Estimates Of Eigenfunctions Research Articles

Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems

Let (M, g) be a compact Riemannian manifold of dimension n and P_1:=-h^2Delta _g+V(x)-E_1 so that dp_1ne 0 on p_1=0. We assume that P_1 is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators P_2,dots P_n with [P_i,P_j]=0, i,j=1,dots n. We study the pointwise bounds for the joint eigenfunctions, u_h of the system {P_i}_{i=1}^n with P_1u_h=E_1u_h+o(1). In Theorem 1, we first give polynomial improvements over the standard Hörmander bounds for typical points in M. In two and three dimensions, these estimates agree with the Hardy exponent h^{-frac{1-n}{4}} and in higher dimensions we obtain a gain of h^{frac{1}{2}} over the Hörmander bound. In our second main result (Theorem 3), under a real-analyticity assumption on the QCI system, we give exponential decay estimates for joint eigenfunctions at points outside the projection of invariant Lagrangian tori; that is at points xin M in the “microlocally forbidden” region p_1^{-1}(E_1)cap dots cap p_n^{-1}(E_n)cap T^*_xM=emptyset . These bounds are sharp locally near the projection of the invariant tori.

Eigenvibrations of an elastic bar with mechanical resonator

The differential eigenvalue problem governing eigenvibrations of an elastic bar with fixed first end and mechanical resonator attached to second end is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We introduce limit differential eigenvalue problems and derive the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as a resonator parameter tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform mesh. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for model problems. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached resonators.

Quadrature finite element method for the problem on eigenvibrations of a bar with elastic support

The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and with elastic support at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate a limit differential eigenvalue problem and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problem as stiffness coefficient tending to infinity. The original differential eigenvalue problem is approximated by the quadrature finite element method of arbitrary order on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with elastic support.

Finite difference approximation of eigenvibrations of a bar with oscillator

The second-order ordinary differential spectral problem governing eigenvibrations of a bar with attached harmonic oscillator is investigated. We study existence and properties of eigensolutions of formulated bar-oscillator spectral problem. The original second-order ordinary differential spectral problem is approximated by the finite difference mesh scheme. Theoretical error estimates for approximate eigenvalues and eigenfunctions of this mesh scheme are established. Obtained theoretical results are illustrated by computations for a model problem with constant coefficients. Theoretical and experimental results of this paper can be developed and generalized for the problems on eigenvibrations of complex mechanical constructions with systems of harmonic oscillators.

Eigenvibrations of a beam with two mechanical resonators attached to the ends

The fourth-order ordinary differential spectral problem describing vertical eigenvibrations of a beam with two mechanical resonators attached to the ends is studied. This problem has positive simple eigenvalues and corresponding eigenfunctions. We define limit differential spectral problem and establish the convergence of the eigenvalues and eigenfunctions of the original spectral problem to the eigenvalues and eigenfunctions of the limit spectral problem as parameters of the attached resonators tending to infinity. The initial fourth-order ordinary differential spectral problem is approximated by the finite difference method. Theoretical error estimates for approximate eigenvalues and eigenfunctions are derived. Obtained theoretical results are illustrated by computations for model problem with constant coefficients. Theoretical and experimental results of this paper can be developed for the problems on eigenvibrations of complex mechanical constructions with systems of resonators.

Reverse Agmon Estimates in Forbidden Regions

Let (M, g) be a compact, Riemannian manifold and $$V \in C^{\infty }(M; {{\mathbb {R}}})$$. Given a regular energy level $$E > \min V$$, we consider $$L^2$$-normalized eigenfunctions, $$u_h,$$ of the Schrodinger operator $$P(h) = - h^2 \Delta _g + V - E(h)$$ with $$P(h) u_h = 0$$ and $$E(h) = E + o(1)$$ as $$h \rightarrow 0^+.$$ The well-known Agmon–Lithner estimates [5] are exponential decay estimates (i.e. upper bounds) for eigenfunctions in the forbidden region $$\{ V>E \}.$$ The decay rate is given in terms of the Agmon distance function $$d_E$$ associated with the degenerate Agmon metric $$(V-E)_+ \, g$$ with support in the forbidden region. The point of this note is to prove a reverse Agmon estimate (i.e. exponential lower bound for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowed region $$\{ V< E \}$$ arbitrarily close to the caustic $$ \{ V = E \}.$$ We then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates.

Error of the Finite Element Approximation for a Differential Eigenvalue Problem with Nonlinear Dependence on the Spectral Parameter

The positive definite ordinary differential nonlinear eigenvalue problem of the second order with homogeneous Dirichlet boundary condition is considered. The problem is formulated as a symmetric variational eigenvalue problem with nonlinear dependence of the spectral parameter in a real infinite-dimensional Hilbert space. The variational eigenvalue problem consists in finding eigenvalues and corresponding eigenfunctions of the eigenvalue problem for a symmetric positive definite bounded bilinear form with respect to a symmetric positive definite completely continuous bilinear form in a real infinite-dimensional Hilbert space. The variational eigenvalue problem is approximated by the mesh scheme of the finite element method on the uniform grid. For constructing the mesh scheme, Lagrangian finite elements of arbitrary order are applied. Error estimates of approximate eigenvalues and error estimates of approximate eigenfunctions in the norm of initial real infinite-dimensional Hilbert space are established. These error estimates coincide in the order with error estimates of mesh scheme of the finite element method for linear eigenvalue problems. Moreover, superconvergence estimates for approximate eigenfunctions in the mesh norm with Gauss quadrature nodes are derived. Investigations of this paper generalize well known results for the eigenvalue problem with linear entrance on the spectral parameter.

