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The non-crossing rule for the energy levels of a parameter-dependent Hamiltonian is revisited and a flaw in a commonly accepted proof is revealed. Some aspects of avoided crossings are illustrated by means of simple models. One of them…
Due to the absence of degeneracy in one dimension, when a parameter, $\lambda$, of a potential is varied slowly the discrete energy eigenvalue curves, $E_n(\lambda)$, cannot cross but they are allowed to come quite close and diverge from…
For bounded domains, eigenvalues and eigenfunctions of double layer potentials are considered. The aim of this paper is to establish some relationships between eigenvalues, eigenfunctions and the geometry of domain boundaries.
Avoided level crossings are associated with exceptional points which are the singularities of the spectrum and eigenfunctions, when they are considered as functions of a coupling parameter. It is shown that the wave function of {\it one}…
At high level density, two states avoid usually crossing at the critical value $a_{\rm cr}$ of the parameter $a$ by which the system is controlled. The wavefunctions of the two states are mixed in a finite parameter range around $a_{\rm…
In order to better understand the occurrence of phase transitions, we adopt an approach based on the study of energy landscapes: The relation between stationary points of the potential energy landscape of a classical many-particle system…
We study quantum localization phenomena in chaotic systems with a parameter. The parametric motion of energy levels proceeds without crossing any other and the defined avoided crossings quantify the interaction between states. We propose…
The effect of non-sphericity of the quantum dot on the eigenvalues and eigenfunctions has been investigated for the case of both the finite and infinite barrier. The ground and excited state energies have been calculated for prolate and…
We analyze the energetics of domains and interfaces in the presence of quenched random fields, particularly at the lower critical dimension of the random field Ising model. The relevance of this study to experiments is also discussed.
The eigenvalue problem for the dressed bound-state of unstable multilevel systems is examined both outside and inside the continuum, based on the N-level Friedrichs model which describes the couplings between the discrete levels and the…
We study the energy levels of carriers confined in a magnetic quantum dot of graphene surrounded by a infinite graphene sheet in the presence of energy gap. The eigenspinors are derived for the valleys $K$ and $K'$, while the associated…
We discuss adiabatic quantum phenomena at a level crossing. Given a path in the parameter space which passes through a degeneracy point, we find a criterion which determines whether the adiabaticity condition can be satisfied. For paths…
Unavoided crossings of the energy levels due to a variation of a real parameter are studied. It is found that after the quantum system in question passes through one of its energy-crossing points {\it alias} Kato's exceptional points (EP),…
In this paper, we define and obtain several properties of the (adjacency) energy of a hypergraph. In particular, bounds for this energy are obtained as functions of structural and spectral parameters, such as Zagreb index and spectral…
We contemplate the pair of the purely imaginary delta-function potentials on a finite interval with Dirichlet boundary conditions. The two parameter model exhibits nicely the expected quantitative features of the unavoided level crossing…
While the phenomenon of the exact crossing of energy levels is a rarely occurring event, in the case of quantum resonances associated with metastable states this phenomenon is much more frequent and various scenarios can occur. When there…
We investigate active electrolytes within the mean-field level of description. The focus is on how the double-layer structure of passive, thermalized charges is affected by active dynamics of all constituting ions. One feature of active…
We investigate the in-plane oscillations of the relaxed shape graphene due to externally applied tensile edge stress along both the armchair and zigzag directions. We show that the total elastic energy density is enhanced with temperature…
I present a review on non relativistic effective energy--density functionals (EDFs). An introductory part is dedicated to traditional phenomenological functionals employed for mean--field--type applications and to several extensions and…
We consider an energy model for harmonic graphs with junctions and study the regularity properties of minimizers and their free boundaries.