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5 New types of solvability in PT symmetric quantum theory 0 0 2 Miloslav Znojil n a Abstract. The characteristic anti-linear (parity/time reversal, PT) symme- J tryofnon-HermitianHamiltonianswithrealenergiesispresentedasasourceof 2 twonew formsofsolvabilityofSchro¨dinger’s bound-state problems. Indetail 2 we describe (1) their very specific semi-exact solvability (SES) and (2) their innovatedvariationaltractability. SEStechnicalitiesarediscussedviacharged 1 oscillator example. In a broader context, speculations are added concerning v possiblerelationshipbetweenPTsymmetry,solvabilityandsuperintegrability. 8 5 0 1 0 PT symmetric quantum mechanics: A brief introduction 5 Prehistory: Isospectral operators. Origins of the popular Bender’s and 0 / Boettcher’sPT symmetricquantummechanics[B0, B2]lieinperturbationtheory h [CG, FG]. For an elementary illustration one may recollect the pioneering pa- p per by Buslaev and Grecchi [BG] who proved the isospectrality of the Hermitian, - h spherically symmetric D dimensional perturbed harmonic oscillator t − a 2 m 1 D D H(PHO)(g)= + x2 +g2 x2 : 2−△ j  j v j=1 j=1 X X i     X (in its m th partial-wave projection at any m = 0,1,...) with its non-Hermitian − r “unstable” anharmonic partner(s) in one dimension, a d2 H(UAO)(g)= +z2(igz 1)2 (D+2m 2)(igz 1/2), g >0. −dz2 − − − − This means that the physics (and, in particular, the reality of energies) remains unchanged while the mathematics itself is significantly simplified when one weak- † ens the Hermiticity of H(PHO)(g) = H(PHO)(g) to the mere PT symmetry of its spectrally equivalent partners [DT]. The latter operators commute with the (cid:2) (cid:3) product PT of parity and time reversal. In the modern language of review [M], one should rather speak about the pseudo-Hermiticity defined by the relation † H(UAO)(g)=P H(UAO)(g) P. h i 1991MathematicsSubjectClassification. Primary34M15,35Q40;Secondary81Q05,81R40. Key words and phrases. non-self-adjoint Hamiltonians, anti-linear symmetries, charged os- cillator,facilitatedquasi-exactsolvability,non-biorthogonal bases,superintegrability. Theauthor wassupportedinpartbyASCRGrant#A1048004. 1 2 MILOSLAVZNOJIL Recent progress: Exactly solvable examples. Paper [BG] extends the proof to the isospectrality between H(PHO)(g) and H(UAO)(g) to the imaginary couplingsg =ih. Thespectrumoftheresultingself-adjointdouble-wellH(UAO)(ih) in one dimension (famous for its perceivably hindered perturbative tractability) may be then deduced from the “unstable” modification H(PHO)(ih) of the central quarticoscillatorinDdimensions. Thelatter,non-HermitianPT symmetricmodel issufficientlysimpleinitsordinarydifferential“radial”representationofref. [BG], d2 ℓ(ℓ+1) H(URO)(h)= + +r2 h2r4, ε −dr2 r2 − r =r(x)=x iε, x ( , ), h>0, ε>0. − ∈ −∞ ∞ In the context of the present proceedings this operator admits an immediate re- interpretation as a specific perturbed and regularized form of the most common superintegrable oscillator H(URO)(0) of Smorodinnsky and Winternitz (SW, [FM, 0 E]) in any number of dimensions and after its separation in cartesian coordinates. The latter observationis inspiring since it is knownthat the concepts of (max- imal) superintegrability and (exact) solvability are closely related at ε = 0 [T0]. In such a setting the weakening ε > 0 of the Hermiticity extends the class of the solvable models [Z0, Z1] as well as their interpretation in terms of Lie algebras [LC] and evokes a number of new open questions. We have addressedthe first few of them in ref. [Z2] where we derived the exact spectrum and wave functions for the complexified SW oscillator H(URO)(0). Our next move in this direction was ε devotedto the closely relatedcomplexified(though stillseparable)Calogeromodel of three particles [ZM] where an overlap of the integrability with PT symmetry actedasageneratorofthe