A multigrid correction scheme for a new Steklov eigenvalue problem in inverse scattering

ABSTRACTWe propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems in fine finite element spaces and a series of eigenvalue problems in the coarsest finite element space. And the coefficient matrices associated with those linear systems are constructed to be symmetric and positive definite. We prove error estimates of eigenvalues and eigenfunctions. Numerical results coincide in theoretical analysis and indicate our scheme is highly efficient in solving the eigenvalue problem.

Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequalities and restricted Sogge-type $L_p$ moment estimates of eigenfunctions.

Error estimates of the quadrature finite element method with biquadratic finite elements for elliptic eigenvalue problems in the square domain

A positive definite eigenvalue problem for second-order self-adjoint elliptic differential operator defined on a square domain in the plane with homogeneous Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange biquadratic finite elements. Error estimates for approximate eigenvalues and eigenfunctions are established.

Error estimates of the finite difference method for eigenvalue problems with nonlinear entrance of the spectral parameter

A positive definite second order ordinary differential eigenvalue problem with coefficients nonlinearly depending on the spectral parameter is studied. This differential nonlinear eigenvalue problem has an increasing sequence of positive simple eigenvalues, which correspond to a normalized system of eigenfunctions. The original differential nonlinear eigenvalue problem is approximated by a mesh scheme of the finite difference method on the uniform grid. New error estimates for approximate eigenvalues and approximate eigenfunctions in dependence on mesh size and eigenvalue size are established. Obtained theoretical results are generalizations of well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.

Investigation of the eigenvalue problem on eigenvibration of a loaded string

In the present paper, the differential eigenvalue problem describing eigenvibrations of a string with fixed ends and attached load at an interior point fixed elastically is investigated. The problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

Approximation of the eigenvalue problem on eigenvibration of a loaded bar

The present paper deals with the investigation of a differential eigenvalue problem describing eigenvibrations of an elastic bar with load rigidly joined to the bar end fixed elastically. The problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

An efficient spectral Galerkin approximation to a Helmholtz transmission eigenvalue problem in spherical geometries

In this paper, we present an efficient spectral method based on Legendre-Galerkin approximation for the Helmholtz transmission eigenvalue problem in spherical geometries. By means of the spherical coordinate transformation and spherical harmonic expansion, we decompose the original problem into a sequence of equivalent one-dimensional generalized eigenvalue problems. Then we establish corresponding weak forms and discrete schemes for each one-dimensional generalized eigenvalue problem. Especially for the case of solid spherical domain, we need to deal with singularities introduced by spherical coordinate transformation. In order to overcome the difficulty, we derive the pole conditions and introduce the weighted Sobolev space according to pole conditions. The corresponding weak forms and discrete schemes are also established according to the weighted Sobolev space. In addition, we prove the error estimates of eigenvalues and eigenfunctions by using the spectral theory of compact operators for each one-dimensional generalized eigenvalue problems.We also provide some numerical experiments to demonstrate the validity of the algorithm and the correctness of the theoretical results.

Error Investigation of a Finite Element Approximation for a Nonlinear Sturm–Liouville Problem

A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter is studied. The problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms depending nonlinearly on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues that correspond to a normalized system of eigenfunctions. The variational problem is approximated by a finite element mesh scheme on a uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are proved depending on the mesh size and the eigenvalue size. The results obtained are generalizations of well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.

$$L^p$$ L p Bilinear Quasimode Estimates

In this paper, we investigate the $$L^p$$ bilinear quasimode estimates on compact Riemannian manifolds. We obtain results in the full range $$p\ge 2$$ on all n-dimensional manifolds with $$n\ge 2$$ . This in particular implies the $$L^p$$ bilinear eigenfunction estimates. We further show that all of these estimates are sharp by constructing various quasimodes and eigenfunctions that saturate our estimates.

Inner radius of nodal domains of quantum ergodic eigenfunctions

In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results (arXiv:1606.02057), which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by the results of Han and Hezari-Riviere small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get logarithmic improvements on the inner radius of such manifolds. We also get improvements for manifolds with ergodic geodesic flows. In addition using the small scale equidistribution results of Lester-Rudnick, one gets polynomial betterments of for toral eigenfunctions in dimensions $n \geq 3$. The results work only for a full density subsequence of eigenfunctions.

Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions

We use Toponogov’s triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya–Nikodym norms introduced in [22] for manifolds of nonpositive sectional curvature. Using these and results from our paper [4] we are able to obtain log-improvements of $L^p (M)$ estimates for such manifolds when $2 \lt p \lt \frac{2(n+1)}{n-1}$. These in turn imply $(\log \lambda)^{\sigma_n} , \sigma_n \approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of the estimates of the second author and Zelditch [28], and using a result from Hezari and the second author [18], under this curvature assumption, we are able to improve the lower bounds for the size of nodal sets of Colding and Minicozzi [12] by a factor of $(\log \lambda)^{\mu}$ for any $\mu \lt \frac{2(n+1)^2}{n-1}$, if $n \geq 3$.

A note on constructing families of sharp examples for 𝐿^{𝑝} growth of eigenfunctions and quasimodes

In this note we analyse L p L^{p} estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular, we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition, we produce a family of flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check L p L^{p} growth estimates against the simple flat model examples. Finally, we present a heuristic that for any subset determines which quasimode in the family is expected to produce sharp examples.

Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of fractional Laplace operators with respect to domain perturbations.