This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

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Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant differential operators, and solve general equivalence problems for both finite-dimensional Lie group actions and infinite Lie pseudo-groups. A wide variety of applications, ranging from differential equations to differential geometry to computer vision are presented. The theoretical justifications for the moving coframe algorithm will appear in the next paper in this series.

Moving coframes. I. A practical algorithm

We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrodinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X1 exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.

http://iopscience.iop.org/article/10.1088/1751-8113/41/39/392001/pdf

Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry

An extension of Bochner's problem: Exceptional invariant subspaces

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.

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Towards a Lie theory of locally convex groups

We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi-exactly solvable Schrodinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi-exactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.

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Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

Finite-dimensional real analytic Lie algebras of vector fields on R 2 are completely classified up to changes of local coordinates

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Lie algebras of vector fields in the real plane

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial flags; the lowest degree polynomial has degree m > 0. In this paper we review the classification of codimension m = 1 exceptional polynomials, and give a novel, compact proof of the fundamental classification theorem for codimension 1 polynomial flags. As well, we describe the mechanism or rational factorizations of 2nd order operators as the analogue of the Darboux transformation in this context. We finish with the example of higher codimension generalization of Jacobi polynomials and perform the complete analysis of parameter values for which these families have non-singular weights.

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On orthogonal polynomials spanning a non-standard flag

The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane

We prove that in the non extreme Kerr Newman black hole geometry the Dirac equation has no normalizable time periodic solutions A key tool is Chandrasekhar s separation of the Dirac equation in this geometry A similar non existence theorem is established in a more general class of stationary axisymmetric metrics in which the Dirac equation is known to be separable These results indicate that in contrast with the classical situation of massive particle orbits a quantum mechanical Dirac particle must either disappear into the black hole or escape to in nity

https://www.mis.mpg.de/preprints/1999/preprint1999_37.pdf

Niky Kamran

Papers

Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

Lie algebras of vector fields in the real plane

On orthogonal polynomials spanning a non-standard flag

The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane

Nonexistence of time‐periodic solutions of the Dirac equation in an axisymmetric black hole geometry