Generalized spiked harmonic oscillator

A variational and perturbative treatment is provided for a family of generalized spiked harmonic oscillator Hamiltonians H = -(d/dx)^2 + B x^2 + A/x^2 + lambda/x^alpha, where B > 0, A >= 0, and alpha and lambda denote two real positive parameters. The method makes use of the function space spanned by the solutions |n> of Schroedinger's equation for the potential V(x)= B x^2 + A/x^2. Compact closed-form expressions are obtained for the matrix elements , and a first-order perturbation series is derived for the wave function. The results are given in terms of generalized hypergeometric functions. It is proved that the series for the wave function is absolutely convergent for alpha <= 2.Comment: 14 page

A Colonel Blotto Game for Interdependence-Aware Cyber-Physical Systems Security in Smart Cities

Smart cities must integrate a number of interdependent cyber-physical systems that operate in a coordinated manner to improve the well-being of the city's residents. A cyber-physical system (CPS) is a system of computational elements controlling physical entities. Large-scale CPSs are more vulnerable to attacks due to the cyber-physical interdependencies that can lead to cascading failures which can have a significant detrimental effect on a city. In this paper, a novel approach is proposed for analyzing the problem of allocating security resources, such as firewalls and anti-malware, over the various cyber components of an interdependent CPS to protect the system against imminent attacks. The problem is formulated as a Colonel Blotto game in which the attacker seeks to allocate its resources to compromise the CPS, while the defender chooses how to distribute its resources to defend against potential attacks. To evaluate the effects of defense and attack, various CPS factors are considered including human-CPS interactions as well as physical and topological characteristics of a CPS such as flow and capacity of interconnections and minimum path algorithms. Results show that, for the case in which the attacker is not aware of the CPS interdependencies, the defender can have a higher payoff, compared to the case in which the attacker has complete information. The results also show that, in the case of more symmetric nodes, due to interdependencies, the defender achieves its highest payoff at the equilibrium compared to the case with independent, asymmetric nodes

Integrating Energy Storage into the Smart Grid: A Prospect Theoretic Approach

In this paper, the interactions and energy exchange decisions of a number of geographically distributed storage units are studied under decision-making involving end-users. In particular, a noncooperative game is formulated between customer-owned storage units where each storage unit's owner can decide on whether to charge or discharge energy with a given probability so as to maximize a utility that reflects the tradeoff between the monetary transactions from charging/discharging and the penalty from power regulation. Unlike existing game-theoretic works which assume that players make their decisions rationally and objectively, we use the new framework of prospect theory (PT) to explicitly incorporate the users' subjective perceptions of their expected utilities. For the two-player game, we show the existence of a proper mixed Nash equilibrium for both the standard game-theoretic case and the case with PT considerations. Simulation results show that incorporating user behavior via PT reveals several important insights into load management as well as economics of energy storage usage. For instance, the results show that deviations from conventional game theory, as predicted by PT, can lead to undesirable grid loads and revenues thus requiring the power company to revisit its pricing schemes and the customers to reassess their energy storage usage choices.Comment: 5 pages, 4 figures, conferenc

Casimir interactions of an object inside a spherical metal shell

We investigate the electromagnetic Casimir interactions of an object contained within an otherwise empty, perfectly conducting spherical shell. For a small object we present analytical calculations of the force, which is directed away from the center of the cavity, and the torque, which tends to align the object opposite to the preferred alignment outside the cavity. For a perfectly conducting sphere as the interior object, we compute the corrections to the proximity force approximation (PFA) numerically. In both cases the results for the interior configuration match smoothly onto those for the corresponding exterior configuration.Comment: 4 pages, 3 figure

Casimir potential of a compact object enclosed by a spherical cavity

We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to the Casimir-Polder result. Although it is not a simple power law, the dependence of the force on the separation of the object from the containing sphere is a universal function of its displacement from the center of the sphere, independent of other details of the object's electromagnetic response. Furthermore, we compute the exact Casimir force between two metallic spheres contained one inside the other at arbitrary separations. Finally, we combine our results with earlier work on the Casimir force between two spheres to obtain data on the leading order correction to the Proximity Force Approximation for two metallic spheres both outside and within one another.Comment: 12 pages, 6 figure