The Resonance Phenomena Associated with the Time Asymmetry in Non-Hermitian Quantum Mechanics

Resonances are defined as the poles of the scattering matrix. The poles are associated with the complex eigenvalues of the Hamiltonian which are embedded in the lower half of the complex plane. The asymptotes of the corresponding eigenfunctions are exponentially diverged. Therefore, the resonance eigenfunctions are not embedded in the Hermitian domain of the Hamiltonian. The time asymmetric problem is discussed for these types of non-Hermitian Hamiltonians and several solutions of this problem are proposed.     

The kernel and the nucleolus of a product of simple games

The kernel and the nucleolus of a product of two simple games are given in terms of the kernels and the nucleoluses of the component games.     

Finding mixed strategies with small supports in extensive form games

The complexity of algorithms that compute strategies or operate on them typically depends on the representation length of the strategies involved. One measure for thesize of a mixed strategy is the number of strategies in itssupport — the set of pure strategies to which it gives positive probability. This paper investigates the existence of “small” mixed strategies in extensive form games, and how such strategies can be used to create more efficient algorithms. The basic idea is that, in an extensive form game, a mixed strategy induces a small set ofrealization weights that completely describe its observable behavior. This fact can be used to show that for any mixed strategy μ, there exists a realization-equivalent mixed strategy µ′ whose size is at most the size of the game tree. For a player with imperfect recall, the problem of finding such a strategy µ′ (given the realization weights) is NP-hard. On the other hand, if μ is a behavior strategy, µ′ can be constructed from μ in time polynomial in the size of the game tree. In either case, we can use the fact that mixed strategies need never be too large for constructing efficient algorithms that search for equilibria. In particular, we construct the first exponential-time algorithm for finding all equilibria of an arbitrary two-person game in extensive form.     

Study of predissociation resonances by the complex coordinate method

The complex coordinate method is applied to the predissociation resonances of two-state model system. The stability of the resonance positions and their widths, as well as the affect of using real and complex basis sets are studied.     

Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality

The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NP-complete, even when all variables are binary. This paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality, and with all variables bounded between 0 andU. For such systems, a 2-approximation algorithm is presented that runs in time O(mnU ² log(Un ² m)), so it is polynomial in the input size if the upper boundU is polynomially bounded. The algorithm works by finding first a super-optimal feasible solution that consists of integer multiples of 1/2. That solution gives a tight bound on the value of the minimum. It furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem. These properties are a generalization of the properties of the vertex cover problem. The algorithm described is, in particular, a 2-approximation algorithm for the problem of minimizing the total weight of true variables, among all truth assignments to the 2-satisfiability problem.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Research supported in part by ONR contracts N00014-88-K-0377 and N00014-91-J-1241.Research supported in part by ONR contract N00014-91-C-0026.Part of this work was done while the author was visiting the International Computer Science Institute in Berkeley, CA and DIMACS, Rutgers University, New Brunswick, NJ.     

Amplified analysis of DNA by the autonomous assembly of polymers consisting of DNAzyme wires

A systematic study of the amplified optical detection of DNA by Mg(2+)-dependent DNAzyme subunits is described. The use of two DNAzyme subunits and the respective fluorophore/quencher-modified substrate allows the detection of the target DNA with a sensitivity corresponding to 1 × 10(-9) M. The use of two functional hairpin structures that include the DNAzyme subunits in a caged, inactive configuration leads, in the presence of the target DNA, to the opening of one of the hairpins and to the activation of an autonomous cross-opening process of the two hairpins, which affords polymer DNA wires consisting of the Mg(2+)-dependent DNAzyme subunits. This amplification paradigm leads to the analysis of the target DNA with a sensitivity corresponding to 1 × 10(-14) M. The amplification mixture composed of the two hairpins can be implemented as a versatile sensing platform for analyzing any gene in the presence of the appropriate hairpin probe. This is exemplified with the detection of the BRCA1 oncogene.     

Visualization of branch points in PT-symmetric waveguides

The visualization of an exceptional point in a PT-symmetric directional coupler (DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schr?dinger perturbation theory we prove that the spectrum of a PT-symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT-symmetric directional coupler to measure the radius of convergence for non-PT-symmetric structures. For such systems the physical meaning of the rather mathematical term radius of convergence is exemplified.     

Electromagnetic realization of orders-of-magnitude tunneling enhancement in a double well system

We report on tunneling enhancement in a periodically perturbed double well system. The double well system was realized by a structure of two optical waveguides. The transfer of light power from one waveguide to the another as induced by the periodic variations of the waveguide geometry was investigated. Our experimental measurements show that, in the presence of periodic perturbation, this transfer of light power can be enhanced by more than 500 times. We use an analogy between electromagnetic wave optics and the quantum wave phenomena to provide an experimental support to the theoretical model of tunneling enhancement of a quantum particle, facilitated by its interaction with auxiliary quantum states.     

Reflection-free complex absorbing potential for electronic structure calculations: Feshbach-type autoionization resonances of molecules

The reflection-free complex absorbing potential (RF-CAP) method has been already applied to the study of the autoionization resonance of helium [Sajeev et al., Chem. Phys. 329, 307 (2006)]. The present work introduces a systematic way for implementing RF-CAP for the electronic structure calculations using Gaussian basis sets for molecules. As a test case study we applied the RF-CAP method to the lowest (1)Sigma(g) (+) and (1)Sigma(u) (+) Feshbach-type autoionization resonances of hydrogen molecule. Since thin RF-CAP absorbs fast electrons much better than the slow ones, a weak dc field has been added to the RF-CAP in the peripheral region of the molecule.