Estimates Of Eigenfunctions Research Articles

A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem

This paper is devoted to a-priori and a-posteriori error analysis of discontinuous Galerkin (DG) method for the Maxwell eigenvalue problem. The discrete compactness of DG space is proved so that the Babuška and Osborn spectral approximation theory can be applicable in the a-priori error analysis. Then we prove the optimal error estimates for DG eigenfunctions in mesh-dependent norm and DG eigenvalues. A special contribution of this work is to prove that the error in L2-norm for smooth eigenfunctions is of higher order than that in mesh-dependent norm, so that the DG eigenvalues can approximate the true eigenvalues from upper. Another contribution of this work is to provide a-posteriori error analysis for the DG method. A reliable a-posteriori error estimator is analyzed. The upper bound property of DG eigenvalues and the robustness of adaptive methods are verified through numerical experiments.

Unique continuation problem on RCD Spaces. I

In this note we establish the weak unique continuation theorem for caloric functions on compact RCD(K, 2) spaces and show that there exists an RCD(K, 4) space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point . We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.

Bound States and Heat Kernels for Fractional-Type Schrödinger Operators with Singular Potentials

We consider non-local Schrödinger operators H=-L-V in L^2(mathbb {R}^d), d geqslant 1, where the kinetic terms L are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and V is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential.

Eigenfunction restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature

For 2 ≤ p > 4 2\leq p>4 , we study the L p L^p norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact 2 2 -dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov [Duke Math. J. 138 (2007), pp. 445–486], and Hu [Forum Math. 21 (2009), pp. 1021–1052] found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For p = 4 p=4 , we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [Comm. Math. Phys. 350 (2017), pp. 1299–1325], Sogge [Math. Res. Lett. 24 (2017), pp. 549–570], and Bourgain [Geom. Funct. Anal. 1 (1991), pp. 147–187].

Sharp Pointwise Weyl Laws for Schrödinger Operators with Singular Potentials on Flat Tori

The Weyl law of the Laplacian on the flat torus $${\mathbb {T}}^n$$ is concerning the number of eigenvalues $$\le \lambda ^2$$ , which is equivalent to counting the lattice points inside the ball of radius $$\lambda $$ in $${\mathbb {R}}^n$$ . The leading term in the Weyl law is $$c_n\lambda ^n$$ , while the sharp error term $$O(\lambda ^{n-2})$$ is only known in dimension $$n\ge 5$$ . Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. This result establishes the sharpness of the general theorems for the Schrödinger operators $$H_V=-\Delta _{g}+V$$ in the previous work (Huang and Zhang (Adv Math, arXiv:2103.05531 )) of the authors, and extends the 3-dimensional results of Frank and Sabin (Sharp Weyl laws with singular potentials. arXiv:2007.04284 ) to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li–Yau’s heat kernel estimates, Blair–Sire–Sogge’s eigenfunction estimates, and Duhamel’s principle for the wave equation.

Pointwise Weyl laws for Schrödinger operators with singular potentials

We consider the Schrödinger operators HV=−Δg+V with singular potentials V on general n-dimensional Riemannian manifolds and study the eigenvalues and eigenfunctions under this perturbation. These singular potentials appear naturally in physics, most notably the Coulomb potential |x|−1. Sogge and the first author [14] proved the sharp Weyl laws for these HV with potentials in the Kato class, which is the minimal assumption to ensure that HV is essentially self-adjoint and bounded from below and the eigenfunctions of HV are bounded. Later, Frank-Sabin [9] studied the problem on the pointwise Weyl laws for these HV in three dimensions by extending the method of Avakumović [2], while it is unknown how to reconstruct this argument in other dimensions. In this paper, we completely solve this problem in any dimensions by using a different argument. First, we establish the pointwise Weyl law for potentials in the Kato class on any n-dimensional manifolds. This extends the 3-dimensional results of Frank-Sabin [9] by a different method. Second, we prove that the pointwise Weyl law with the standard sharp error term O(λn−1) holds for potentials in Ln(M). This extends the classical results for smooth potentials by Avakumović [2], Levitan [18] and Hörmander [12] to critically singular potentials. In three dimensions, this L3 condition also naturally appears in Boccato-Brennecke-Cenatiempo-Schlein [6] on the ground state energy of the Hamilton operator in the Gross-Pitaevskii regime. These two results are sharp, and our proof exploits Li-Yau's heat kernel bounds and Blair-Sire-Sogge's eigenfunction estimates.

Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius

Consider the class of optimal partition problems with long range interactions $$\begin{aligned} \inf \left\{ \sum _{i=1}^k \lambda _1(\omega _i):\ (\omega _1,\ldots , \omega _k) \in \mathcal {P}_r(\Omega ) \right\} , \end{aligned}$$ where $$\lambda _1(\cdot )$$ denotes the first Dirichlet eigenvalue, and $$\mathcal {P}_r(\Omega )$$ is the set of open k-partitions of $$\Omega $$ whose elements are at distance at least r: $${{\,\mathrm{dist}\,}}(\omega _i,\omega _j)\ge r$$ for every $$i\ne j$$ . In this paper we prove optimal uniform bounds (as $$r\rightarrow 0^+$$ ) in $$\mathrm {Lip}$$ –norm for the associated $$L^2$$ -normalized eigenfunctions, connecting in particular the nonlocal case $$r>0$$ with the local one $$r \rightarrow 0^+$$ . The proof uses new pointwise estimates for eigenfunctions, a one-phase Alt–Caffarelli–Friedman and the Caffarelli-Jerison-Kenig monotonicity formulas, combined with elliptic and energy estimates. Our result extends to other contexts, such as singularly perturbed harmonic maps with distance constraints.

Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian

We study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian (-Delta )^s in bounded open Lipschitz sets in the small order limit s rightarrow 0^+. While it is easy to see that all eigenvalues converge to 1 as s rightarrow 0^+, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with Fourier symbol 2log |xi |. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that L^2-normalized Dirichlet eigenfunctions of (-Delta )^s corresponding to the k-th eigenvalue are uniformly bounded and converge to the set of L^2-normalized eigenfunctions of the logarithmic Laplacian. In order to derive these spectral asymptotics, we establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.

Multiplicative Dirac System

We define a Dirac system in multiplicative calculus by some algebraic structures. Asymptotic estimates for eigenfunctions of the multiplicative Dirac system are obtained. Eventually, some fundamental properties of the multiplicative Dirac system are examined in detail.

A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

Interior estimates for the eigenfunctions of the fractional Laplacian on a bounded domain

This paper is devoted to interior estimates for eigenfunctions of the restricted fractional Laplacian on a bounded domain in Rd. We prove that the eigenfunctions satisfy the expected Lp bounds analogous to the classical results by Sogge [24]. As the fractional Laplacian is nonlocal, the standard method for Laplacian eigenfunction estimates can no longer work here. In the proof, we mainly reduce Lp bounds to a kind of L2 commutator estimates, which can be handled by the explicit integral expression of the restricted fractional Laplacian and its heat kernel estimates.

The Local and Parallel Finite Element Scheme for Electric Structure Eigenvalue Problems

In this paper, an efficient multiscale finite element method via local defect-correction technique is developed. This method is used to solve the Schrödinger eigenvalue problem with three-dimensional domain. First, this paper considers a three-dimensional bounded spherical region, which is the truncation of a three-dimensional unbounded region. Using polar coordinate transformation, we successfully transform the three-dimensional problem into a series of one-dimensional eigenvalue problems. These one-dimensional eigenvalue problems also bring singularity. Second, using local refinement technique, we establish a new multiscale finite element discretization method. The scheme can correct the defects repeatedly on the local refinement grid, which can solve the singularity problem efficiently. Finally, the error estimates of eigenvalues and eigenfunctions are also proved. Numerical examples show that our numerical method can significantly improve the accuracy of eigenvalues.

