We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for multipoint potentials in two and three dimensions. For single point potentials in 3D such formulas were obtained in an old unpublished work of L.D
. Faddeev. For single point potentials in 2D such formulas were given recently by the authors in arXiv:1110.3157 .
The Complete Manifold of Ground State Eigenfunctions for the Purely Magnetic 2D Pauli Operator is considered as a by-product of the new reduction found by the present authors few years ago for the Algebrogeometric Inverse Spectral Data (i.e. Riemann
Surfaces and Divisors). This reduction is associated with the (2+1) Soliton Hierarhy containing a 2D analog of the famous Burgers System. This article contains also exposition of the previous works made since 1980 including the first topological ideas in the space of quasimomenta. We present here also new results dedicated to the self-adjoint boundary problems for Pauli Operator. The 2D zero level nonspectral Bloch-Floquet functions give discrete points of additional spectrum similar to the boundary states of finite-gap 1D potentials in the gaps.
The Theory of (2+1) Systems based on 2D Schrodinger Operator was started by S.Manakov, B.Dubrovin, I.Krichever and S.Novikov in 1976. The Analog of Lax Pairs introduced by Manakov, has a form $L_t=[L,H]-fL$ (The $L,H,f$-triples) where $L=partial_xpar
tial_y+Gpartial_y+S$ and $H,f$-some linear PDEs. Their Algebro-Geometric Solutions and therefore the full higher order hierarchies were constructed by B.Dubrovin, I.Krichever and S.Novikov. The Theory of 2D Inverse Spectral Problems for the Elliptic Operator $L$ with $x,y$ replaced by $z,bar{z}$, was started by B.Dubrovin, I.Krichever and S.Novikov: The Inverse Spectral Problem Data are taken from the complex Fermi-Curve consisting of all Bloch-Floquet Eigenfunctions $Lpsi=const$. Many interesting systems were found later. However, specific properties of the very first system, offered by Manakov for the verification of new method only, were not studied more than 10 years until B.Konopelchenko found in 1988 analogs of Backund Transformations for it. He pointed out on the Burgers-Type Reduction. Indeed, the present authors quite recently found very interesting extensions, reductions and applications of that system both in the theory of nonlinear evolution systems (The Self-Adjoint and 2D Burgers Hierarhies were invented, and corresponding reductions of Inverse Problem Data found) and in the Spectral Theory of Important Physical Operators (The Purely Magnetic 2D Pauli Operators). We call this system GKMMN by the names of authors who studied it.
