Pointwise reconstruction of wave functions from their moments through weighted polynomial expansions: an alternative global-local quantization procedure
Carlos R. Handy, Daniel Vrinceanu, Carl Marth, Harold A. Brooks
PPointwise reconstruction of wave functions fromtheir moments through weighted polynomialexpansions: an alternative global-local quantizationprocedure
Carlos R. Handy , Daniel Vrinceanu , Carl Marth , andHarold A. Brooks Department of Physics, Texas Southern University, Houston, Texas 77004; Dulles High School,Sugar Land, Texas 77459E-mail: [email protected]
Abstract.
Many quantum systems admit an explicit analytic Fourierspace expansion, besides the usual analytic Schrodinger configuration spacerepresentation. We argue that the use of weighted orthonormal polynomialexpansions for the physical states (generated through the power moments)can define an L convergent, non-orthonormal, basis expansion with sufficientpoint-wise convergent behaviors enabling the direct coupling of the global(power moments) and local (Taylor series) expansions in configuration space.Our formulation is elaborated within the orthogonal polynomial projectionquantization (OPPQ) configuration space representation previously developed( J. Phys. A: Math. Theor. Submitted to:
Physics Letters
PACS numbers: 03.65.Ge, 02.30.Hq, 03.65.Fd a r X i v : . [ m a t h - ph ] N ov lobal-Local Quantization
1. Introduction
Many quantum systems are defined by analytic solutions which in turn compell us togenerate them through (approximate) analytic methods. Although one is generallymore interested in the hierarchical large scale structure leading to quantization (for thebound states), recovering the local structure is also important and the principal focusof this work. Our methods have the important advantage that they are essentiallyalgebraic in nature, allowing for high precision calculations through multiple precisionalgorithms such as Mathematica.Of interest to us are those systems for which the power series expansion inmomentum space, and the local power series expansion in configuration space, areknown in the sense that their recursive structure can be generated from the verynature of the underlying Schrodinger equation. Our objective is to define a robustquantization procedure that directly couples both sets of power series coefficients.Since the momentum space expansion is governed by the power moments ofthe solution, µ p = (cid:82) dxx p Ψ( x ), our approach is tantamount to defining an effectivefunction-moment reconstruction ansatz, which is generally a difficult problem. Oneadvantage we have is that we know the algebraic structure for all the power moments,and the underlying solutions are generally smooth and bounded, as opposed to themore general problem. In keeping with this, mathematicians have known that the useof weighted (orthonormal) polynomial expansions,Ψ( x ) = (cid:88) n Ω n P n ( x ) R ( x ) , particularly for Freudian weights, R ( x ) = e −| x | q , q >
1, can converge (pointwise) tothe types of solutions encountered for physical systems [1]. The expansion coefficients,Ω n , are determined by the power moments, as discussed below. We stress that theexpressions { P n ( x ) R ( x ) } represent a non-orthonormal basis, which is an essentialcomponent to the flexibility of the above representation as used here.Despite the extensive mathematical literature on the importance of weightedpolynomial expansions, its relevance for quantizing physical systems has not beenappreciated. We believe that this stems from the preferance physicists have forconfiguration space based bound state quantization analysis, as opposed to those basedin Fourier space and (when appropriate) dependent on the underlying power momentstructure.Recently, the Orthogonal Polynomial Projection Quantization (OPPQ) methodwas developed [2], using weighted polynomial expansions for quantization, andemphasizing a numerical approximation to the asymptotic condition lim n →∞ Ω n =0, which is a defining condition for the bound states. A particularly importantachievement of the OPPQ analysis is that it is exceptionally stable and rapidlyconverges for weights (or reference functions) inaccesible to other methods, such asthe Hill determinant method [3]. As argued by Hautot [4], and confirmed by