Uniform Resolvent Estimates Research Articles

Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.

Uniform Resolvent Estimates for Laplace–Beltrami Operator on the Flat Euclidean Cone

Uniform Resolvent Estimates for Laplace–Beltrami Operator on the Flat Euclidean Cone

From spectral cluster to uniform resolvent estimates on compact manifolds

It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, Sogge's universal spectral cluster estimates for the Laplace–Beltrami operator on closed Riemannian manifolds directly imply uniform resolvent estimates outside a parabolic region, without any reference to parametrices. The method is purely functional analytic and takes full advantage of the known spectral cluster bounds. This yields new resolvent estimates for manifolds with boundary or with low-regularity metrics, among other examples. Moreover, we show that the resolvent estimates are stable under perturbations and use this to establish uniform Sobolev and spectral cluster inequalities for Schrödinger operators with singular potentials.

Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields

Abstract In this paper, we combine the arguments of [L. Fanelli, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operators in critical magnetic fields, Int. Math. Res. Not. IMRN 2023), 10.1093/imrn/rnac362] and [Y. Sire, C. D. Sogge, C. Wang and J. Zhang, Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications, Comm. Partial Differential Equations 47 2022, 6, 1124–1132] to prove the maximal estimates for fractional Schrödinger equations ( i ⁢ ∂ t + ℒ 𝐀 α 2 ) ⁢ u = 0 {(i\partial_{t}+\mathcal{L}_{\mathbf{A}}^{\frac{\alpha}{2}})u=0} in the purely magnetic fields which includes the Aharonov–Bohm fields. The proof is based on the cluster spectral measure estimates. In particular, for α = 1 {\alpha=1} , the maximal estimate for wave equation is sharp up to the endpoint.

Local potential operator and uniform resolvent estimate for generalized Schrödinger operator in Orlicz spaces

AbstractThe local potential operator with integral kernel restricted in a ball of radius less than some fixed number has appeared frequently in the spectral estimates of the Schrödinger operator. In this paper, we establish a good‐λ inequality for this operator and characterize its uniform boundedness on the weighted Orlicz space in both strong and weak senses. The uniformity in r of this boundedness enables us to recover the classical boundedness of “global” potential operator, by letting . As an application, we establish uniform estimate for the resolvent of some generalized Schrödinger operator on the Orlicz space. An explicit representation in its operator norm on the dependence of is also given.

Uniform Resolvent Estimates for Critical Magnetic Schrödinger Operators in 2D

Abstract We study the $L^{p}-L^{q}$-type uniform resolvent estimates for 2D-Schrödinger operators in scaling-critical magnetic fields, involving the Aharonov–Bohm model as a main example. As an application, we prove localization estimates for the eigenvalue of some non–self-adjoint zero-order perturbations of the magnetic Hamiltonian.

On the Oseen-Type Resolvent Problem Associated with Time-Periodic Flow past a Rotating Body

Consider the time-periodic flow of an incompressible viscous fluid past a body performing a rigid motion with non-zero translational and rotational velocity. We introduce a framework of homogeneous Sobolev spaces that renders the resolvent problem of the associated linear problem well posed on the whole imaginary axis. In contrast to the cases without translation or rotation, the resolvent estimates are merely uniform under additional restrictions, and the existence of time-periodic solutions depends on the ratio of the rotational velocity of the body motion to the angular velocity associated with the time period. Provided that this ratio is a rational number, time-periodic solutions to both the linear and, under suitable smallness conditions, the nonlinear problem can be established. If this ratio is irrational, a counterexample shows that in a special case there is no uniform resolvent estimate and solutions to the time-periodic linear problem do not exist.

On the Stokes-Type Resolvent Problem Associated with Time-Periodic Flow Around a Rotating Obstacle

Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.

Decay and Strichartz estimates in critical electromagnetic fields

We study the L1→L∞-decay estimates for the Klein-Gordon equation in the Aharonov-Bohm magnetic fields, and further prove Strichartz estimates for the Klein-Gordon equation with critical electromagnetic potentials. The novel ingredients are the Schwartz kernels of the spectral measure and heat propagator of the Schrödinger operator in Aharonov-Bohm magnetic fields. In particular, we explicitly construct the representation of the spectral measure and resolvent of the Schrödinger operator with Aharonov-Bohm potentials, and prove that the heat kernel in critical electromagnetic fields satisfies Gaussian boundedness. In future papers, this result on the spectral measure will be used to (i) study the uniform resolvent estimates, and (ii) prove the Lp-regularity property of wave propagation in the same setting.

Uniform resolvent estimates for the discrete Schrödinger operator in dimension three

In this note, we prove the uniform resolvent estimate of the discrete Schrödinger operator with dimension three. To do this, we show a Fourier decay of the surface measure on the Fermi surface.

Kato Smoothing, Strichartz and Uniform Sobolev Estimates for Fractional Operators With Sharp Hardy Potentials

Let $0<\sigma<n/2$ and $H=(-\Delta)^\sigma +V(x)$ be Schr\"odinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2\sigma}$ with a sharp coupling constant $a\ge -C_{\sigma,n}$ ($C_{\sigma,n}$ is the best constant of Hardy's inequality of order $\sigma$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schr\"odinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{\sigma,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all $0<\sigma<n/2$, which turn out to be equivalent to {\it Kato smoothing estimates} for the Cauchy problem. We then establish {\it Strichartz estimates} for $\sigma>1/2$ and {\it uniform Sobolev estimates} of Kenig--Ruiz--Sogge type for $\sigma\ge n/(n+1)$. These extend the same properties for the Schr\"odinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtain {\it improved Strichartz estimates with a gain of regularities} for general initial data if $1<\sigma<n/2$ and for radially symmetric data if $n/(2n-1)<\sigma\le1$, which extends the corresponding results for the free evolution to the case with Hardy potentials. These arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator. Finally, in the critical coupling constant case (i.e. $a=-C_{\sigma,n}$), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions.

