Reviews in Mathematical Physics

A systematic formulation of the higher genus expansion in topological string theory is considered. We also develop a simple way of evaluating genus zero correlation functions. At higher genera we derive some interesting formulas for the free energy in the A₁ and A₂ models. We present some evidence that topological minimal models associated with Lie algebras other than the A-D-E type do not have a consistent higher genus expansion beyond genus one. We also present some new results on the CP¹ model at higher genera.

We consider a hydrogen atom in a constant and very large magnetic field. For bounded angular momentum about the direction of the magnetic field we consider resonances in the sense of dilation analyticity. We find a class of resonances corresponding to quasi-stationary states which behave in a certain sense like ground states. The position of these resonances in the complex plane is estimated. Further it is shown that under the time evolution generated by the Hamiltonian for this system these quasi-stationary states exhibit exponential decay in time.

In radiative decay the coupled system, electron bound by some external potential plus radiation field, relaxes to the ground state in the long time limit. Our central issue is to prove such a behaviour on the basis of the corresponding Schrödinger equation. We argue that the spin-boson Hamiltonian is a simple, but physically still acceptable test case. We relate radiative decay to scattering theory and prove the existence of wave operators. Other approaches are reviewed and compared. Some challenging open problems are listed.

We propose a general framework for constructing systematically the Lax formulation of the soliton equations using the bi-Hamiltonian formalism. The method is applied to several examples, both classical and supersymmetric.

A generalization of the method of martinez on phase space tunneling is discussed. In particular, the key estimate is formulated for a class of ħ-pseudodifferential operators.

We define total scattering cross-sections for magnetic Schrödinger operators in two dimensions and prove the shadow scattering (the quantum total cross-sections double the classical ones in the semi-classical limit) under some assumptions.

The main aim of this paper is to provide a proper mathematical framework for the theory of topological non-compact quantum groups, where we have to deal with non-unital C*-algebras. The basic concepts and results related to the affiliation relation in the C*-algebra theory are recalled. In particular natural topologies on the set of affiliated elements and on the set of morphisms are considered. The notion of a C*-algebra generated by a finite sequence of unbounded elements is introduced and investigated. It is generalized to include continuous quantum families of generators. An essential part of the duality theory for C*-algebras is presented including complete proofs of many theorems announced in [17]. The results are used to develop a presentation method of introducing non-unital C*-algebras. Numerous examples related mainly to the quantum group theory are presented.