A relativistic analogue of the quantum adiabatic approximation is developed for Klein-Gordon fields minimally coupled to electromagnetism, gravity and an arbitrary scalar potential. The corresponding adiabatic dynamical and geometrical phases are calculated. The method introduced in this paper avoids the use of an inner product on the space of solutions of the Klein-Gordon equation. Its practic...

This paper establishes certain existence and classification results for solutions to SU(n) Toda systems with three singular sources at 0, 1, and∞. First, we determine the necessary conditions for such an SU(n) Toda system to be related to an nth order hypergeometric equation. Then, we construct solutions for SU(n) Toda systems that satisfy the necessary conditions and also the interlacing condi...

McKean and Trubowitz [2] showed that the theory of the KdV equation ∂ ∂t g(x, t) = ∂ 3 ∂x 3 g(x, t) − 6g(x, t) ∂g ∂x (x, t). is intimately related to the geometry of a related hyperelliptic curve of infinite genus, the Bloch spectrum B g t of the operator L g t : ψ → d 2 dx 2 ψ(x) + g(x, t)ψ(x), where g t = g(x, t). As was known classically, B g t is independent of t, when g(x, t) evolves accor...

Four integrable symplectic maps approximating two Hamiltonian flows from the relativistic Toda hierarchy are introduced. They are demostrated to belong to the same hierarchy and to examplify the general scheme for symplectic maps on groups equiped with quadratic Poisson brackets. The initial value problem for the difference equations is solved in terms of a factorization problem in a group. Int...

We study a modified version of an equation of the continuous Toda type in 1+1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter ǫ. We apply the prolongation method, the symmetry and the approximate symmetry approach. This strategy allows us to get insight into both the equations for ǫ = 0 and ǫ 6= 0, whose properties arising in th...

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The equation of motion of affine Toda field theory is a coupled equation for r fields, r is the rank of the underlying Lie algebra. Most of the theories admit reduction, in which the equation is satisfied by fewer than r fields. The reductions in the existing literature are achieved by identifying (folding) the points in the Dynkin diagrams which are connected by symmetry (automorphism). In thi...

We introduce a family of compatible Poisson brackets on the space of rational functions with denominator of a fixed degree and use it to derive a multi-Hamiltonian structure for a family of integrable lattice equations that includes both the standard and the relativistic Toda lattices. PACS 02.30, 05.45, 11.30.N

We develop relativistic wave equations in the framework of the new non-hermitian PT quantum mechanics. The familiar Hermitian Dirac equation emerges as an exact result of imposing the Dirac algebra, the criteria of PT -symmetric quantum mechanics, and relativistic invariance. However, relaxing the constraint that in particular the mass matrix be Hermitian also allows for models that have no cou...