Scalet Equation

A similar equation exists, in the translation variable space, dbjt(a, b) = e a, b)j (a, b) however, it will be implicitly incorporated within our scalet equation analysis when we specify the form of the initial, infinite scale, scalet configurations, (0,6). 

Instead of these conditions, from the scalet equation s (and CWT s) perspective, the more natural, and asymptotically equivalent, quantization conditions correspond to... 

A casual inspection of Eq.(1.9) or Eq.(l.lO) would suggest that a TPQ analysis can be done totally within a scalet representation (i.e. within the solution space corresponding to Eq.(l.S)), without the need to explicitly make use of wavelets. This is true, provided only approximate estimates for the discrete state energies are required. It turns out that for any E value (physical or not), the scalet equation admits a large subset of scalet solutions which, in the zero scale limit, become a solution of the Schrodinger equation. 

One can realize an exact TPQ analysis, capable of achieving arbitrary quantization precision, if one works within a combined scalet-wavelet representation. In this case, one uses the scalet equation as a generator for the required wavelet transform coefficients within the dual-wavelet basis expansion for the wavefunction. Imposing the TPQ conditions (either Eq.(1.9) or Eq.(l.lO)) on such an expansion defines an exact quantization formalism. This is discussed in Sec. 1.5. 

In principle, the scalet equation can be solved in order to generate , provided a convenient scaling function is chosen (i.e. one whose algebraic structure generates a scalet equation). 

Alternatively, the scalet equation can be used to generate the wavelet transform for mother wavelets that C6in be generated from corresponding scaling functions (i.e. derivatives of such expressions, etc.). In this case, appropriate differences of the scalets generate the wavelet transform. For instance, in the case of the Mexican hat wavelet, the wavelet transform corresponds to the difference between the zeroth order and second order scalet functions ... 

In the earlier works by Handy and Murenzi, the initial value problem character of the scalet equation was used to recover the pointwise structure of the physical solution based upon EMM or MRF estimates for the physical energy and the associated (infinite scale) missing moments. Thus, given the physical missing moments, /if(a = oo, = 0) = /i( ) 0 < C < m , one can then generate all the infinite scale, i>-dependent, moments through the relation... 

For reasons clarified in the following discussions, the development of a purely scalet, TPQ, analysis can only yield approximate results. This is because the scalet equation, in the zero scale limit, at a fixed b position, cannot distinguish between physical (bounded) and unphysical (unbounded) solutions to the Schrodinger equation. That is, for any E value, the scalet equation admits solutions whose zero scale limit becomes a solution to the Schrodinger equation. 

In order to achieve an exact TPQ analysis, capable of attauning arbitrary precision, one must access the zero scale limit in a manner that uniquely singles out the physical solutions. This can only be done by using the scalet equation to generate the CWT or DCWT coefficients required in the signal( )-wavelet inversion formulas, followed by the imposition of the TPQ conditions, on the CWT/DCWT representation. This is discussed in the final section. 

For a given, arbitrary, value of the energy variable, E, the scalet equation admits rus + 1 independent basic solutions, B E,e-,a,b) (or l a,b), where the E, e dependence is implicitly assumed, for reasons of compactness of notation in the relations presented below) each satisfying the initial conditions i.e. take i ic and n(iv) = in Eq.(1.80))... 

In order to better understand the structure of the scalet equation formalism, we consider the problem of the quartic anharmonic oscillator potential... 

The analytic structure of the matrix, with respect to both its a and e dependence, assures us that any solution to the scalet equation will be analytic in these variables as well. 

The scalet equation integrates along the positive a direction, utilizing the initial configuration in Eq.(1.80), which depend, linearly, on the missing moments, p( ) = p (0,0). 

Both the scalet equation and the above (translation direction differential) equation are satisfied by physical solutions. It is not clear whether the same holds for unphysical scalet solutions, although preliminary numerical results suggest that it does. 

Inserting this in the scalet equation (i.e. da = —a da) we obtain (note that we have implicitly made use of J/ = 1, for this problem)... 

A more comprehensive understanding of the zero scale asymptotic behavior of the Ut solutions can be obtained by transforming the scalet equation (four coupled, first order, differential equations) into one fourth order differential equation for fj,o a,b). From lowest order JWKB analysis, one obtains the four basic modes (Handy and Brooks (2000))... 

The (convergent) asymptotic series results agree with the direct integration of the scalet equation, where the missing moments and energies can be obtained by MRF analysis. Specifically, for the quartic anharmonic oscillator, Egr = 1.392351642, the nonzero missing moments are /x(0) =. 6426706223, and p(2) =. 3573293777. For the quartic double well potential, Egr = -3.410142761, p(0) = 0.3223013271, and p(2) = 0.6776986729. 

The scalet equation is analytic at e = 0. This is immediately apparent from the M matrix s regular e dependence in Eq.(1.92). This is true in general, for any rational fraction potential, as long as the underlying scaling function dies off more slowly than the lowest order JWKB estimate for the physical wavefunction. 

Using Eqs.(1.91-1.92) by way of example, the infinite scale, initial, configuration, /t(0, b), is to be regarded as e independent (i.e. of zeroth order in e). The zeroth order scalet equation becomes... 

Since the compactly supported moments are defined for any solution of the Schrodinger equation, we see that the original scalet equation is capable of generating solutions of the Schrodinger equation, for any E. 

As argued previously, we then have ag(T<) < Ou,(rf). This condition was used within the TPQ-MRF representation to discriminate between the physical and spurious solutions generated (HMBB (2000)). The effective quantization scale in that case corresponded to the smallest scale up to which the MRF generated solutions satisfied the scalet equation. 

The scalet equation representation incorporates the Moment Quantization formalism, with its explicit anal rtic (regular) dependence on the kinetic energy expansion parameter, c, and all the (complex) turning points. 

The total number of all (complex) turning points is equal to the dimension of the scalet equation. 

In general, it is difficult to develop a TPQ analysis within the scalet representation because the (approximate) conditions to be imposed (i.e zero kinetic energy) can only be done at scales, oq, that can be close to asch- At the Schrodinger scale asch-, the scalet equation solutions are close to the corresponding Schrodinger equation solution they are converging to, and the local structure of the TPQ conditions would be incapable of efficiently distinguishing between physical and unphysical solutions. 

Having established that a direct TPQ analysis in the scalet equation representation is very difficult, we present an alternative procedure which can yield exact results, based on combining the scalet equation with the DCWT (or CWT) signal ( )-wavelet inversion formula. 

In this exact TPQ-scalet/wavelet formulation, we shall use the scalet equation as the generator for the DCWT coefficients. In keeping with the basic philosophy of the TPQ approach, we assume that the more important DCWT coefficients, e those for which p and... 

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