Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.

The aim of this short lecture course is to develop Selberg’s trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the Laplace-Beltrami operator − ∆ on the surface M represents the quantum Hamiltonian of a particle, whose classical dynamics is governed by the (strongly chaotic) geodesic flow on the unit tangent bundle of M. The trace formula is currently the only available tool to analyze the fine structure of the spectrum of −∆; no individual formulas for its eigenvalues are known. In the case of more general quantum systems, the role of Selberg’s formula is taken over by the semiclassical Gutzwiller trace formula [10], [7]. We begin by reviewing the trace formulas for the simplest compact manifolds, the circle S 1 (Section 1) and the sphere S 2 (Section 2). In both cases, the corresponding geodesic flow is integrable, and the trace formula is a consequence of the Poisson summation formula. In the remaining sections we shall discuss the following topics: the Laplacian on the hyperbolic plane and isometries (Section 3); Green’s functions (Section 4); Selberg’s point pair invariants (Section 5); The ghost of the sphere (Section 6); Linear operators on hyperbolic surfaces (Section 7); A trace formula for hyperbolic cylinders and poles of the scattering matrix (Section 8); Back to general hyperbolic surfaces (Section 9); The spectrum of a compact surface, Selberg’s pre-trace and trace formulas (Section 10); Heat kernel and Weyl’s law (Section 11); Density of closed geodesics (Section 12); Trace of the resolvent (Section 13); Selberg’s zeta function (Section 14); Suggestions for exercises and further reading (Section 15). Our main references are Hejhal’s classic lecture notes [12, Chapters one and two], Balazs and Voros ’ excellent introduction [1], and Cartier and Voros’ nouvelle interprétation [6]. Section 15 comprises a list of references for further reading. These notes are based on lectures given at the International School Quantum

Abstract. The curvature effect on a quantum dot with impurity is investigated. The model is considered on the Lobachevsky plane. The confinement and impurity potentials are chosen so that the model is explicitly solvable. The Green function as well as the Krein Q-function are computed.

We consider a charged quantum particle living in the Lobachevsky plane and interacting with a homogeneous magnetic field perpendicular to the plane and a point interaction which is transported adiabatically along a closed loop C in the plane. We show that the bound-state eigenfunction acquires at that the Berry phase equal to 2 times the number of the flux quanta through the area encircled by C. 1 Introduction The phenomena arising from a geometric phase called Berry phase [Ber] have been put in evidence in many quantum mechanical systems. Recently such a "Berry phase effect" has been observed in some magnetic systems [LSG, MHK]. Moreover, it was shown that the geometric phase can emerge even in a time-independent homogeneous magnetic field when a potential 1 well trapping a two-dimensional particle is transported along a closed loop. An example in which the potential is of zero range is worked out in [EG], where a formula was proved showing that in the absence of an additional conf...

We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential V (̺) = (a 2 ω 2 /4)sinh(̺/a) 2 where a is the curvature radius and ̺ is the geodesic distance from a fixed center. Thus the potential is rotationally symmetric and unbounded likewise as in the Euclidean case. The eigenvalue equation leads to the differential equation of spheroidal functions. We provide a basic numerical analysis of eigenvalues and eigenfunctions in the case when the value of the angular momentum, m, equals 0. 1

In this paper I discuss by means of path integrals the quantum dynamics of a charged particle on the hyperbolic plane under the influence of an Aharonov–Bohm gauge field. The path integral can be solved in terms of an expansion of the homotopy classes of paths. I discuss the interference pattern of scattering by an Aharonov–Bohm gauge field in the flat space limit, yielding a characteristic oscillating behavior in terms of the field strength. In addition, the cases of the isotropic Higgs-oscillator and the Kepler–Coulomb The Aharonov–Bohm gauge field has a long history, beginning in 1959 by a classical paper by Aharonov and Bohm [Aharonov and Bohm (1959)]. The effect has been well studied and well confirmed [Anandan and Safko (1994)], but not necessarily well understood. It describes the motion of charged particles, i.e. electrons, which are scattered by an infinitesimal thin solenoid.