In this preliminary Brief Report we explore a method for examining relativistic effects in one-electron molecules that avoids the evaluation of multicenter integrals. Here the Dirac equation for a simplified version of the ${{\mathrm{H}}_{2}}^{+}$ ion is solved exactly for the ground-state Schr\"odinger and Dirac energy levels. The model consists of representing the electron-proton Coulomb interaction as a projection operator onto a linear subspace of Hilbert space.In this preliminary Brief Report we explore a method for examining relativistic effects in one-electron molecules that avoids the evaluation of multicenter integrals. Here the Dirac equation for a simplified version of the ${{\mathrm{H}}_{2}}^{+}$ ion is solved exactly for the ground-state Schr\"odinger and Dirac energy levels. The model consists of representing the electron-proton Coulomb interaction as a projection operator onto a linear subspace of Hilbert space.

Physical Review AMay 13, 20141 citation

Hilbert Spaces and Bounded Operators

Valter Moretti

UnitextJan 1, 20170 citations

Eigenstate thermalization in ( 2 + 1 )-dimensional SU(2) lattice gauge theory

Lukas Ebner, Andreas Schäfer A, Clemens Seidl +2 more

We present preliminary numerical evidence for the hypothesis that the Hamiltonian SU(2) gauge theory discretized on a lattice obeys the eigenstate thermalization hypothesis (ETH). To do so we study three approximations: (a) a linear plaquette chain in a reduced Hilbert space limiting the electric field basis to j = 0 , 1 2 , (b) a two-dimensional honeycomb lattice with periodic or closed boundary condition and the same Hilbert space constraint, and (c) a chain of only three plaquettes but such a sufficiently large electric field Hilbert space ( j ≤ 7 2 ) that convergence of all energy eigenvalues in the analyzed energy window is observed. While an unconstrained Hilbert space is required to reach the continuum limit of SU(2) gauge theory, numerical resource constraints do not permit us to realize this requirement for all values of the coupling constant and large lattices. In each of the three studied cases we check first for random matrix theory (RMT) behavior in the eigenenergy spectrum and then analyze the diagonal as well as the off-diagonal matrix elements between energy eigenstates for a few operators. Within current uncertainties all results for (a), (b) and (c) agree with ETH predictions. Furthermore, we find the off-diagonal matrix elements of the electric energy operator exhibit RMT behavior in frequency windows that are small enough in (b) and (c). To unambiguously establish ETH behavior and determine for which class of operators it applies, an extension of our investigations is necessary. Published by the American Physical Society 2024We present preliminary numerical evidence for the hypothesis that the Hamiltonian SU(2) gauge theory discretized on a lattice obeys the eigenstate thermalization hypothesis (ETH). To do so we study three approximations: (a) a linear plaquette chain in a reduced Hilbert space limiting the electric field basis to j = 0 , 1 2 , (b) a two-dimensional honeycomb lattice with periodic or closed boundary condition and the same Hilbert space constraint, and (c) a chain of only three plaquettes but such a sufficiently large electric field Hilbert space ( j ≤ 7 2 ) that convergence of all energy eigenvalues in the analyzed energy window is observed. While an unconstrained Hilbert space is required to reach the continuum limit of SU(2) gauge theory, numerical resource constraints do not permit us to realize this requirement for all values of the coupling constant and large lattices. In each of the three studied cases we check first for random matrix theory (RMT) behavior in the eigenenergy spectrum and then analyze the diagonal as well as the off-diagonal matrix elements between energy eigenstates for a few operators. Within current uncertainties all results for (a), (b) and (c) agree with ETH predictions. Furthermore, we find the off-diagonal matrix elements of the electric energy operator exhibit RMT behavior in frequency windows that are small enough in (b) and (c). To unambiguously establish ETH behavior and determine for which class of operators it applies, an extension of our investigations is necessary. Published by the American Physical Society 2024

arXiv:2308.16202Physical review. D/Physical review. D.Jan 9, 202427 citations

Hilbert spaces and bounded operators

Valter Moretti

UnitextJan 1, 20130 citations

{{\mathrm{H}}_{2}}^{+}

ion is solved exactly for the ground-state Schr\"odinger and Dirac energy levels. The model consists of representing the electron-proton Coulomb interaction as a projection operator onto a linear subspace of Hilbert space.