newsolvableself-adjointHamiltonians[ZT]. Thisexpe- rience makes the further study of PT symmetric models of (super)integrable type very promising. In the present paper, we are going to describe some new results achieved in this direction. Runningproject: Partiallysolvablemodels. Inplaceofthe(notsoeasily feasible)analysisofthe traditionalquarticperturbations,weshallpayattentionto the slightly simpler charged oscillators characterized, first of all, by their incom- pletely or quasi-exactly solvable1 (QES) status. From the historical perspective, the acceptance of this concept was comparatively dramatic. In a prelude, vari- ouspotentials havebeen shownpartiallysolvablein termsofelementaryfunctions. Thirty years ago, this period has been initiated by a two-page remark by Andr´e Hautot [H] who noticed that the charged oscillator possesses arbitrary finite mul- tiplets of exact Sturmian solutions in two (D = 2) and three (D = 3) dimensions. The Hautot’s choice of the elementary wave-function ansatz N ϕ(r)=e−r2/2−gr h rn+κ n n=0 X (with arbitrary N) preceded the discoveries of the QES sextic oscillator [SB], of non-polynomial QES anharmonicities [WF] etc. During the “golden age” of the developmentofthe subject, the existence of the commonLie-algebraicbackground of all the QES systems has been emphasized [T1]. A summary of the “state of the art”up tothe earlynineties hasbeenofferedbyAlex Ushveridzeinhis monograph 1termcoinedbyA.Turbiner,meaninganelementarysolvabilityforfinitemultipletsofstates NEW TYPES OF SOLVABILITY IN PT SYMMETRIC QUANTUM THEORY 3 [U]wherethequarticpolynomialoscillatorismentionedasatypicalsystemwithout quasi-exact solvability. A change of the approach to QES models has been initiated by Bender and Boettcher [B1] who revealed and described the QES solutions for the “unstable” quartic polynomial oscillators. Their construction has been extended to all the partial waves in ref. [Z3]. In paper [Z4] we made the next move and turned attention to PT symmetrization of the Hautot’s charged oscillator (cf. Appendix A). In the present continuation of this effort, we shall complete the picture by its extension, i.a., to all the partial waves (cf. Appendix B). In section 1 we describe our main result, viz., the quasi-even solutions which remained unnoticed in [Z4]. Unexpectedly,these“facilitatedQES”statesareavailableininfinitemultiplets2,all theelementsofwhichare,basically,non-numerical. Inawayemphasizedinsection 2, these sets may easily serve as certain non-standard bases in some “sufficiently large” subspaces of Hilbert space. For this reason, we suggest to call them semi- exactly solvable (SES, cf. Table 1). Table 1. Tentative classification of solvability. class quasi exact semi exact exact − − solutions available finite set infinitely many all range of couplings restricted restricted any illustrative example H(BBO)(a,b) see below H(URO)(0) ε 1. Quasi-exact solvability on complex contours 1.1. Three-term recurrences. Let us consider the Schr¨odinger equation d2 ℓ(ℓ+1) F (1.1) + +i +2ibr(x)+r2(x) ψ (x)=E ψ (x) −dx2 r2(x) r(x) n n n (cid:20) (cid:21) ℓ=ℓ(L)=(L 1)/2, L=D 2,D,D+2,... . − − It contrast to the Hautot’s QES problem of Appendix A it works with the purely imaginarychargeandwithaconstantcomplexshiftofcoordinatesr =r(x)=x iε − (cf. [A]). It also generalizesthe ℓ=0 problemof ref. [Z4]so that we must employ the more powerful ansatz N (1.2) ψ (r)=e−r2/2−ibr (ir)n−ℓ p . n n n=0 X 2suchafeatureismuchmorecharacteristicforthecompletelysolvablemodels 4 MILOSLAVZNOJIL After its insertion, the differential formof our PT symmetric bound state problem (1.1) is replaced by the finite-dimensional matrix equation, B C 0 0 A B C p 1 1 1 0   (1.3)  ... A... B... C  p...2 =0 , N <∞  N−1 N−1 N−1    A B  p   N N  N   AN+1     with the real matrix elements, A =b2+2n L E, B = (2n+1 L)b F, n n − − − − − C =(n+1)(n+1 L), L=2ℓ+1, n=0,1,... . n − The last line of (1.3) represents