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Register a new userEquation for the Nakanishi weight function using the inverse Stieltjes transform
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The bound state Bethe-Salpeter amplitude was expressed by Nakanishi in terms of a smooth weight function g. By using the generalized Stieltjes transform, we derive an integral equation for the Nakanishi function g for a bound state case. It has the standard form g= Vg, where V is a two-dimensional integral operator. The prescription for obtaining the kernel V starting with the kernel K of the Bethe-Salpeter equation is given.
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The bound state Bethe-Salpeter amplitude was expressed by Nakanishi using a two-dimensional integral representation, in terms of a smooth weight function $g$, which carries the detailed dynamical information. A similar, but one-dimensional, integral representation can be obtained for the Light-Front wave function in terms of the same weight function $g$. By using the generalized Stieltjes transform, we first obtain $g$ in terms of the Light-Front wave function in the complex plane of its arguments. Next, a new integral equation for the Nakanishi weight function $g$ is derived for a bound state case. It has the standard form $g= N g$, where $N$ is a two-dimensional integral operator. We give the prescription for obtaining the kernel $ N$ starting with the kernel $K$ of the Bethe-Salpeter equation. The derivation is valid for any kernel given by an irreducible Feynman amplitude.
The Bethe-Salpeter amplitude $Phi(k,p)$ is expressed, by means of the Nakanishi integral representation, via a smooth function $g(gamma,z)$. This function satisfies a canonical equation $g=Ng$. However, calculations of the kernel $N$ in this equation, presented previously, were restricted to one-boson exchange and, depending on method, dealt with complex multivalued functions. Although these difficulties are surmountable, but in practice, they complicate finding the unambiguous result. In the present work, an unambiguous expression for the kernel $N$ in terms of real functions is derived. For the one-boson scalar exchange, the explicit formula for $N$ is found. With this equation and kernel, the binding energies, calculated previously, are reproduced. Their finding, as well as calculation of the Bethe-Salpeter amplitude in the Minkowski space, become not more difficult than in the Euclidean one. The method can be generalized to any kernel given by irreducible Feynman graph. This generalization is illustrated by example of the cross-ladder kernel.
The inverse scattering transform is extended to investigate the Tzitz{e}ica equation. A set of sectionally analytic eigenfunctions and auxiliary eigenfunctions are introduced. We note that in this procedure, the auxiliary eigenfunctions play an important role. Besides, the symmetries of the analytic eigenfunctions and scattering data are discussed. The asymptotic behaviors of the Jost eigenfunctions are derived systematically. A Riemann-Hilbert problem is constructed to study the inverse scattering problem. Lastly, some novel exact solutions are obtained for reflectionless potentials.
As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse Scattering Transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation $v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0$, in this paper we establish the following. 1. The non-local term $partial_x^{-1}$ arising from its evolutionary form $v_{t}= v_{x}v_{y}-partial^{-1}_{x},partial_{y},[v_{y}+v^2_{x}]$ corresponds to the asymmetric integral $-int_x^{infty}dx$. 2. Smooth and well-localized initial data $v(x,y,0)$ evolve in time developing, for $t>0$, the constraint $partial_y {cal M}(y,t)equiv 0$, where ${cal M}(y,t)=int_{-infty}^{+infty} left[v_{y}(x,y,t) +(v_{x}(x,y,t))^2right],dx$. 3. Since no smooth and well-localized initial data can satisfy such constraint at $t=0$, the initial ($t=0+$) dynamics of the Pavlov equation can not be smooth, although, as it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results, should be successfully used in the study of the non-locality of other basic examples of integrable dispersionless PDEs in multidimensions.
A systematic approach for the model building of Generalized Parton Distributions (GPDs), based on their overlap representation within the DGLAP kinematic region and a further covariant extension to the ERBL one, is applied to the valence-quark pions case, using light-front wave functions inspired by the Nakanishi representation of the pions Bethe-Salpeter amplitudes (BSA). This simple but fruitful pions GPD model illustrates the general model building technique and, in addition, allows for the ambiguities related to the covariant extension, grounded on the Double Distribution (DD) representation, to be constrained by requiring a soft-pion theorem to be properly observed.
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