Taterand Turbiner (for the sextic anahrmonic potential) [5] if one uses reference functionsthat mimic the asymptotic form of the physical states, the Hill determinant methodcan become unstable, nonconvergent, or converge to the wrong solution. The OPPQapproach is not plagued by these problems. Since the Tater and Turbiner analysisdoes not clearly show these behaviors, we reproduce them here, as given in Fig 1 (Hilldeterminant results). The relevant discussion is given below, as well as the comparativepower of OPPQ and the alternate“global-local” quantization method presented here lobal-Local Quantization N a (cid:82) dx S ( xa ) = 1) [6]. It also couples the power momentsto the local structure of the wavefunction. Consider the scaling transform of thewavefunction, for some apppropriate, bounded, scaling function, S ( x ) ≡ (cid:80) n σ n x n : S Ψ( a, b ) = N a (cid:90) dx S ( x − ba )Ψ( x ) . (1)Depending on the asymptotic decay of the scaling function relative to the physicalsolution, the scaling transform will be analytic in the inverse scale variable, asgenerated from the power moments: S Ψ( a, b ) = N a (cid:88) n σ n a n µ n ( b ) , (2)where µ n ( b ) = (cid:82) dxx n Ψ( x + b ), involving a linear combination of the power momentsfor b = 0. Thus, the power moments control the large scale structure of the scalingtransform. Recovery of the local properties then requires the small scale asymptoticanalysis S Ψ( a, b ) = a N a (cid:90) dx S ( x )Ψ( ax + b )lim a → S Ψ( a, b ) ≈ ν (cid:88) n a n Ψ ( n ) ( b ) ν n n ! , (3)where ν n = (cid:82) dxx n S ( x ).For all one dimensional systems of the type considered here, upon solving forthe physical moments of the bound state solutions, one can analytically continuethe scaling transform and recover excellent pointwise results for the wavefunction[6]. That is, knowledge of the physical power moments (derived by other means)coupled with the appropriate analytic continuation strategy, proved very effective inrecovering the local solution. However, imposing the local structure (at the turningpoints) on the scaling transform representation, in order to determine the physicalpower moments from the local derivatives, proved ineffective. As demonstrated here,the OPPQ representation does allow for this type of “global-local” analysis. lobal-Local Quantization
2. OPPQ and Global-Local Quantization
In order to make more precise the above claims, consider the one dimensional Fouriertransform, assumed to be analytic (usually entire), with a corresponding k -space powerseries: ˆΨ( k ) = 1 √ π (cid:90) dx e − ikx Ψ( x ) (4)= 1 √ π ∞ (cid:88) p =0 µ p p ! ( − ik ) p . (5)We limit our analysis to (multidimensional) quantum systems for which the momentscan be generated through a linear recursive relation of order 1 + m s (in the onedimensional case), referred to as the moment equation . In many cases, a coordinatetransformation may be necessary to realize this. Its structure will take on the form µ p = m s (cid:88) (cid:96) =0 M p,(cid:96) ( E ) µ (cid:96) , (6)where m s is problem dependent. The M p,(cid:96) ( E )’s are known functions of the energy.The unconstrained moments { µ , . . . , µ m s } are referred to as the missing moments .The configuration space wavefunction is to be represented asΨ( x ) = (cid:16) ∞ (cid:88) n =0 c n x n (cid:17) R ( x ) , (7)for some positive weight function, R ( x ) >
0. The nature of the potential, and domain,usually dictates the possible choices for the weight. We are assuming that this is donein such manner ensuring that the series expression in Eq.(7) corresponds to an analyticexpansion near the origin (or any other desired point). Under these assumptions, the c n coefficients satisfy a recursive, second order, Frobenius method relation c n = T n, ( E ) c + T n, ( E ) c . (8)Given that the global power moments, µ p , and the local c n , satisfy known, energydependent constraints, respectively, how can we quantize by constraining both sets ofvariables? What is clearly needed is a robust, wavefunction representation ansatz capable ofrecovering (in a stable manner) the local structure of the wavefunction from the global,power moments. Weighted polynomial expansions,as implemented within the OPPQrepresentation, can provide this. Specifically, we will work withΨ( x ) = ∞ (cid:88) j =0 Ω j P j ( x ) R ( x ) , (9)where the orthonormal polynomials, P j ( x ) ≡ (cid:80) ji =0 Ξ ( j ) i x i , satisfy (cid:104) P j |R| P j (cid:105) = δ j ,j . (10) lobal-Local Quantization j = (cid:90) dx Ψ( x ) P j ( x ) , (11)= j (cid:88) i =0 Ξ ( j ) i µ i , (12)or Ω j ( µ , . . . , µ m s ) = m s (cid:88) (cid:96) =0 (cid:16) j (cid:88) i =0 Ξ ( j ) i M i,(cid:96) ( E ) (cid:17) µ (cid:96) . (13)If the weight R ( x ) satisfies the condition that (cid:82) dx Ψ ( x ) R ( x ) < ∞ , then one canargue that [2] Lim j →∞ Ω j = 0 . (14)This analysis is repeated, in a more complete manner, in Sec. IV. These conditionshold, in particular, if the weight decays, asymptotically, no faster than the physicalsolutions: lim | x |→∞ | Ψ( x ) |R ( x ) < ∞ .Within the original OPPQ quantization analysis, we approximate this asymptoticlimit by taking Ω n = 0 for N − m s ≤ n ≤ N , N → ∞ . This leads to an( m s +1) × ( m s +1) determinantal constraint on the energy, yielding rapidly convergingapproximations to the physical energies.This approach allows for great flexibility in how the weight, or reference function ,is chosen. In particular, one can allow the weight to mimic the asymptotic form ofthe physical solution, or take on the form of any positive solution (i.e. the bosonicground state, if known, even approximately). This is in sharp contrast to the popularHill determinant approach corresponding to taking c N = 0 , c N − = 0 in Eq.(8), andletting N → ∞ .Within the Hill determinant approach, truncating the c -power series is relevantif the system is exactly, or quasi-exactly, solvable. One might then believe that thisworks as an approximation in the general case. Initial studies of the Hill determinantapproach confirmed this, for weights that did not mimic the physical asymptotic formof the solution.One simple observation that suggests potential problems with the Hilldeterminant approach is that the asymptotic behavior of the c ’s cannot be directlyrelated to the normalizable, or non-normalizable, behavior of the physical orunphysical states, respectively. Indeed, it was pointed out by Tater and Turbinerthat the Hill determinant fails to converge, or converges to the wrong energy levels, incases where the reference function is chosen to mimic the physical asymptotic form.They used the sextic anharmonic potential as an example: V ( x ) = ax + bx + x .The asymptotic form of the wavefunction is R ( x ) = e − ( x + bx ) / . The correspondingrecursion relation for the c ’s is c n +2 = ( a + 2 n − − b / c n − + ( b ( n + 1 / − E ) c n ( n + 1)( n + 2) . (15)Consistent with the Tater and Turbiner results [5], Hautot [4] had argued thatsuch finite difference relations, coupled with the Hill determinant conditions ( c N = 0, lobal-Local Quantization N = even , N → ∞ , for parity invariant systems) can fail to take into account certaindominant solutions essential to quantization. He proposed a complicated procedurefor fixing this problem. We believe that OPPQ is a more transparent solution thatachieves the same result.
10 20 30 40 50 60 70 80 (cid:45) (cid:45) (cid:45) (cid:45) e n e r gy (cid:72) a (cid:76)
20 40 60 80 100 (cid:45) (cid:45) e n e r gy (cid:72) b (cid:76)
20 40 60 80 100 (cid:45) e n e r gy (cid:72) c (cid:76) Figure 1.
Hill determinant results for anomalous cases of the sextic anharmonicoscillator for parameter values: (a) (a=-18, b=0), (b) (a=-8, b=0) and (c) (a=-4,b=0).
For illustrative purposes we note that in Fig 1(a-c) we give the convergence ofthe first two even energy levels for three parameter cases: ( a = − b = 0), ( a = − b = 0) and ( a = − b = 0). In the first case both energy levels appear to convergeto the correct limit, albeit at a very slow rate. For the second case only the groundstate shows correct convergence, while the first excited state converges to the wronglimit. In the third situation the Hill determinant method fails for all energy levels,there is no convergence and in some cases there are no real solutions, represented by avalue of “0” in Figure. 1c. All three cases are correctly recovered by the global-localOPPQ variant developed in this paper (described in Sec. 2.2), as given in Table 1.The indicated limits are in keeping with a pure OPPQ analysis as publshed elsewhere[2]. lobal-Local Quantization Table 1.
