Abstract:
For a class of singular potentials, including the Coulomb potential (in three and less dimensions) and $V(x) = g/x^2$ with the coefficient $g$ in a certain range ($x$ being a space coordinate in one or more dimensions), the corresponding Schr dinger operator is not automatically self-adjoint on its natural domain. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The article discusses how the self-adjoint domains can be identified in terms of a boundary condition for the asymptotic behaviour of the wave functions around the singularity, and what physical differences emerge for different self-adjoint versions of the Hamiltonian. The paper reviews and interprets known results, with the intention to provide a practical guide for all those interested in how to approach these ambiguous situations.

Abstract:
For a class of singular potentials, including the Coulomb potential (in three and less dimensions) and $V(x) = g/x^2$ with the coefficient $g$ in a certain range ($x$ being a space coordinate in one or more dimensions), the corresponding Schr\"odinger operator is not automatically self-adjoint on its natural domain. Such operators admit more than one self-adjoint domain, and the spectrum and all physical consequences depend seriously on the self-adjoint version chosen. The article discusses how the self-adjoint domains can be identified in terms of a boundary condition for the asymptotic behaviour of the wave functions around the singularity, and what physical differences emerge for different self-adjoint versions of the Hamiltonian. The paper reviews and interprets known results, with the intention to provide a practical guide for all those interested in how to approach these ambiguous situations.

Abstract:
We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the canonical energy-momentum tensor and the energy functional defined as the expectation value of the corresponding nonlinear Hamiltonian operator. Some examples of such ambiguity are presented for a particularly simple model and some known modifications. However, we point out that there exist a class of nonlinear modifications of the Schr\"{o}dinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schr\"{o}dinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schr\"{o}dinger equation or its underlying Lagrangian density which is assumed to be ``bilinear'' in the wave function in some rather general sense. Yet, it is only for a particular form of this density that the ambiguity in question does not arise. A salient feature of this form is the presence of phase functionals. The present paper thus introduces a new class of modifications characterized by this desirable and rare property.

Abstract:
We propose a variant formulation of Hamiltonian systems by the use of variables including redundant degrees of freedom. We show that Hamiltonian systems can be described by extended dynamics whose master equation is the Nambu equation or its generalization. Partition functions associated with the extended dynamics in many degrees of freedom systems are given. Our formulation can also be applied to Hamiltonian systems with first class constraints.

Abstract:
This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by A.M. Gleason to prove his celebrated theorem. In particular, the problem of associating quantum state with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical observables representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical objects representing quantum ones result to be frame functions. The requirements of $U(n)$ covariance and (convex) linearity play a central r\^ole in the proof of those theorems. A new characterization of classical observables describing quantum observables is presented, together with a geometric description of the $C^*$-algebra structure of the set of quantum observables in terms of classical ones.

Abstract:
An investigation of classical fields with fractional derivatives is presented using the fractional Hamiltonian formulation. The fractional Hamilton's equations are obtained for two classical field examples. The formulation presented and the resulting equations are very similar to those appearing in classical field theory.

Abstract:
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schr\"odinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features---such as uncertainties and state vector reductions---which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric---a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.

Abstract:
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory defined on a symplectic geometry. However, after invention of general relativity, we are convinced that geometry is physical and affect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observable are real-valued functions on the manifold, and quantum evolution is governed by the symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.

Abstract:
We derive the Hamiltonian formulation of classical mechanics directly, without reference to Lagrangian mechanics. We start from the definition of states in terms of labels used to identify them, and show how, under a deterministic and reversible process, the conservation of the cardinality of the labels leads to Hamilton's equations.