Related papers: Eigenvalue Bounds for Double Saddle-Point Systems
We obtain eigenvalues and eigenfunctions of the Schr\"{o}dinger equation with a hyperbolic double-well potential. We consider exact polynomial solutions for some particular values of the potential-strength parameter and also numerical…
Given a Schr\"odinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary…
In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $\delta$- and $\delta'$-interactions are studied. We show the self-adjointness of these…
The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963] for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to N-dimensions. In particular a simple formula is derived which bounds the…
We establish the Schauder estimates at the boundary away from the characteristic points for the Dirichlet problem by means of the double layer potential in a Heisenberg-type group $\mathbb{G}$. Despite its singularity we manage to invert…
For elliptic systems with block structure in the upper half-space and t-independent coefficients, we settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal…
We obtain sharp uniform bounds on the low lying eigenfunctions for a class of semiclassical pseudodifferential operators with double characteristics and complex valued symbols, under the assumption that the quadratic approximations along…
We bring in some new notions associated with $2\times 2$ block positive semidefinite matrices. These notions concern the inequalities between the singular values of the off diagonal blocks and the eigenvalues of the arithmetic mean or…
Estimation of the degree of stability and the bounds of solutions to non-autonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques are frequently yield overconservative conditions which are…
This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is…
In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition…
We give upper and lower bounds on the largest singular value of a matrix using analogues to walks in graphs. For nonnegative matrices these bounds are asymptotically tight. In particular, we improve a bound due to I. Schur.
Design matrices are sparse matrices in which the supports of different columns intersect in a few positions. Such matrices come up naturally when studying problems involving point sets with many collinear triples. In this work we consider…
Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically…
For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple…
We obtain a tail bound for the least non-zero singular value of $A-z$ when $A$ is a random matrix and $z$ is an eigenvalue of $A$ in a neighbourhood of a given point $z_0$ in the bulk of the spectrum. The argument relies on a resolvent…
We consider the large-sparse symmetric linear systems of equations that arise in the solution of weak constraint four-dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can…
Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or…
We show how $\ell$-ifications, which are companion forms of matrix polynomials, namely, lower order matrix polynomials with the same eigenvalues as a given complex square matrix polynomial, can be used in combination with other recent…
Unconditional guaranteed lower and upper eigenvalue bounds are mandatory for the understanding of the Schr\"odinger eigenvalue spectrum and its spectral gaps. While upper eigenvalue bounds are naturally induced by conforming…