Convergence properties for V ( x ) = ax + bx + x using R ( x ) = e − x / by using the global-local quantization. N E E E E a = − b = 010 -23.943914597 -17.917277160 -11.400078722 -8.48010452120 -21.324952739 -21.322438503 -7.599903464 -7.35950343730 -21.323394711 -21.321841616 -7.599035456 -7.36065799340 -21.323394694 -21.321841620 -7.599035461 -7.360657990 a = − b = 010 -5.477684656 -3.531662882 1.811585056 6.57067728620 -3.900838586 -3.534341976 2.086375864 6.05451002530 -3.900635158 -3.534354171 2.086528016 6.05540508740 -3.900635159 -3.534354170 2.086528012 6.055405205 a = − b = 010 -0.625342803 1.038625070 5.801273017 8.72554762420 -0.523263742 1.005832318 5.374951631 10.561437390230 -0.523268623 1.005768335 5.374969926 10.57258545840 -0.523268622 1.005768340 5.374970009 10.572585045In general, within the OPPQ ansatz, the better the reference function mimics theasymptotic form of the solution, the faster the convergence to the physical energies.Other redeeming features of the OPPQ procedure are that the reference function neednot be analytic. Thus, for the quartic anharmonic potential, V ( x ) = x + mx ,converging results are obtained if R ( x ) = e − | x | . In principle, for this case, adifferentiable form of the reference function would be R ( x ) = exp ( x )+ exp ( − x ) .Also, for generating the energies, one does not need the explicit, configurationspace, representation for the reference function. Thus one could use the (unknown)bosonic ground state (which must be positive), provided its power moments can begenerated to high accuracy, enabling the generation of its corresponding orthonormalpolynomials. In principle, the Eigenvalue Moment Method could be used for such cases(generating high precision values for the power moments of the ground state, as wellas the energy, through converging bounds) [7-9]. This convex optimization proceduredefines the first [10] application of semidefinite programming analysis to quantumoperators [7], and its computational implementation through linear programming[8,9,11].We note that a more conventional analysis involving expanding Ψ( x ) in terms ofan orthonormal basis P n ( x ) R ( x ), or Ψ( x ) = (cid:80) n γ n P n ( x ) R ( x ) , does not providethe flexibility of OPPQ, since the generation of the projection coefficients involvesthe integrals γ n = (cid:82) dx P n ( x ) R ( x )Ψ( x ), which cannot be expressed, generally, as aknown (i.e. in closed form) linear combination of the power moments of R ( x )Ψ( x ),except for special weights. One good example is the aforementioned observation thatOPPQ allows the use of the (accurately determined) power moments of the (bosonic)ground state, for quantizing the excited states. This type of analysis is not possiblewithin the more conventional, ortho-normal basis, approach.A final observation is that in selecting the appropriate R ( x ) that duplicatesthe asymptotic behavior of the configuration space solution, we expect its Fourier lobal-Local Quantization The OPPQ expansion in Eq.(9) was originally developed in the spirit of a (non-orthonormal) basis expansion where the expansion coeffcients are given by Eq.(11)and the following integral expression is finite: (cid:82) dx Ψ ( x ) R ( x ) = (cid:80) j Ω j < ∞ . Therewas no demand for pointwise convergence. However, there are good mathematicalreasons for expecting the OPPQ representation to be (non-uniformly) convergent ina pointwise manner. As previoulsy noted, this representation is a specific case ofthe more general problem of representing analytic functions by weighted families ofpolynomials. For Freudian weights of the form R ( x ) = e −| x | q , q >
1, it is known [1]that the representation in Eq.(9) converges within an infinite strip in the complex- x plane whose width is determined by the closest singularity (of the physical solution) tothe real axis. If we assume this, in general, for the types of physical systems of interest,then the natural question is to test the pointwise convergence of such representationsat the origin (among other possibilities):Ψ( x ) R ( x ) = ∞ (cid:88) j =0 Ω j P j ( x ) = (cid:16) ∞ (cid:88) n =0 c n x n (cid:17) , (16)where ∞ (cid:88) j = n Ξ ( j ) n Ω j = c n = T n, ( E ) c + T n, ( E ) c . (17)We approximate this through the truncation N (cid:88) j = n Ξ ( j ) n Ω j ( µ , . . . , µ m s ) = c n = T n, ( E ) c + T n, ( E ) c , (18)where we make explicit Ω’s linear dependence on the missing moments. Since thereare m s + 3 linear variables { c , c , µ , . . . , µ m s } , we must take n = 0 , . . . , m s + 2, inEq.(18), although N → ∞ . An energy dependent, ( m s + 3) × ( m s + 3) determinantalequation ensues, yielding converging results for the physical energies, as N → ∞ .The above analysis was implemented on the anomalous parameter values for thesextic anharmonic oscillator, as calculated through the Hill determinant approachgiven in Fig. 1. Table I shows the exceptional stability of the above “global-local”quantization procedure. These results agree with a pure OPPQ analysis as given inreference [2]; thereby strongly affirming the reliability of the OPPQ representationin capturing the local behavior of the physical solutions. A second example is givenbelow. lobal-Local Quantization
