Related papers: On Pointwise Products of Elliptic Eigenfunctions

We construct a generalization of the multiplicative product of distributions presented by L. H\"ormander in [L. H\"ormander, {\it The analysis of linear partial differential operators I} (Springer-Verlag, 1983)]. The new product is defined…

We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…

This article is about two types of restrictions of eigenfunctions $\phi_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $\gamma_H \phi_j$ in eigenfunctions $e_k$ of…

Let $H(\mathbb{B})$ denote the space of all holomorphic functions on the unit ball $\mathbb{B}\in \mathbb{C}^n$. In this paper we investigate the boundedness and compactness of the products of radial derivative operator and the following…

Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected submanifold. Let…

This paper is devoted to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L = --$\Delta$+V (x)\cdot \nabla with Dirichlet boundary conditions, where V is a bounded vector field. In the…

We obtain upper bounds for the eigenvalues of the Schr\"odinger operator $L=\Delta_g+q$ depending on integral quantities of the potential $q$ and a conformal invariant called the min-conformal volume. Moreover, when the Schr\"odinger…

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…

Assuming that $\phi(t)=o(t^2)$ as $t\to0$, we establish a lemma on simultaneous polynomial approximation in Orlicz-Beurling-Sobolev spaces $\ell_a^{\phi}$. These spaces, endowed with the Luxemburg norm $\Vert \cdot \Vert_{\ell^{\phi}}$,…

We study a class of potentials $f$ on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous)…

We solve the following problems associated with Toeplitz operators $T_{\Phi}$ on Hilbert space-valued Hardy spaces $H_{\mathcal{E}}^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$. $(I)$ Given operator-valued bounded analytic…

Analysis of non-compact manifolds almost always requires some controlled behavior at infinity. Without such, one neither can show, nor expect, strong properties. On the other hand, such assumptions restrict the possible applications and…

We prove the existence of a family of compact subdomains $\Omega$ of the flat cylinder $\mathbb{R}^N\times \mathbb{R}/2\pi\mathbb{Z}$ for which the Neumann eigenvalue problem for the Laplacian on $\Omega$ admits eigenfunctions with constant…

We develop arguments on the critical point theory for locally Lipschitz functionals on Orlicz-Sobolev spaces, along with convexity and compactness techniques to investigate existence of solution of the multivalued equation $\displaystyle -…

We study the growth of Laplacian eigenfunctions $ -\Delta \phi_k = \lambda_k \phi_k$ on compact manifolds $(M,g)$. H\"ormander proved sharp polynomial bounds on $\| \phi_k\|_{L^{\infty}}$ which are attained on the sphere. On a `generic'…

We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we…

We prove local bounds on the amplitude of eigen- functions of complex constant-coefficient elliptic operators with a smooth potential on an arbitrary open subset of \R^d by estimating it in terms of the number of solutions of a diophantine…

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the…

This note concerns the nodal sets of eigenfunctions of semiclassical Schr\"odinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. We prove that if H is a separating hypersurface that lies inside the classically…

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and…