Sharp endpoint estimates for eigenfunctions restricted to submanifolds of codimension 2

Burq-Gérard-Tzvetkov [10] and Hu [26] established Lp estimates (2≤p≤∞) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint L2 estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be removed in general, while the problem is still open. So this paper is devoted to the study of sharp endpoint restriction estimates for eigenfunctions in this case. Chen-Sogge [17] removed the log loss for the geodesics on 3-dimensional manifolds. In this paper, we generalize their result to higher dimensions and prove that the log loss can be removed for totally geodesic submanifolds of codimension 2. Moreover, on 3-dimensional manifolds, we can remove the log loss for curves with nonvanishing geodesic curvatures, and more general finite type curves. The problem in 3D is essentially related to Hilbert transforms along curves in the plane and a class of singular oscillatory integrals studied by Phong-Stein [35], Ricci-Stein [38], Pan [32], Seeger [40], Carbery-Pérez [15].

Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities

We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. An elliptic operator on spherical domains with coefficients singular on boundary plays an important role. Key step is the study of the eigenvalue and eigenfunction estimates of such operators.

Estimates of eigenvalues and eigenfunctions in elliptic homogenization with rapidly oscillating potentials

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.

Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

A locking‐free shifted inverse iteration based on multigrid discretization for the elastic eigenvalue problem

We propose a locking‐free shifted inverse iteration based on multigrid discretization and prove the error estimates of eigenvalues and eigenfunctions. Besides, we also present a multigrid correction scheme based on the multilevel correction. Numerical experiments coincide in our theoretical analysis and validate that our scheme is highly efficient.

Upper bounds of nodal sets for eigenfunctions of eigenvalue problems

The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey–Nirenberg and Carleman estimates.

Integrated Variational Approach to Conformational Dynamics: A Robust Strategy for Identifying Eigenfunctions of Dynamical Operators.

One approach to analyzing the dynamics of a physical system is to search for long-lived patterns in its motions. This approach has been particularly successful for molecular dynamics data, where slowly decorrelating patterns can indicate large-scale conformational changes. Detecting such patterns is the central objective of the variational approach to conformational dynamics (VAC), as well as the related methods of time-lagged independent component analysis and Markov state modeling. In VAC, the search for slowly decorrelating patterns is formalized as a variational problem solved by the eigenfunctions of the system's transition operator. VAC computes solutions to this variational problem by optimizing a linear or nonlinear model of the eigenfunctions using time series data. Here, we build on VAC's success by addressing two practical limitations. First, VAC can give poor eigenfunction estimates when the lag time parameter is chosen poorly. Second, VAC can overfit when using flexible parametrizations such as artificial neural networks with insufficient regularization. To address these issues, we propose an extension that we call integrated VAC (IVAC). IVAC integrates over multiple lag times before solving the variational problem, making its results more robust and reproducible than VAC's.

A robust approach to sharp multiplier theorems for Grushin operators

We prove a multiplier theorem of Mihlin–Hörmander-type for operators of the form − Δ x − V ( x ) Δ y -\Delta _x - V(x) \Delta _y on R x d 1 × R y d 2 \mathbb {R}^{d_1}_x \times \mathbb {R}^{d_2}_y , where V ( x ) = ∑ j = 1 d 1 V j ( x j ) V(x) = \sum _{j=1}^{d_1} V_j(x_j) , the V j V_j are perturbations of the power law t ↦ | t | 2 σ t \mapsto |t|^{2\sigma } , and σ ∈ ( 1 / 2 , ∞ ) \sigma \in (1/2,\infty ) . The result is sharp whenever d 1 ≥ σ d 2 {d_1} \geq \sigma {d_2} . The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schrödinger operators, which are stable under perturbations of the potential.