a separate condition A = 0 which gives our N+1 energies in closed form, (1.4) E =E =2N +2 L+b2, N =0,1,... . N − Thekeyconsequenceofthiseasybutimportantsimplificationliesintheemergence of a zero C = 0 in the upper diagonal of the square matrix of the simplified L−1 system (1.3). In Appendix B the presence of this zero enables us to construct the quasi-odd solutions at all ℓ. 1.2. The new, quasi-even solutions. All the coefficients inthe polynomial wave functions (1.2) may be defined by closed formula (4.2) displayed in Appen- dix B. Let us now turn our attention to the quasi-even states. The superscript (+) will be introduced to mark their even quasi-paritywhich maybe characterized,for ourpresentpurposes,bythesimplenegationoftheoddquasi-paritycondition(4.3), (1.5) p(+) + p(+) +...+ p(+) >0. | 0 | | 1 | | L−1| For quasi-even states, the vanishing matrix element C(+) = 0 plays a different L−1 role. It separates again the two subsets of equations but the upper one ceases to be trivial. This means that the related subdeterminant must vanish, B C 0 0 A B C 1 1 1   (1.6) S(+) =det ... ... ... =0.    AL−2 BL−2 CL−2   AL−1 BL−1    Uptotheexceptional,degeneratecaseswhereeqs. (1.6)and(4.4)holdatthesame time,wemaydropthenon-vanishingfactorS(−) =0anddetermineallthecharges 6 F =F(+) as roots of polynomial (1.6). N,k The (+) superscripted wave function coefficients are determined by eq. (4.2), − this time in the full range of indices j =N,N 1,...,1. As long as the dimension − ofthesedeterminantsgrowswiththedifferenceN j,weshallrecommendareturn − NEW TYPES OF SOLVABILITY IN PT SYMMETRIC QUANTUM THEORY 5 to the recurrences (+) 2N β1 F 4 2L p0 − 2 −2N β− F 9 3L p(+)  − 2− −  1  ... ... ... ... =0  −4 βN−−12−F Nβ2N−−NFL  pp(N(N++−))1     to be read from below upwards3. The process must be initiated at p(+) = 0. The N 6 vanishing CL−1 =0 does not play any role in it. Roots of the SES secular determinants 1.3. Dimension-independence. Ineverypartialwave,thesizeLofthema- trix ineq. (1.6) coincideswith the degreeofthe resulting secularpolynomial. This value does not change with the growth of the dimension N +1 of the vector ~p. Thus,the degreeN ofourpolynomialwavefunctions entersourtridiagonalsecular equation as a mere parameter. Containing extremely elementary matrix elements, the fully explicit form of this equation reads (L 1)b F 1 L − − − ..  2N (L 3)b F .  − − − det 2N +2 ... 4 2L =0  − −   ... (3 L)b F 1 L   − − −   2(N +2 L) (1 L)b F   − − − −    and specifies the family of the admissible charges F = F(+) at any integer L > 0 N,k and index k = 1,2,...,L. The first few partial waves are exceptional since their eigenchargesmayin principle be definedby closedformulaeatL=1, L=2,L=3 and L = 4. At the simplest choice of L = 1 (which may mean both the s wave − in three dimensions and an even state at D = 0), one does not obtain anything new. Secular equation (1.6) provides the single root F(+) which is equal to zero N,1 at all N. In the limit ε 0, the quasi-even L = 1 solutions converge to the well → known Hermite polynomials. One just re-discovers the exactly solvable harmonic oscillatorbasisatevenparity. Wemustre-emphasizethatatanyL 0thenumber ≥ of our quasi-even roots does not change with the growthof the dimension N. This is of paramount importance since the practical determination of eigencharges is performed for all the indices N at once. Each of these families numbered by k contains infinitely many elements numbered by the first subscript. This might facilitate their future applications (cf. section 2 below). 1.4. The first nontrivial generalization of oscillator basis: L = 2. Let usmovetotheindexL=2givingthep waveintwodimensionsortothes wavein − − four dimensions. The related ℓ=1/2 wave functions may be chosen as compatible with the even-quasi-paritycriterion (1.5), 1 p(+)(N,k)= [b+F ] p(+)(N,k), k =1,2 . 