3. Global-Local Quantization for a Rational Anaharmonic Oscillator (cid:45) (cid:45) (cid:45) (cid:45) V (cid:72) x (cid:76) V (cid:72) x (cid:76) (cid:61) x (cid:45) (cid:73) (cid:43) x (cid:77) E (cid:61) (cid:45) Ψ (cid:72) x (cid:76) (cid:61) e (cid:45) x (cid:145) (cid:43) (cid:144) x Figure 2.
The anharmonic potential (solid line) and the unperturbed harmonicpotential energy curves and the ground state wave function corresponding toground state energy E = − Let us now consider the rational anharmonic oscillator potential V ( x ) = x − x ) , (19)for which the ground state is exactly knownΨ gr ( x ) ≡ e − x / (1 + x ) . This potential and the ground state associated with energy E = − µ p +6 = ( E − µ p +4 + (cid:16) p ( p + 7) + 3 E + 394 (cid:17) µ p +2 + (cid:16) p ( p + 3) + 94 E + 18 (cid:17) µ p + 94 p ( p − µ p − , (20)with m s = 5. The corresponding recursion relation for Taylor’s coefficients is c n +2 = 4(2 n − E − c n − + 4( n (11 − n ) − E − c n − − E + 2 n (3 n − c n n + 1)( n + 2) . (21)Three representative calculations are given in Table 2. The first is OPPQ usingthe weight R ( x ) = e − x / , the asymptotic form for the bound states. The second is lobal-Local Quantization Table 2.
Convergence for energy levels of the rational anharmonic potential forvarious methods as a function of truncation order.
N E E E E Direct OPPQ method with reference function e − x /
20 -2.919286247 0.910167637 3.603889662 5.91394800340 -2.996045597 0.799435213 3.437942536 5.72022536460 -2.999705662 0.792968365 3.426931961 5.70555177080 -2.999970901 0.792460681 3.426034090 5.704282111100 -2.999996463 0.792409589 3.425942480 5.704148274Matching local and global behavior through Eq. (18)20 -3.192388811 -0.534923547 3.546551239 5.74380212940 -3.008483327 0.735594478 3.407202508 5.60449728460 -3.000567710 0.789212601 3.424198982 5.69674028080 -3.000052234 0.792139749 3.425751715 5.703466230100 -3.000006028 0.792374246 3.425907672 5.704054764Ground state wave function as reference function10 -3.000000000 0.757672016 3.265699150 5.40772234720 -3.000000000 0.790563451 3.418884509 5.69113049330 -3.000000000 0.792220915 3.425272253 5.70287944240 -3.000000000 0.792377300 3.425841380 5.70395735450 -3.000000000 0.792398005 3.425914514 5.70409923860 -3.000000000 0.792401433 3.425926384 5.704122686obtained by implementing the global-local ansatz given in Eq.(18), ensuring not onlyaccurate and fast converging energy eigenvalues, but also faithful representation ofthe wave function that has implicitly both the correct local and global behavior. Thethird set of results illustrates the freedom of choice of the reference function withinOPPQ method, by taking it to be the exactly known ground state.Figure 3 demonstrate the pointwise convergence properties for the wave functioncalculated using the global-local quantization. The results presented are obtained byusing Eqs. (9) and (18) with truncation order N = 80. It is also instructive to comparethe power expansion of the reconstructed ground state wave function, scaled by thereference function R ( x ) = e − x / , which is given byΨrec( x ) /e − x = 1 − . x + 0 . x − . x + O ( x ) , with the corresponding exact expansion of the ground state:1(1 + x ) = 1 − . x + 0 . x − . x + O ( x ) . This shows that the solution provided by the global-local quantization has the claimedpointwise convergence properties of the wave function together with fast convergenceof energy levels.It is instructive to compare the (non-uniform) global convergence (on the realline) of the OPPQ representation with the local Taylor series expansion. Specifically,the wave function representation in terms of orthogonal polynomials as in Eq. (16)has global convergence properties, as opposed to the local convergence of Taylor’spower expansion which is always restricted to a disk. This point is illustrated in Fig.4 where the convergence domain in the complex x -plane of Ψ gr covers an increasing lobal-Local Quantization (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) x e rr o r Figure 3.
