Besov Spaces for the Schrödinger Operator with Barrier Potential

Let H = ?d ²/dx ² + V be a Schrödinger operator on the real line, where \({V=c\chi_{[a,b]}}\) , c > 0. We define the Besov spaces for H by developing the associated Littlewood–Paley theory. This theory depends on the decay estimates of the spectral operator \({{\varphi}_j(H)}\) for the high and low energies. We also prove a Mihlin multiplier theorem on these spaces, including the L ^p boundedness result. Our approach has potential applications to other Schrödinger operators with short-range potentials.     

A One-Dimensional Schrödinger Operator with Square-Integrable Potential

We study the spectral properties of a one-dimensional Schrödinger operator with squareintegrable potential whose domain is defined by the Dirichlet boundary conditions. The main results are concerned with the asymptotics of the eigenvalues, the asymptotic behavior of the operator semigroup generated by the negative of the differential operator under consideration. Moreover, we derive deviation estimates for the spectral projections and estimates for the equiconvergence of the spectral decompositions. Our asymptotic formulas for eigenvalues refine the well-known ones.     

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.     

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.     

Weighted Spectral Gap for Magnetic Schrödinger Operators with a Potential Term

We prove that singular Schrödinger equations with external magnetic field admit a representation with a positive Lagrangian density whenever their “nonmagnetic” counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L ^p -norm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of a magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrödinger equation with the periodic magnetic field.     

Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

The Quenched Asymptotics for Nonlocal Schrödinger Operators with Poissonian Potentials

Potential Analysis - We study the quenched long time behaviour of the survival probability up to time t, $mathbf {E}_{x}left [e^{-{{int }_{0}^{t}} V^{omega }(X_{s})mathrm {d}s}right ],$ of a...     

Schrödinger Operator with Morse Potential and Zeros of the Riemann Zeta Function

Mathematical Notes -     

Lieb–Thirring Type Inequalities for Schrödinger Operators with a Complex-Valued Potential

We present a possible generalization of the Lieb–Thirring inequalities for eigenvalues of Schrödinger operators to the case of complex potentials. We ask for a proof or disproof of this generalization. Some weaker results which have been obtained are reviewed.     

Newton's Polyhedron and Weyl's Formula for the Spectrum of the Schrödinger Operator with Polynomial Potential

Introduce the notation: V is a nonnegative polynomial, Q(V) is the corresponding Newton polyhedron, L= -+V is the Schrodinger operator on , N(,L) is the number of eigenvalues of L that are less than . The upper estimate for N(,L) is derived in terms of Q(V), and the Weyl asymptotic formula is established under certain assumptions on the geometry of Q(V). The case where the behavior of V(x) is not regular as is also considered. Bibliography: 6 titles.     

Asymptotic Eigenfunctions of the “Bouncing Ball” Type for the Two-Dimensional Schrödinger Operator with a Symmetric Potential

Theoretical and Mathematical Physics - We construct asymptotic eigenfunctions for the two-dimensional Schrödinger operator with a potential in the form of a well that is mirror-symmetric with...     

A Poisson Formula for the Schrödinger Operator

We prove a Poisson type formula for the Schrödinger group. Such a formula had been derived in a previous article by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H ¹-critical nonlinearities are allowed.     

Maximum Principle for the Regularized Schrödinger Operator

In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger–Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger–Günter problem on a class of conformally flat cylinders and tori.     

Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

We consider the discrete spectrum of the two-dimensional Hamiltonian H = H ₀ + V, where H ₀ is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H.     

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L ^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder’s inequality, which we believe to be new.     

On Pointwise Convergence for Schrödinger Operator in a Convex Domain

Journal of Fourier Analysis and Applications - In this paper, we prove that the maximal inequality $$begin{aligned} big Vert sup _{|t|<1}|e^{itDelta _D}f(x,y)|big Vert...     

Dispersion for Schrödinger Equation with Periodic Potential in 1D

We extend a result on dispersion for solutions of the linear Schrödinger equation, proved by Firsova for operators with finitely many energy bands only, to the case of smooth potentials in 1D with infinitely many bands. The proof consists in an application of the method of stationary phase. Estimates for the phases, essentially the band functions, follow from work by Korotyaev. Most of the paper is devoted to bounds for the Bloch functions. For these bounds we need a detailed analysis of the quasimomentum function and the uniformization of the inverse of the quasimomentum function.     

On Schrödinger Propagator for the Special Hermite Operator

We establish a Strichartz type estimate for the Schrödinger propagator e ^it? for the special Hermite operator ? on ? ⁿ . Our method relies on a regularization technique. We show that no admissibility condition is required on (q,p) when 1≤q≤2.