Complex Classical Mechanics of a QES Potential

A complex periodic QES potential and exceptional points

We show that the complex PT -symmetric periodic potential V (x) = −(iξ sin 2x+ N)2, where ξ is real and N is a positive integer, is quasi-exactly solvable. For odd values of N ≥ 3, it may lead to exceptional points depending upon the strength of the coupling parameter ξ. The corresponding Schrödinger equation is also shown to go over to the Mathieu equation asymptotically. The limiting value of...

Complex Variables in Classical Hamiltonian Mechanics

O ct 2 00 7 A complex periodic QES potential and exceptional points

We show that the complex PT -symmetric periodic potential V (x) = −(iξ sin 2x+ N)2, where ξ is real and N is a positive integer, is quasi-exactly solvable. For odd values of N ≥ 3, it may lead to exceptional points depending upon the strength of the coupling parameter ξ. The corresponding Schrödinger equation is also shown to go over to the Mathieu equation asymptotically. The limiting value of...

Classical Mechanics

here xi = (xi1, . . . , xid) are coordinates of the i-th particle and ∂xi is the gradient (∂xi1 , . . . , ∂xid); d is the space dimension (i.e. d = 3, usually). The potential energy function will be supposed “smooth”, i.e. analytic except, possibly, when two positions coincide. The latter exception is necessary to include the important cases of gravitational attraction or, when dealing with ele...

Polyakov's String Classical Mechanics

This paper is almost an exercise in which the Hamiltonian scheme is developed for Polyakov's classical string, by following the usual framework suggested by Dirac and Bergman for the reduction of gauge theories to their essential physical degrees of freedom. The results collected here will be useful in some forthcoming papers, where strings will be studied in the unusual context of Wigner-covar...