Related papers: A note on eigenvalues estimates for one-dimensiona…
We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of…
We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the…
A method for estimating the spectral gap along with higher eigenvalues of nonequilateral quantum graphs has been introduced by Amini and Cohen-Steiner recently: it is based on a new transference principle between discrete and continuous…
In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\ge 2$ with constant real coefficients. Under suitable growth…
Based on the recent work \cite{KKK} for compact potentials, we develop the spectral theory for the one-dimensional discrete Schr\"odinger operator $$ H \phi = (-\De + V)\phi=-(\phi_{n+1} + \phi_{n-1} - 2 \phi_n) + V_n \phi_n. $$ We show…
In this paper we discuss spectral properties of operators associated with the least-squares finite element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the…
We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. Using random walks and exit times, we associate to certain subgraphs (domains) a pair of sequences, each of which…
We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schr\"odinger operator with convex potential and Dirichlet boundary data on a convex domain is…
In this paper, we give upper estimates for the number and sum of eigenvalues below the bottom of the essential spectrum counting multiplicities of quantum waveguides in two dimensions. We consider both straight and curved waveguides of…
In the present work we show that the joint probability distribution of the eigenvalues can be expressed in terms of a differential operator acting on the distribution of some other matrix quantities. Those quantities might be the diagonal…
In this paper, we investigate an eigenvalue problem associated with an age-structured operator incorporating random diffusion and advection. Our primary focus is on examining the asymptotic behaviors of the principal eigenvalue with respect…
In this study, singular diffusion operator with jump conditions is considered. Integral representations have been derived for solutions that satisfy boundary conditions and jump conditions. Some properties of eigenvalues and eigenfunctions…
Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)_n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered…
We present an efficient method to compute the eigenvalues and eigenmodes of the diffusion operator $\nabla(D\nabla)$ on one-dimensional heterogeneous structures with multiple semi-permeable barriers. This method allows us to calculate the…
We give a lower estimate of the gap of the first two eigenvalues of the Schrodinger operator in the case when the potential is strongly convex. In particular, if the Hessian of the potential is bounded from below by a positive constant, the…
The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where $B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with the Dirichlet boundary condition at…
We show that the interior transmission eigenvalues are discrete by proving that the interior transmission operator has upper triangular compact resolvent, and that the spectrum of these operators share many of the properties of operators…
We prove a unique continuation principle or uncertainty relation valid for Schr\"odinger operator eigenfunctions, or more generally solutions of a Schr\"odinger inequality, on cubes of side $L\in 2\NN+1$. It establishes an equi-distribution…
We study the spectra of non-selfadjoint first-order operators on the interval with non-local point interactions, formally given by ${i\partial_x+V+k\langle \delta,\cdot\rangle}$. We give precise estimates on the location of the eigenvalues…
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…