The improved decay rate for the heat semigroup with local magnetic field in the plane

We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta. The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schrödinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behaviour of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov–Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables. Since no assumptions about the symmetry of the magnetic field are made in the present work, it gives a normwise variant of the recent pointwise results of Kova?ík (Calc Var doi:10.1007/s00526-011-0437-4) about large-time asymptotics of the heat kernel of magnetic Schrödinger operators with radially symmetric field in a more general setting.     

Variation Operators for Semigroups and Riesz Transforms on BMO in the Schrödinger Setting

In this paper we prove that the variation operators of the heat semigroup and the truncations of Riesz transforms associated to the Schrödinger operator are bounded on a suitable BMO type space.     

Weak Dispersive Estimates for Schrödinger Equations with Long Range Potentials

We prove local smoothing estimates for the Schrödinger initial value problem with data in the energy space L ²(? ^d ), d ≥ 2 and a general class of potentials. In the repulsive setting we have to assume just a power like decay (1 + |x|)^?γ for some γ > 0. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.     

Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of {\mathbb{R}^d}

Let L = ?Δ + V be a Schrödinger operator and Ω be a strongly Lipschitz domain of ${\mathbb R^{d}}Let L = −Δ + V be a Schr?dinger operator and Ω be a strongly Lipschitz domain of \mathbb R^d{\mathbb R^{d}} , where Δ is the Laplacian on \mathbb R^d{\mathbb R^{d}} and the potential V is a nonnegative polynomial on \mathbb R^d{\mathbb R^{d}} . In this paper, we investigate the Hardy spaces on Ω associated to the Schr?dinger operator L.     

Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold

We consider the eigenvalue problem of the Schrödinger operator with the magnetic field on a compact Riemannian manifold. First we discuss the least eigenvalue. We give a representation of the least eigenvalue by the variational formula and give a relation to the least eigenvalue of the Schrödinger operator without the magnetic field. Second, we discuss the asymptotic distribution of eigenvalues by obtaining the asymptotic expansion of the kernel of semigroup. Here we use the theory of asymptotic expansion for Wiener functionals.     

The Schrödinger Equation Type with a Nonelliptic Operator

We consider some linear Schrödinger equation with variable coefficients associated to a smooth symmetric metric g which can be degenerate, without sign and such that g has a submatrix of fixed rank v which is uniformly nondegenerate. In this general setting we prove Strichartz estimates with a loss of derivative on the solution. We also discuss the problem of the control of high frequencies. In particular, we prove that if the equation preserves the H ^s norm for all s ≥ 0, then we obtain almost the same Strichartz estimates as those for the Schrödinger equation associated to a Riemannian metric of dimension 2d ? v.     

Dunkl-Schrödinger semigroups and applications

We obtain dispersive estimates for the linear Dunkl–Schrödinger equations with and without quadratic potential. As a consequence, we prove the local well-posedness for semilinear Dunkl–Schrödinger equations with polynomial nonlinearity in certain magnetic field. Furthermore, we study many applications: as the uncertainty principles for the Dunkl transform via the Dunkl–Schrödinger semigroups, the embedding theorems for the Sobolev spaces associated with the generalized Hermite semigroup. Finally, almost every where convergence of the solutions of the Dunkl–Schrödinger equation is also considered.     

Linear Restriction Estimates for Schrödinger Equation on Metric Cones

In this paper, we study some modified linear restriction estimates of the dynamics generated by Schrödinger operator on metric cone M, where the metric cone M is of the form M = (0, ∞) _r  × Σ, with the cross section Σ being a compact (n ? 1)-dimensional Riemannian manifold (Σ, h) and the equipped metric being g = dr ² + r ² h. Assuming the initial data possesses additional regularity in angular variable θ ∈ Σ, we show some linear restriction estimates for the solutions. In terms of their applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initial data scattering theory for the mass-critical nonlinear Schrödinger equation on two-dimensional metric cones.     

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (?i? _x  ? A(x))² + V(x) and the related inverse problem in an exterior domain Ω in R ² with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω ^c is convex.     

