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For Dirac operators, which have discrete spectra, the concept of eigenvalues gradient is given and formulae for this gradients are obtained in terms of normalized eigenfunctions. It is shown how the gradient is being used to describe…
The similarity renormalization group is used to transform Dirac Hamiltonian into a diagonal form, which the upper (lower) diagonal element becomes an operator describing Dirac (anti-)particle. The eigenvalues of the operator are verfied to…
In this chapter we focus first on the theoretical methods and relevant computational approaches to calculate the electronic structure of atoms, molecules, and clusters containing heavy elements for which relativistic effects become…
We study the surface plasmon modes of an arbitrarily shaped nanoparticle in the electrostatic limit. We first deduce an eigenvalue equation for these modes, expressed in terms of the Dirichlet-Neumann operators. We then use the properties…
The aim of the present work is to derive a error estimates for the Laplace eigenvalue problem in mixed form, by means of a virtual element method. With the aid of the theory for non-compact operators, we prove that the proposed method is…
Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double…
In the standard treatment of particle oscillations the mass eigenstates are implicitly assumed to be scalars and, consequently, the spinorial form of neutrino wave functions is not included in the calculations. To analyze this additional…
The main issues of the spectral theory of Dirac operators are presented, namely: transformation operators, asymptotics of eigenvalues and eigenfunctions, description of symmetric and self-adjoint operators in Hilbert space, expansion in…
We open out one of incorrect solutions of the Driac equation in the Coulomb field given in a published paper. By introducing a transformation of function, the paper transformed the original radial first-order Dirac-Coulomb equation into two…
Perturbation theory is used systematically to investigate the symmetries of the Dirac Hamiltonian and their breaking in atomic nuclei. Using the perturbation corrections to the single-particle energies and wave functions, the link between…
The use of complexified quaternions and $i$-complex geometry in formulating the Dirac equation allows us to give interesting geometric interpretations hidden in the conventional matrix-based approach.
The Dirac eigenvalues form a subset of observables of the Euclidean gravity. The symplectic two-form in the covariant phase space could be expressed, in principle, in terms of the Dirac eigenvalues. We discuss the existence of the formal…
We introduce the confluent version of the quantum-mechanical supersymmetry (SUSY) formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the…
A procedure to calculate the radiation spectrum emitted by an arbitrarily prepared Dirac wave packet is developed. It is based on the Dirac charge current and classical electrodynamic theory. Apart from giving absolute intensity values, it…
We uncover a remarkable quantum scattering phenomenon in two-dimensional Dirac material systems where the manifestations of both classically integrable and chaotic dynamics emerge simultaneously and are electrically controllable. The…
As a continuation of previous investigations, the formalism used there is extended to the case when an external electric field is present and the covariant formulation is performed again. The equation system obtained allows no restriction…
Covergent eigensolutions of the Dirac Equation for a relativistic electron in an external Coulomb potential are obtained using the Lanczos Algorithm. A tri-diagonal matrix representation of the Dirac Hamiltonian operator is constructed…
Spurious velocities are unphysical currents that appear close to curved interfaces in diffuse interface methods. We analyse the causes of these spurious velocities in the free energy lattice Boltzmann algorithm. By making a suitable choice…
Scattering resonances arise in wave phenomena and play an important role in many applications. While extensive theoretical studies have been conducted, effective numerical computation remains limited, and most existing methods suffer from…
We show that a natural explanation for characteristic features (angular momentum and radial quantum numbers) of pseudospin doublets and intruder levels in nuclei can be obtained by combining the relativistic attributes of pseudospin…