The difference between the ground state wave function Ψgr and itsreconstruction obtained by using Eqs. (9) and (18). horizontal strip, as the OPPQ truncation order in Eq. (16) increases from N = 20 to N = 140. The strip is determined by the ± i (cid:113) singularities. On the other hand, theconvergence of Taylor’s expansion of the ground state is always local, limited by a diskaround the expansion center (chosen in Fig. 4 to be centered at 1.4) and bordered bythe singularities.The process of analytic continuation of a function, f ( x ), results in power seriesexpansions for the local expression f ( τ + χ ) = (cid:80) j c j ( τ ) χ j , involving non-globalexpansion coefficients (i.e. the c j ( τ ) are τ dependent). The orthonormal polynomialexpansion in Eq.(19) transcends this into a global statement, particularly close to thereal axis. Of course, one advantage of the conventional analytic continuation processis that one can control the uniformity of convergence of the analytic continuationto the target function. For our purposes, the non-uniform nature of the pointwiseconvergence of the OPPQ expansion is not a problem, since we are using the localinformation (i.e. the derivatives, etc.) at a chosen point.
4. General Considerations
Traditional orthonormal basis expansion methods in quantum mechanics, Ψ( x ) = (cid:80) n d n B n ( x ) (i.e. (cid:104)B m |B n (cid:105) = δ m,n ) are not designed to recover the pointwize structureof the approximated solution, since they emphasize an L convergence. That is, the N -th partial sum, Ψ N ( x ) = (cid:80) Nn =0 d n B n ( x ), converges to the physical solution accordingto lim N →∞ (cid:82) dx | Ψ( x ) − Ψ N ( x ) | = 0. This does not imply pointwize convergencesince in the infinite limit one could have Ψ( x j ) − Ψ ∞ ( x j ) (cid:54) = 0 on a subset of measurezero. The OPPQ representation has a greater chance of pointwize convergence becauseits underlying structure mimics the usual power series expansion. In addition, itcan simultaneously recover the local, global, and asymptotic features of the desiredphysical solution. lobal-Local Quantization -4 -2 0 2 4-202 +++ Re(x) I m ( x ) i√ − -i√ − Figure 4.
