International Journal of Geometric Methods in Modern Physics

We consider a Schrödinger differential expression L₀ = Δ_M + V₀ on a Riemannian manifold (M,g) with metric g, where Δ_M is the scalar Laplacian on M and V₀ is a real-valued locally square integrable function on M. We consider a perturbation L₀ + V, where V is a non-negative locally square-integrable function on M, and give sufficient conditions for L₀ + V to be essentially self-adjoint on . This is an extension of a result of T. Kappeler.

In this paper we find sufficient conditions for a Bergman Einstein metric on a complex manifold to be balanced in terms of its Bochner's coordinates.

The bosonic string theory evolved as an attempt to find a physical/quantum mechanical model capable of reproducing Euler's beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In several recent publications, attempts were made to restore the missing links. As discussed in these publications, the existing mathematical interpretation of the multidimensional analogue of Euler's beta function as one of the periods associated with the corresponding differential form "living" on the Fermat-type (hyper) surface, happens to be crucial for restoration of the quantum/statistical mechanical models reproducing such generalized beta function. There is a number of nontraditional models — all interrelated — capable of reproducing the Veneziano amplitudes. In this work we would like to discuss two of such new models: symplectic and supersymmetric. The symplectic model is based on observation that the Veneziano amplitude is just the Laplace transform of the generating function for the Ehrhart polynomial. Such a polynomial counts the number of lattice points inside the rational polytope (i.e. polytope whose vertices are located at the nodes of a regular lattice) and at its boundaries. In the present case, the polytope is a regular simplex. It is a deformation retract for the Fermat-type (hyper) surface (perhaps inflated, as explained in the text). Using known connections between polytopes and dynamical systems, the quantum mechanical system associated with such a dynamical system is found. The ground state of this system is degenerate with degeneracy factor given by the Ehrhart polynomial. Using some ideas by Atiyah, Bott and Witten we argue that the supersymmetric model related to the symplectic can be recovered. While recovering this model, we demonstrate that the ground state of such a model is degenerate with the same degeneracy factor as for earlier obtained symplectic model. Since the wave functions of this model are in one to one correspondence with the Veneziano amplitudes, this exactly solvable supersymmetric (and, hence, also symplectic) model is sufficient for recovery of the partition function reproducing the Veneziano amplitudes thus providing the exact solution of the Veneziano model.

We show that graphs, networks and other related discrete model systems carry a natural supersymmetric structure, which, apart from its conceptual importance as to possible physical applications, allows us to derive a series of spectral properties for a class of graph operators which typically encode relevant graph characteristics.

A geometric framework for describing and solving time-dependent implicit differential equations F(t,x,x′) = 0 is studied, paying special attention to the linearly singular case, where F is affine in the velocities: A(t,x)x′ = b(t,x). This framework is based on the jet bundle of a time-dependent configuration space, and is an extension of the geometric framework of the autonomous case. When A is a singular matrix, the solutions can be obtained by means of constraint algorithms, either directly or through an equivalent autonomous system that can be constructed using the vector hull functor of affine spaces. As applications, we consider the jet bundle description of time-dependent lagrangian systems and the Skinner–Rusk formulation of time-dependent mechanics.

We study the action of vector field Vect(S¹) on the space of higher order differential operators , and its connection to projective structure on S¹. In particular, we study the Euler–Poincaré flows on the space of third order differential operators and its relation to Drienfeld–Sokolov and coupled KdV type systems. We also discuss the Boussinesq equation associated with the third order operator. We study the factorization of higher order operators and its compatibility with the action of Vect(S¹). We obtain the generalized Miura transformation and its connection to the modified Boussinesq equation. Finally, we also discuss flows on the special higher order differential operators for all u_i = f(u,u_x,u_xx…) and its connection to KdV equation.

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space. For that purpose we consider Kähler phase spaces, endowed with a dynamics whose Hamiltonian equals the local Kähler potential.

In a recent paper, a variational principle for Liouville dynamics (vector fields preserving a volume form) was introduced. In the present paper, we discuss the relation between symmetries and conservation laws in Liouville dynamics. We recall a result by Crampin relating symmetries and conserved quantities in Liouville dynamics, and discuss how symmetry reduction is implemented in the frame of the variational principle for Liouville dynamics.

An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, two-component spinor calculus, conformal gravity, α-planes in Minkowski space-time, α-surfaces and twistor geometry, anti-self-dual space-times and Penrose transform, spin-3/2 potentials, heaven spaces and heavenly equations.