0 −2N N,k 1 3weabbreviatedβn≡−(2n+1−L)bforthesakeofbrevity 6 MILOSLAVZNOJIL Secular equation (1.6) reads b F 1 det − − =F2 b2 2N =0 2N b F − − (cid:18) − − − (cid:19) and has two roots (1.7) F(+) = (b2+2N), F(+) = (b2+2N). N,1 N,2 − [Note the contrastwith thpe necessity of searching forproots of the polynomial (4.4) oftheN L+1stdegree!] Weencounterthenon-vanishingchargesforthefirsttime. − They grow with the increasing size of the shift b and with the quantum number N (i.e., with the energy). One finds their set more similar to the “F =0 line” of the harmonicoscillatorthantotheQESquasi-oddrootsofAppendix B.Indeed, inthe (−) lattercasetherootsF mustbedeterminedbythe methodwhichdependsonN. N,k Inthissensewemaynowspeakaboutthemostnaturalanduniquenon-Hermitean generalizationof the harmonic oscillator even-parity basis of L (0, ). 2 ∞ 1.5. Cardano charges at L=3. At L=3 the comparatively compact form of our secular equation 2b F 2 0 − − (1.8) det 2N F 2 =0  − − −  0 2N +2 2b F − − −   enables us to search for the triplet of charges F ,F ,F via the closed (so N,1 N,2 N,3 { } calledCardano)formulae. One of alternative strategies may consist in a transition from the SES eigencharges F = FN,k(b) to the inverse functions b = bN′,k′(F). Sucha trick lowers(by one)the degree ofthe secular polynomialatthe odd values of L and, hence, extends the solvable class up to L = 5. A serious shortcoming of such an approach may be seen in the parallel deformation of the energies (1.4) which would change with the shift b. In the similar spirit, another simplification of formulae may be based on the formaleliminationofN. Theconsequencesmaybeillustratedbythe secularL=3 determinant(1.8)whichrepresentsthe mere linearproblemforN,withthe unique solution 1 1 1 F 2 1 2 (1.9) N = 4Fb2+8b F3 4F = + b+ . −8F − − 2"(cid:18)F 2(cid:19) −(cid:18) F(cid:19) # (cid:0) (cid:1) Thisformulasuggestsasimultaneoustuningofboththeshiftsb andtherelated [1,2] charges F . This may be achieved, say, by their hyperbolic re-parametrization [1,2] with F(t) = √2et and cosht = √N coshα(t) etc. In terms of the parameters t = t and/or α = α , the spectrum of energies becomes deformed by the N,k N,k induced parametric dependence of b = b (t) = 2cosh2t 2N e−t/√2. [1,2] ± − − Purely formally, the eliminaton of N might extend the use of closed formulae up p to L=9. Higher partial waves 1.6. Numerical methods. A shortcoming of the present SES construction lies in the growth of its complexity at the large L. At L 5 one already cannot ≥ generate closed formulae for eigencharges,and the purely numerical search for the NEW TYPES OF SOLVABILITY IN PT SYMMETRIC QUANTUM THEORY 7 rootsF isnecessaryatallthe verylargeL 1. InanillustrationusingL=3,and ≫ b=5 one gets the three eigencharges F = 10.757, 10.400, 0.35755 { − − } at the smallest possible N =2. They smoothly grow to the values (1.10) F = 89.98, 89.975, 0.0049407 { − − } evaluated at very large N = 1000. This type of calculation is very quick and its results exhibit a smooth N dependence sampled in Table 2. − Table 2. N dependence of eigencharges at L=4 and b=5 − N F N,k 3 15.611 5.9279 4.8887 16.651 − − 30 27.149 9.2909 8.9294 27.511 − − 300 74.856 24.984 24.936 74.904 − − 3000 232.82 77.610 77.605 232.83 − − 30000 734.99 245.00 245.00 734.99 − − 1.7. Large N expansions. Any informationconcerningthe eigenvaluesF = F(b)and/orb=b(F)mayshortenthenecessarycomputations. Attheintermediate L = 5 (which corresponds to the d wave in three dimensions) we may eliminate, − for example, the product 512FN± which is equal to 768b 256Fb2+768F +40F3 24 (1024b2+192bF3+512F2+F6) − − ± andhuse this formula asa constraint. Fortupnately, closedformulaeof this typeimay also open the door to the perturbative methods. Empirically,manynumericallycomputedrootsF areverysmoothfunctions N,k of N, especially in the asymptotic domain where they may be well approximated by their available large N estimates. For example, the exact result (1.10) already − lies very close to its leading-order analytic estimate F √8N, √8N, b/N 89.44, 89.44, 0.005 . ≈{ − − }≈{ − − } At the intermediate values of the dimension N, precision may be still insufficient but one can easily evaluate the large N corrections. For illustration, let us return to eq. (1.9) and re-write it, in the c−ase of its smallest root F = ˆbFˆ/N, as the − strictly equivalent formula β ˆb3 b Fˆ =1+ Fˆ3, β = , ˆb= , L=3 . N3 8b 1+(b2 1)/2N − It is suitable for iterations which represent our root as the following power series, β β2 β3 β4 Fˆ =1+ +3 +12 +28 +... . N3 N6 N9 N12 AlsotheothertwoL=3rootsmayberepresentedbythesimilarasymptoticseries. 8 MILOSLAVZNOJIL At the higher partial waves the same method works with a comparable effi- ciency. At L = 4, for example, the use of the variable M = √2N +b2 2 > 0 − compactifies secular equation (1.6), F2 M2 F2 9M2 =36 48bF . − − − In accord with Table(cid:0)2, its asy(cid:1)m(cid:0)ptotically d(cid:1)ominant O(M2) part determines the four distinct leading-order asymptotics of F ̺M where ̺ = 1, 3. Once we ∼ ± ± re-normalize F =̺M√1+R, the four exact charges obey the relation 48b√1+R 1 36 1 R= + , L=4 . 8̺ 16+̺2R ̺M3 8̺ 16+̺2R ̺2M4 | |− | |− Its iterations evaluate the correction term R with an astounding efficiency. 2. Quasi-variational solvability on model spaces “Sufficiently large” finite subspaces in Hilberet space 2.1. Left and right SES solutions. A full-fledgedapplicability of the stan- dard sets of QES wave functions is weakened by their incompleteness. Moreover, all the Hermitian QES Hamiltonians are usually interpreted as possessing just a finite multiplet of elementary bound states (at a given, special QES coupling) or Sturmians (cf. their example in Appendix B). For this reason, even the study of theirownsmallperturbationsisnoteasyatall[Z5]. Still,thesetsofQESsolutions themselves become large enough when we take into consideration both their fixed- coupling and fixed-energy subsets. In practice, such a philosophy proveduseful for approximationpurposes [BC]. Inthis section, wearegoingto returnto the latter ideain the presentbroader, PT symmetric SES context. For the sake of definiteness let us re-consider Hamil- tonianofeq. (1.1)asifitweredefinedatanychargeF standingattheCoulomb-like force W(r) = ir−1. Then we may denote H = H(F) = H(0)+F W and, in the spiritoftheDirac’sbraandketnotation,abbreviate ψ(−) = M,k . Thisenables | M,ki | i us to re-write the special SES version of eq. (1.1) in shorthand, (2.1) H(0) M,k +W M,k F = M,k E M,k M | i | i | i k =1,...,L, M =0,1,... . Our PT symmetric Hamiltonian is non-Hermitean so that its left and right eigen- states will differ in general. One has to complement eq. (2.1) by its counterpart (2.2) N,j H(0)+F N,j W =E N,j , N,j N hh | hh | hh | j =1,...,L, N =0,1,... where the same operators act to the left. After a return to the differential form of this problem (1.1), a transition between Hermitian-conjugate equations (2.1) and (2.2)maybere-interpretedasacertainreflectionR inthe spaceoftheparameters, (2.3) R : F F, b b, ε ε. . { ←→ − ←→ − ←→ − } In an illustration using L = 2, our SES states of section 1 may be represented by a discrete set of points in the energy-chargeplane. The values of their coordinates NEW TYPES OF SOLVABILITY IN PT SYMMETRIC QUANTUM THEORY 9 (E ,F ) are specified by the respective closed formulae (1.4) and (1.7). These N N,k points may be perceived as located on the two (viz. left and right) branches of a single hyperbolic curve. These two infinitely large multiplets are mutually related by the transformation R of eq. (2.3)so that R N,1 = N,2 and vice versa. | i | ii 2.2. Generalized Sturmian SES states as a basis. There is no a priori reason for a bi-orthogonality between our multi-indexed bras N,k A and hh | ≡ hh | ket vectors N′,k′ b so that their overlaps | i ≡ | i Q = Ab A,b hh | i form a non-diagonal and asymmetric matrix in