Resolvent Estimates for the Lamé Operator and Failure of Carleman Estimates

In this paper, we consider the Lamé operator \(-\Delta ^*\) and study resolvent estimate, uniform Sobolev estimate, and Carleman estimate for \(-\Delta ^*\). First, we obtain sharp \(L^p\)–\(L^q\) resolvent estimates for \(-\Delta ^*\) for admissible p, q. This extends the particular case \(q=\frac{p}{p-1}\) due to Barceló et al. [4] and Cossetti [8]. Secondly, we show failure of uniform Sobolev estimate and Carleman estimate for \(-\Delta ^*\). For this purpose we directly analyze the Fourier multiplier of the resolvent. This allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp \(L^p\)–\(L^q\) bounds for the resolvent of \(-\Delta ^*\). Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator \(-\Delta ^*\) even though the uniform resolvent estimates for \(-\Delta ^*\) are valid for certain range of p, q. This contrasts with the classical result regarding the Laplacian \(\Delta \) due to Kenig, Ruiz, and Sogge [23] in which the uniform resolvent estimate plays a crucial role in proving the uniform Sobolev and Carleman estimates for \(\Delta \). We also describe locations of the \(L^q\)-eigenvalues of \(-\Delta ^*+V\) with complex potential V by making use of the sharp \(L^p\)–\(L^q\) resolvent estimates for \(-\Delta ^*\).

On the numerical range of second-order elliptic operators with mixed boundary conditions in $$L^p$$

We consider second-order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on $$L^p$$ in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in Chill et al. (C R Acad Sci Paris 342:909–914, 2006). Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterisation of elements of the form domains inducing mixed boundary conditions.

Absence of eigenvalues of non‐self‐adjoint Robin Laplacians on the half‐space

By developing the method of multipliers, we establish sufficient conditions which guarantee the total absence of eigenvalues of the Laplacian in the half-space, subject to variable complex Robin boundary conditions. As a further application of this technique, uniform resolvent estimates are derived under the same assumptions on the potential. Some of the results are new even in the self-adjoint setting, where we obtain quantum-mechanically natural conditions.

Uniform Resolvent Estimates on Manifolds of Bounded Curvature

We establish $$L^{q*}\rightarrow L^q$$ bounds for the resolvent of the Laplacian on compact Riemannian manifolds assuming only that the sectional curvatures of the manifold are uniformly bounded. When the resolvent parameter lies outside a parabolic neighborhood of $$[0,\infty )$$, the operator norm of the resolvent is shown to depend only on upper bounds for the sectional curvature and diameter and lower bounds for the volume. The resolvent bounds are derived from square-function estimates for the wave equation, an approach that admits the use of paradifferential approximations in the parametrix construction.

Uniform resolvent estimates for Schrödinger operator with an inverse-square potential

We study the uniform resolvent estimates for Schrödinger operator with a Hardy-type singular potential. Let LV=−Δ+V(x) where Δ is the usual Laplacian on Rn and V(x)=V0(θ)r−2 where r=|x|,θ=x/|x| and V0(θ)∈C1(Sn−1) is a real function such that the operator −Δθ+V0(θ)+(n−2)2/4 is a strictly positive operator on L2(Sn−1). We prove some new uniform weighted resolvent estimates and also obtain some uniform Sobolev estimates associated with the operator LV.

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

In this paper we are concerned with resolvent estimates for the Laplacian $\Delta$ in Euclidean spaces. Uniform resolvent estimates for $\Delta$ were shown by Kenig, Ruiz and Sogge \cite{KRS} who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp $L^p$--$L^q$ bounds depending on $z$ has not been considered in a general framework which admits all possible $p,q$. In this paper, we present a complete picture of sharp $L^p$--$L^q$ resolvent estimates, which may depend on $z$. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner--Riesz operators of negative index.

Eigenvalue Estimates for Bilayer Graphene

Recently, Ferrulli-Laptev-Safronov (2016arXiv161205304F) obtained eigenvalue estimates for an operator associated to bilayer graphene in terms of $L^q$ norms of the (possibly non-selfadjoint) potential. They proved that for $1<q<4/3$ all non-embedded eigenvalues lie near the edges of the spectrum of the free operator. In this note we prove this for the larger range $1\leq q\leq 3/2$. The latter is optimal if embedded eigenvalues are also considered. We prove similar estimates for a modified bilayer operator with so-called "trigonal warping" term. Here, the range for $q$ is smaller since the Fermi surface has less curvature. The main tool are new uniform resolvent estimates that may be of independent interest and are collected in an appendix (in greater generality than needed).

A limiting absorption principle for the Helmholtz equation with variable coefficients

We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions (L+\lambda)v=f, \quad \lambda\in \mathbb{R} under a Sommerfeld radiation condition at infinity. The operator L is a second order elliptic operator with variable coefficients; the principal part is a small, long range perturbation of -\Delta , while lower order terms can be singular and large. The main tool is a sharp uniform resolvent estimate, which has independent applications to the problem of embedded eigenvalues and to smoothing estimates for dispersive equations.

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

AbstractExplicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L2 restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.