Related papers: Lower bounds to eigenvalues of one-electron Hamilt…
The computational cost of quantum algorithms for physics and chemistry is closely linked to the spectrum of the Hamiltonian, a property that manifests in the necessary rescaling of its eigenvalues. The typical approach of using the 1-norm…
A method is presented to compute approximate solutions for eigenequations in quantum mechanics with an arbitrary kinetic part. In some cases, the approximate eigenvalues can be analytically determined and they can be lower or upper bounds.…
When an eigenvector of a semi-bounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-like…
In this paper we present results of the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids…
Measuring the expectation value of the molecular electronic Hamiltonian is one of the challenging parts of the variational quantum eigensolver. A widely used strategy is to express the Hamiltonian as a sum of measurable fragments using…
We apply classical algorithms for approximately solving constraint satisfaction problems to find bounds on extremal eigenvalues of local Hamiltonians. We consider spin Hamiltonians for which we have an upper bound on the number of terms in…
This paper proposes hybrid high-order eigensolvers for the computation of guaranteed lower eigenvalue bounds. These bounds display higher order convergence rates and are accessible to adaptive mesh-refining algorithms. The involved…
We provide two new methods for computing lower bounds of eigenvalues of symmetric elliptic second-order differential operators with mixed boundary conditions of Dirichlet, Neumann, and Robin type. The methods generalize ideas of Weinstein's…
We propose a method for obtaining rigorous and accurate upper and lower bounds on the eigenvalues of ordinary and partial differential operators in bounded regions of Euclidean space. It uses a boundary condition homotopy method starting…
Methods for calculating lower bounds to the exact energy using the variance of the upper bound energy are discussed and explored. All the matrix elements of the Hamiltonian squared are collected and considered, and those for which no known…
We establish Lieb-Thirring type inequalities for non self-adjoint relatively compact perturbations of certain operators of mathematical physics. We apply our results to quantum Hamiltonians of Schr{\"o}dinger and Pauli with constant…
The eigenfunctions of electronic Hamiltonians determine the stable structures and dynamics of molecules through the local distributions of their densities. In this paper an a priori upper bound for such local distributions of the densities…
An approximate relativistic two-component Hamiltonian for use in molecular electronic structure calculations is derived in the form of a sum of fixed atom-centered kinetic and spin-orbit operators added to the non-relativistic Hamiltonian.…
It is shown how mixed finite element methods for symmetric positive definite eigenvalue problems related to partial differential operators can provide guaranteed lower eigenvalue bounds. The method is based on a classical compatibility…
For bound states of atoms and molecules of $N$ electrons we consider the corresponding $K$-particle reduced density matrices, $\Gamma^{(K)}$, for $1 \le K \le N-1$. Previously, eigenvalue bounds were obtained in the case of $K=1$ and…
We present a new method for calculating electronic states in low-dimensional semiconductor heterostructures, which is based on the real-space Hamiltonian in the envelope function approximation. The numerical implementation of the method is…
Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &\Delta {\mathbf u}+ \alpha{\rm grad}(\text{div}{\mathbf…
By application of a straightforward variational procedure we derive a simple, analytic upper bound on the ground-state energy eigenvalue of a semirelativistic Hamiltonian for (one or two) spinless particles which experience some…
Several techniques for deriving semianalytical bounds on the energy eigenvalues of the spinless Salpeter equation and for estimating the quality of the corresponding approximate eigenstates are reviewed.
If a single particle obeys non-relativistic QM in R^d and has the Hamiltonian H = - Delta + f(r), where f(r)=sum_{i = 1}^{k}a_ir^{q_i}, 2\leq q_i < q_{i+1}, a_i \geq 0$, then the eigenvalues E = E_{n\ell}^{(d)}(\lambda) are given…