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The purpose of this article is to develop a general parametric estimation theory that allows the derivation of the limit distribution of estimators in non-regular models where the true parameter value may lie on the boundary of the…
We study inverse boundary problems for semilinear Schr\"odinger equations on smooth compact Riemannian manifolds of dimensions $\ge 2$ with smooth boundary, at a large fixed frequency. We show that certain classes of cubic nonlinearities…
New exactly solvable quantum models are obtained with the help of the supersymmetric extencion of the nonstationary Schr/"odinger equation.
In this paper, we provide the boundedness property of the Riesz transforms associated to the Schr\"odinger operator $\mathcal{L}=-\Delta + \mathbf{V}$ in a new weighted Morrey space which is the generalized version of many previous Morrey…
We consider the Bayesian approach to linear inverse problems when the underlying operator depends on an unknown parameter. Allowing for finite dimensional as well as infinite dimensional parameters, the theory covers several models with…
This paper introduces \textbf{Measure Learning}, a paradigm for modeling ambiguity via non-linear expectations. We define Neural Expectation Operators as solutions to Backward Stochastic Differential Equations (BSDEs) whose drivers are…
We give a distribution-dependent concentration inequality for functions of independent variables. The result extends Bernstein's inequality from sums to more general functions, whose variation in any argument does not depend too much on the…
In this paper, we study Strichartz estimates for the Schr\"odinger equation on a metric cone $X$, where $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$. For the metric $g$…
The paper deals with (multidimensional and one-dimensional) Bochner-Phillips functional calculus. Bounded perturbations of Bernstein functions of (one or several commuting) semigroup generators on Banach spaces are considered, conditions…
We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is…
We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schr\"odingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y.…
Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity…
We propose a scalable variational Bayes method for statistical inference for a single or low-dimensional subset of the coordinates of a high-dimensional parameter in sparse linear regression. Our approach relies on assigning a mean-field…
Schr\"odinger equations with nonlinearities concentrated in some regions of space are good models of various physical situations and have interesting mathematical properties. We show that in the semiclassical limit it is possible to…
We consider a linear Schr\"odinger equation, on a bounded interval, with bilinear control, that represents a quantum particle in an electric field (the control). We prove the controllability of this system, in any positive time, locally…
We introduce a self-consistent theoretical framework associated with the Schwinger unitary operators whose basic mathematical rules embrace a new uncertainty principle that generalizes and strengthens the Massar-Spindel inequality. Among…
We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To…
In this paper, we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities, which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results…
In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schr\"odinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic…
The first author established in [8] a quantitative Borg-Levinson theorem for the Schr\"odinger operator with unbounded potential. In the present work, we extend the results in [8] to the magnetic Schr\"odinger operator. We discuss both the…