)) Combiig ow iequalities 17) ad 19), oe arrives at formula 18) Propositio 2.- It is satisfied the followig iequality: Sˆ 1 + Γ ˆ Tˆ 2) where, obviously, the semiorm of Ŝ is the same as for ˆΓ, V ˆ ad ˆT. Proof: Expressio 2) comes straightforwardly from Defiitios 1 ad 2 combied with formulas 5), 6) ad 7) Corollary.- By Defiitio 2 ad formulae 6) ad 7), oe gets Φ Ŝ which, together with iequality 2), 17). Fially, we cosider the followig tight-bidig basis [1,2]: ψ = ψ ψ ψ m m 21) where = 1,2,... The oe has: Propositio 3.- It is verified that [1,2]: ψ ψ ψ ψ m m 18) 19) 22) Proof: By summig over the rage = 1,2,... the product of the et vectors o the left-had side of 21) by the correspodig bra vectors ad iterchagig summatio with respect to by summatio relative to 2, the the left-had side of 22) becomes ψ ψ ψ ψ with m m ψ ψ = 1 so expressio 22) is satisfied Similarly to formula 21), we also cosider the tight-bidig basis: ϕ = ϕ ϕ ϕ m m 23) so that, as i relatioship 22), we have: ϕ ϕ = ϕ ϕ 24) m m Expressios 21) to 24) represet a useful formulatio from the poit of view of disordered systems i Codesed Matter Physics [1-3]. 3 Discussio It is well-ow that problems upo operator theory cocerig quatum perturbed systems [14-19] ad related issues [2-25] have a great relevace i Mathematical Physics; our precedig formulatio is a example of this. As a matter of fact, by usig relatios 6), 7), 9) ad 1), we have developed a approach whose mai results are relatioships 15) ad 18). I our formulatio, it has bee employed see relatios 6) ad 7)) a semiorm that ca be regarded as uusual give that, of course, the usual orm o the Hilbert space of the wavefuctios cosidered here is 12 ψ ψ = 1. However, the aforemetioed semiorm ad the orm defied i formulae 9) ad 1) which, of course, correspod to o-hilbert space but to semi-baach or Baach space, the costitute, say, ey sigatures or figures of merit stregths ) to characterize the matrices ad Φ. We may spea of a semi-baach space as a complete semiormed real or complex vector space. O the other had, from iequality 16) oe gets the Copyright 218 Isaac Scietific Publishig

5 New Horizos i Mathematical Physics, Vol. 2, No. 1, March relative deviatio Φ ) Γ ˆ V ˆ so that a upper boud of this relative deviatio equals the product of the stregths of ˆΓ ad V ˆ. 4 Cocludig Remars We have show that, by utilizig the four liear operators ˆΓ, V ˆ, ˆT ad Ŝ, a elegat mathematical represetatio has bee achieved withi a uusual framewor because the usual Hilbert-space structure has ot bee cosidered but the Baach or, more precisely, semi-baach) space structure. This fact opes ew aveues to treat a umber of relevat issues o Mathematical Physics i a alterative way which could be very attractive for future studies. Refereces 1. R.M. White i Tetrahedrally boded amorphous semicoductors M.H. Brodsy, S. Kirpatric, ad D. Weaire, Eds.), AIP Cof. Proc. No ) M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B 18 24) M.A. Grado-Caffaro, M. Grado-Caffaro, Act. Pass. Electroic Comp ) N.H. Duc, D. Givord, C. Lacroix, C. Piettes, Europhys. Lett ) M.A. Grado-Caffaro, M. Grado-Caffaro, Act. Pass. Electroic Comp ) M.A. Grado-Caffaro, M. Grado-Caffaro, Superlat. Microstruct ) A. Soldatov, N. Bogolubov Jr.), S. Kruchii, Codes. Matter Phys. 9 26) V. Ermaov, S. Kruchii, H. Hori, A. Fujiwara, It. J. Mod. Phys. B 21 27) S. Kruchii, H. Nagao, S. Aoo, It. J. Mod. Phys. B ) M. Apostol, Phys. Lett. A ) J.J. Saurai, J.J. Napolitao: Moder Quatum Mechaics Addiso-Wesley, secod editio, 21). 12. M.A. Grado-Caffaro, M. Grado-Caffaro, Mod. Phys. Lett. B ) M.A. Grado-Caffaro, M. Grado-Caffaro, Adv. Stud. Theor. Phys ) T. Kato: Perturbatio theory for liear operators Spriger-Verlag, 1966). 15. L. Parovsi, A.V. Sobolev, A. H. Poicaré 2 21) A.V. Sobolev, A. H. Poicaré 6 25) J. Feldma, H. Körrer, E. Trubowitz, Ivet. Math ) J. Feldma, H. Körrer, E. Trubowitz, Commet. Math. Helvetici ) Yu. Karpeshia: Perturbatio theory for the Schrödiger operator with a periodic potetial. Lecture Notes i Mathematics, vol Spriger-Verlag, Berli, 1997). 2. A. Mohamed, J. Math. Phys ) E. Korotyaev, A. Pushitsi, Bull. Lodo Math. Soc ) B.E.J. Dahlberg, E. Trubowitz, Commet. Math. Helvetici ) M. Srigaov, Ivet. Math ) D. She, M. Shubi, Math. USSR Sbori ) F. Schwabl: Advaced Quatum Mechaics Spriger-Verlag, Heidelberg, 24). Copyright 218 Isaac Scietific Publishig