general. We have to assume that afterafinite-dimensionaltruncation,thisoverlapmatrixisinvertible. Onlyinsuch a case we may define the inverse R=Q−1 and introduce an identity projector in a “sufficiently large” subspace of Hilbert space, (2.4) I = a R B . a,B | i hh | a∈JkeXt,B∈Jbra Inaconstructivemood,letus returnto equations(2.1)and(2.2)andimaginethat theysharealltheireigen-energiesandeigen-charges. Thisenablesustowritedown the following two alternative matrix equations (2.5) N,j H(0) M,k = N,j M,k E N,j W M,k F M M,k hh | | i hh | i −hh | | i (2.6) N,j H(0) M,k =E N,j M,k F N,j W M,k N N,j hh | | i hh | i− hh | | i with (N,j) J and (M,k) J . Their subtraction gives the constraint bra ket ∈ ∈ (2.7) (F F ) N,j W M,k =(E E ) Q . M,k N,j M N (N,j),(M,k) − hh | | i − Due to the way of its derivation, this relation may be understood as an immediate generalizationofbi-orthogonality. Inparticular,weseethatwithinthe subspaceof a single Sturmian multiplet (i.e., for M = N), the left-hand side expression must be a diagonal matrix with respect to its second indices. For our present purposes we shall abbreviate N,j W N,j w . N,j hh | | i≡ Non-QES bound-state problems 2.3. Matrix equations. The knowledge of a basis canfacilitate the study of many perturbed Hamiltonians via textbook perturbation recipes [F]. The use of the present SES bases might lead to a new progress in this area. Let us recall the respective right and left forms of any Schr¨odinger non-QES bound state problem with the above-mentioned structure, [H(0)+F W] Ψ =E(F) Ψ , | i | i Ψ [H(0)+FW]=E(F) Ψ . hh | hh | Assuming that F =F(QES) and using eq. (2.4) we may insert 6 Ψ = a R B Ψ = a h , a,B a | i | i hh | i | i a∈JkeXt,B∈Jbra a∈XJket Ψ = Ψa R B = g B a,B B hh | hh | i hh | hh | a∈JkeXt,B∈Jbra BX∈Jbra 10 MILOSLAVZNOJIL and arrive at the double eigen-problem A H(F) b R C Ψ =E AΨ , A J b,C bra hh | | i hh | i hh | i ∈ b∈JkeXt,C∈Jbra (plus its – omitted – conjugated companion), i.e., at the two conjugate linear alge- braic systems of equations written in terms of the same matrix, (2.8) Z(E,F)~h=0, ~g†Z(E,F)=0 Z (E,F)= A H(0) b E Ab +F A W b . A,b hh | | i− hh | i hh | | i Now,one couldintroduce a small parameterλ=F F(QES) andtry to construct, − say, E =E(λ) in the form of a power series in λ. 2.4. Matrix elements. Long before any numerical determination of bound states,wemustevaluateallthenecessarymatrixelementsasaninput. Inpractice, the latter step is usually the most time-consuming part of the algorithm. One has to optimize it in the present QES setting, therefore. In the first step we recall eq. (2.6) and eliminate all the matrix elements of H(0). Thismeansthatineq. (2.8)wereducethecostlyinputtothemereevaluation of the matrix elements of the weakly singular Coulomb potential W(r)=i/r, Z (E,F)=(F F ) A W b (E E ) Ab . A,b A A − hh | | i− − hh | i In the second step we keep M =N (i.e., we stay out of the Sturmian subspaces or 6 diagonalblocksinthematrixZ)andpostulatetheabsenceofarandomdegeneracy of charges. This means F = F so that we are permitted to re-arrange the M,k N,j 6 bi-orthogonality-like relation (2.7) into a further reduction of the necessary input information, E E M N N,j W M,k = − Q , M =N . hh | | i F F (N,j),(M,k) 6 M,k N,j − Inthis waywe arriveatthe finalformofourlinearSchr¨odingernon-QESalgebraic problem for the right eigenvectors, E E N K w h + − Q h = N,j N,j F F (N,j),(K,p) K,p N,j K,p K(X6=N),p − E E N = − Q h , F F (N,j),(M,k) M,k N,j − M,k X j =1,2,...,L, N =0,1,... . Forlefteigenvectors,the systemofequationsisverysimilarthoughnotequivalent, E E N K w g + − Q g = N,j N,j F F (K,p),(N,j) K,p N,j K,p K(X6=N),p − E E N = − Q g , F F (M,k),(N,j) M,k N,j − M,k X j =1,2,...,L, N =0,1,... . To solve any of these systems, say, by a perturbation technique, we just need to know the overlaps Q and the vector of Coulombic elements w . N,j

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