Converge domains in the complex plane for Taylor’s expansionand OPPQ expansion for the ground state wave function scaled by the referencefunction (Ψ gr / R ) of the rational anharmonic oscillator in Eq. (19). The OPPQexpansion is carried out for the sequence of truncation orders N = 20 , ,
100 and140, showing that in the N → ∞ limit, its convergence domain is a horizontalstrip bordered by the singular point of the wave function. In contrast, Taylor’sexpansion does not converge outside the disk centered at the chosen expansioncenter, in the calculated case x = 1 . The OPPQ basis functions P n ( x ) R ( x ) are non-orthonormal. The advantage ofthis representation is that the Ω j projection coefficients are easily determined for thetypes of (multidimensional) systems of interest to us, although the present work islimited to one dimensional problems.We note that P n ( x ) (cid:112) R ( x ) is expected to correspond to a complete orthonormalbasis; however the expansion Ψ( x ) = (cid:80) n γ n P n ( x ) (cid:112) R ( x ) does not lead to an easy,algebraic (closed form) generation of the projection coefficients, γ n = (cid:104) P n ( x ) √R| Ψ (cid:105) ,for arbitrary R , as pursued here. Despite this, Ψ √R = (cid:80) ∞ n =0 Ω n P n ( x ) (cid:112) R ( x ),will correspond to a conventional orthonormal basis expansion, with expected L convegence : Lim N →∞ (cid:90) dx | Ψ( x ) (cid:112) R ( x ) − N (cid:88) n =0 Ω n P n ( x ) (cid:112) R ( x ) | = 0 . (22)Given that M in x R ( x ) > Lim N →∞ (cid:90) dx | Ψ( x ) − Ψ N ( x ) | = 0 , (23)where the OPPQ N -th partial sum is defined byΨ N ( x ) = N (cid:88) n =0 Ω n P n ( x ) R ( x ) . (24)That is, the OPPQ representation will also be L convergent. lobal-Local Quantization Lim N →∞ N (cid:88) n =0 Ω n = (cid:90) dx Ψ R , (25)or Lim n →∞ Ω n = 0 . (26)That is, if the basis { P n ( x ) (cid:112) R ( x ) } is complete, then the limit in Eq. (26) holds.An alternative representation for the orthonormality relations of the P n ’s is (cid:90) dx x p P n ( x ) R ( x ) = 0 , (27)for p < n , leading to µ p = (cid:90) dx x p Ψ N ( x ) , ≤ p ≤ N. (28)Thus the N -th OPPQ partial sum has its first 1 + N moments identical to that ofthe physical state. This is the more general interpretation of the equality in Eq.(9).Thus, the Ψ N ( x ) contain physical information.Depending on the asymptotic decay of the reference function, as compared to thecomplex extension of the Fourier kernel e − ikx , the Fourier transform of the truncatedOPPQ expression, ˆΨ N ( k ), can be an entire function, bounded along the real axis.Furthermore, both the Fourier transform of the truncated OPPQ expression, ˆΨ N ( k ),and the actual Fourier transform of the physical solution, ˆΨ( k ), will have identicalpower series expansions up to order k N . It is therefore reasonable to expect that | ˆΨ( k ) − ˆΨ N ( k ) | < (cid:15) N over some interval | k | < κ N , where Lim N →∞ (cid:15) N = 0 and Lim N →∞ κ N = ∞ . Therefore, the error in local approximation, near the origin inconfiguration space, is controlled by | Ψ( x ) − Ψ N ( x ) | < √ π | (cid:82) dk e ixk ( ˆΨ( k ) − ˆΨ N ( k )) | < √ π (cid:16) (cid:15) N κ N + (cid:82) | k | >κ N dk | ˆΨ( k ) − ˆΨ N ( k ) | (cid:17) . If Lim N →∞ (cid:15) N κ N = 0, and ˆΨ N ( k ) cancapture the form of the physical solution’s decay, for | k | > κ N , then we can expect goodlocal approximations for x ≈
0. In other words, how the reference function is chosenwill lead to enhanced convergence rates to the local properties of the wavefunction inconfiguration space. More generally, it is to be expected that the OPPQ representationwill generally converge, pointwise, to the physical solution.
5. Conclusion
The importance of moment representations in algebratizing many quantizationproblems is often overlooked. The importance, and flexibility, of weighted orthonormalpolynomial representations has been amply demonstrated here and in previous workswith regards to their ability to generate the discrete energies to arbitrary precision.The relevance of such representations for reconstructing the wavefunctions is stronglysuggested by the present work, including our ability to quantize by imposing global-local constraints on the OPPQ, weighted polynomial, expansion. lobal-Local Quantization Acknowledgments
Discussions with Dr. Daniel Bessis, Dr. Donald Kouri, and Dr. Walter Gautschi aregreatly appreciated. Additional correspondences with Dr. H. N. Mhaskar and Dr. D.Lubinsky are also appreciated. One of the authors (DV) is grateful for the supportreceived from the National Science Foundation through a grant for the Center forResearch on Complex Networks (HRD-1137732).
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