An inverse spectral problem for a fractional Schrödinger operator

Archiv der Mathematik - We establish that the potential appearing in a fractional Schrödinger operator is uniquely determined by an internal spectral data.     

Dirichlet problem for Schrödinger equation with the boundary value in the BMO space

Science China Mathematics - Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q > 1) and an L2-Poincaré inequality. Let $${\cal L} = {\cal L} + V$$ be a...     

An inverse problem for a time-dependent Schrödinger operator in an unbounded strip

In this Note we prove a stability result for two independent coefficients (each one depending on only one space variable and the potential also depending on the time variable) for a time-dependent Schrödinger operator in an unbounded strip with one observation on an unbounded subset of the boundary and the data of the solution at a fixed time on the whole domain.     

Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions

We study the inverse problem of determining a real-valued potential in the two-dimensional Schrödinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this article, we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy. In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases with increasing energy (in modulus).     

Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential μ⁻¹ V((x − x ₀)ɛ⁻¹), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛ^δ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.     

Irregular boundary value problem for the Sturm–Liouville operator

We consider an eigenvalue problem for the Sturm–Liouville operator with nonclassical asymptotics of the spectrum. We prove that this problem, which has a complete system of root functions, is not almost regular (Stone-regular) but its Green function has a polynomial order of growth in the spectral parameter.     

Logarithmic stability in the dynamical inverse problem for the Schrödinger equation by arbitrary boundary observation

In this paper, we investigate the inverse problem of determining the potential of the dynamical Schrödinger equation in a bounded domain from the data of the solution in a subboundary over a time interval. Assuming that in a neighborhood of the boundary of the spatial domain, the potential is known and without any assumption on the dynamics (i.e. without the geometric optics condition for the observability), we prove a logarithmic stability estimate for the inverse problem with a single measurement on an arbitrarily given subboundary.     

An inverse scattering problem for the Schrödinger equation with a potential having a periodic asymptotics

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = –d ^z/dx ²+q(x)(–<x<)with a potential q(x), satisfying the condition

Fourth-order Schrödinger type operator with singular potentials

We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form \({{\mathbb S}^{n+1}_{p,1}}\) induces a Hamiltonian variation of the normal congruence in the space \({{\mathbb L}({\mathbb S}^{n+1}_{p,1})}\) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in \({{\mathbb L}({\mathbb S}^{n+1})}\) (resp. \({{\mathbb L}({\mathbb H}^{n+1})}\)) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface \({\Sigma}\) in \({{\mathbb S}^{n+1}}\) (resp. \({{\mathbb H}^{n+1}}\)) that is a critical point of the functional \({{\mathcal W}(\Sigma) = \int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}}\), where k_i denote the principal curvatures of \({\Sigma}\) and \({\epsilon \in \{-1, 1\}}\). In addition, for \({n = 2}\), we prove that every Hamiltonian minimal surface in \({{\mathbb L}({\mathbb S}^{3})}\) (resp. \({{\mathbb L}({\mathbb H}^{3})}\)), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in \({{\mathbb S}^{3}}\) (resp. \({{\mathbb H}^{3}}\)) that is a critical point of the functional \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K+1}}\) (resp. \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K-1}}\)), where H and K denote, respectively, the mean and Gaussian curvature of \({\Sigma}\).     

On a class of nonlinear Schrödinger equations with nonnegative potentials in two space dimensions

Mathematical Notes - This paper discusses a class of critical nonlinear Schrödinger equations which are closely related to several applications, in particular to Bose-Einstein condensates with...     

On the inverse spectral problem for the quasi-periodic Schrödinger equation

We study the quasi-periodic Schrödinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \quad x \in{ \mathbf {R}} $$ in the regime of “small” V. Let $(E_{m}',E''_{m})$ , m∈Z ^ν , be the standard labeled gaps in the spectrum. Our main result says that if $E''_{m} - E'_{m} \le\varepsilon\exp(-\kappa_{0} |m|)$ for all m∈Z ^ν , with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then the Fourier coefficients of V obey $|c(m)| \le \varepsilon^{1/2} \exp(-\frac{\kappa_{0}}{2} |m|)$ for all m∈Z ^ν . On the other hand we prove that if |c(m)|≤εexp(?κ ₀|m|) with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then $E''_{m} - E'_{m} \le2 \varepsilon\exp(-\frac {\kappa_{0}}{2} |m|)$ .     

On the asymptotics of solutions to the second initial boundary value problem for Schrödinger systems in domains with conical points

In this paper, for the second initial boundary value problem for Schrödinger systems, we obtain a performance of generalized solutions in a neighborhood of conical points on the boundary of the base of infinite cylinders. The main result are asymptotic formulas for generalized solutions in case the associated spectrum problem has more than one eigenvalue in the strip considered.