Related papers: Spectral inequalities for elliptic pseudo-differen…

We study the solvability of boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +$\infty$, X being a UMD complex Banach space. The originality of this work lies in the fact that we…

Let $(E,\mathcal F,\mu)$ be a probability space, and let $P$ be a Markov operator on $L^2(\mu)$ with $1$ a simple eigenvalue such that $\mu P=\mu$ (i.e. $\mu$ is an invariant probability measure of $P$). Then $\hat P:=\ff 1 2 (P+P^*)$ has a…

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator…

In this paper we introduce new characterizations of spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We…

This paper is concerned with the construction of discrete counterparts of the Laplace-Beltrami operator on Riemannian manifolds that can be effectively used in the numerical solution of partial differential equations. Since existing…

We study the null-controllability of parabolic equations associated to non-autonomous Ornstein-Uhlenbeck operators. When a Kalman type condition holds for some positive time $T>0$, these parabolic equations are shown to enjoy a Gevrey…

In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy…

We show that tensoring the Laplace and Dolbeault-Dirac operators of a K\"ahler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the…

This work is devoted to the study of uncertainty principles for finite combinations of Hermite functions. We establish some spectral inequalities for control subsets that are thick with respect to some unbounded densities growing almost…

We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[\tau f = \frac{1}{r} \left(- \big(p[f' + s…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential…

We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of R^N (N $\in$ N *), assumed to be an unknown perturbation of a reference domain. We are interested…

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be…

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the…

Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function $e_{h,\varepsilon}(x,x,\lambda)$ for a scalar operator \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*}…

We consider the inverse problem to determine a smooth compact Riemannian manifold $(M,g)$ from a restriction of the source-to-solution operator, $\Lambda_{\mathcal{S,R}}$, for the wave equation on the manifold. Here, $\mathcal{S}$ and…

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y),…

We consider a strongly elliptic differential expression of the form $b(D)^* g(x/\varepsilon) b(D)$, $\varepsilon >0$, where $g(x)$ is a matrix-valued function in ${\mathbb R}^d$ assumed to be bounded, positive definite and periodic with…

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$…