Stability of an interconnected Schrödinger–heat system in a torus region

In this paper, we study the exponential stability of a two‐dimensional Schrödinger–heat interconnected system in a torus region, where the interface between the Schrödinger equation and the heat equation is of natural transmission conditions. By using a polar coordinate transformation, the two‐dimensional interconnected system can be reformulated as an equivalent one‐dimensional coupled system. It is found that the dissipative damping of the whole system is only produced from the heat part, and hence, the heat equation can be considered as an actuator to stabilize the whole system. By a detailed spectral analysis, we present the asymptotic expressions for both eigenvalues and eigenfunctions of the closed‐loop system, in which the eigenvalues of the system consist of two branches that are asymptotically symmetric to the line Reλ =? Imλ. Finally, we show that the system is exponentially stable and the semigroup, generated by the system operator, is of Gevrey class δ > 2. Copyright © 2016 John Wiley & Sons, Ltd.     

Low Frequency Estimates and Local Energy Decay for Asymptotically Euclidean Laplacians

For Riemannian metrics G on ? ^d which are long range perturbations of the flat one, we prove estimates for (? Δ _G  ? λ ?iε)^?n as λ → 0, which are uniform with respect to ε, for all n ≤ [d/2] +1 in odd dimension and n ≤ d/2 in even dimension. We also give applications to the time decay of Schrödinger and Wave (or Klein–Gordon) equations.     

Local well-posedness of the nonlinear Schrödinger equations on the sphere for data in modulation spaces

In this paper, the authors discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schrödinger on the sphere S². Exploring suitable a priori estimates, the authors prove the existence of solution for data whose regularity is s = 1/4.     

Semiclassical Pseudodifferential Calculus and the Reconstruction of a Magnetic Field

We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R ⁿ , n ≥ 3. The magnetic potential is assumed to be continuous with L ^∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces.     

Structures of the twisted N = 1 Schrödinger–Neveu–Schwarz algebra

In this paper, we determine the universal central extension, derivation algebra and automorphism group of the twisted N = 1 Schrödinger–Neveu–Schwarz algebra. Furthermore, we generalize these results to the generalized twisted N = 1 Schrödinger–Neveu–Schwarz algebra in the final section.     

The sharp Strichartz and Sobolev–Strichartz inequalities for the fourth‐order Schrödinger equation

In this paper, we will obtain that there exists a maximizer for the non‐endpoint Strichartz inequalities for the fourth‐order Schrödinger equation with initial data in the L²( R ^d) space in all dimensions, and then we obtain a maximizer also for the non‐endpoint Sobolev–Strichartz inequality for the fourth‐order Schrödinger equation with initial data in the homogeneous Sobolev space. Our analysis derived from the linear profile decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

A Spectral Multiplier Theorem Associated with a Schrödinger Operator

We establish a Hörmander type spectral multiplier theorem for a Schrödinger operator \(H=-\Delta +V(x)\) in \(\mathbb {R}^3\), provided V is contained in a large class of short range potentials. This result does not require the Gaussian heat kernel estimate for the semigroup \(e^{-tH}\), and indeed the operator H may have negative eigenvalues. As an application, we show local well-posedness of a 3d quintic nonlinear Schrödinger equation with a potential.     

On non–symmetric generalized schrodinger semigroup

In this paper some general phenomena are described for not necessarily systemeric so–called generalized Schrödinger semigroups (or generalized absorption/exciatation semigroups). These results are also applicable in case we consider Schrödinger semigroups on R ^v. In particular we describe some results on integral kernels: continuity, pointwise inequalities, ultracontractivity etc.For these inequalities we use a kind of stochastic bridge measure. The operator H is a closed linear extension of the operator H ⁰ + V in the space C ⁰(E) Here E is a locally compact second countable Hausdorff space and –H ₀ is supposed to generate a Feller semigroup in C ₀(E). Results in L^p (E,m) are also availale. Some examples are given     

A remark on the Lifshitz tail for Schrödinger operator with random magnetic field

In this note, we consider the Lifshitz singularity for Schrödinger operator with ergodic random magnetic field. A key estimate is an energy bound for magnetic Schrödinger operators as discussed in Nakamura [8]. Here we remove a technical assumption in [8], namely, the uniform boundedness of the magnetic field.     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.