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In this paper, we show that for all triangles in the plane, the equilateral triangle maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This is an extension of work by Siudeja, who proved the inequality in the case of acute triangles. The proof utilizes inequalities due to Siudeja and Freitas, together with improved variational bounds., Comment: 16 pages, 4 figures

We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 \Delta u + \xi u$ in $\mathbb R^2$ with a Gaussian (space) white-noise potential $\xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $\log U(t)\sim \chi t \log t$ for $t \to \infty$, with the deterministic constant $\chi$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour $\boldsymbol \lambda_1(Q_t)\sim\chi\log t$ of the principal eigenvalue $\boldsymbol\lambda_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t= [-\frac{t}{2},\frac{t}{2}]^2$.

We prove uniform Sobolev estimates for the resolvent of Schr\"odinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some non-admissible retarded estimates, a H\"ormander type spectral multiplier theorem, and Keller type eigenvalue bounds with complex-valued potentials are also obtained., Comment: 31 pages, 1 figure; revised version

In 1954, G. Polya conjectured that the counting function $N(Ω,Λ)$ of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set $Ω\subset R^d$ is lesser (resp. greater) than $(2π)^{-d} ω_d |Ω| Λ^{d/2}$. Here $Λ$ is the spectral parameter, and $ω_d$ is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We heavily use the ideas from \cite{LPS}., merged into "P\'olya's conjecture for Euclidean balls" by N. Filonov, M. Levitin, I. Polterovich, D. Sher, arXiv:2203.07696v3

This paper is a mathematical analysis of conduction effects at interfaces between insulators. Motivated by work of Haldane-Raghu , we continue the study of a linear PDE initiated in papers of Fefferman-Lee-Thorp-Weinstein. This PDE is induced by a continuous honeycomb Schrodinger operator with a line defect. This operator exhibits remarkable connections between topology and spectral theory. It has essential spectral gaps about the Dirac point energies of the honeycomb background. In a perturbative regime, Fefferman-Lee-Thorp-Weinstein construct edge states: time-harmonic waves propagating along the interface, localized transversely. At leading order, these edge states are adiabatic modulations of the Dirac point Bloch modes. Their envelops solve a Dirac equation that emerges from a multiscale procedure. We develop a scattering-oriented approach that derives all possible edge states, at arbitrary precision. The key component is a resolvent estimate connecting the Schrodinger operator to the emerging Dirac equation. We discuss topological implications via the computation of the spectral flow, or edge index., Comment: 61 pages; 11 figures

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012), 815--838] as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic -- rather than numerical -- results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in [Conti \textit{et al}, Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\ Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

The paper by G. Liu [arxiv:2109.02561] contains an error. In this note, I give a brief review of the problem and indicate what the error is.

We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semi-discretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary $\theta$-method in time (including the forward and backward Euler methods, and a second-order method by choosing $\theta\in [0,1]$ suitably). The full discretization generates a two-parameter family of circulant matrices $M\in\mathbb{R}^{m\times m}$, where each matrix entry is a rational function in $\theta$ and $\nu$. Here, $\nu$ denotes the CFL number, being proportional to the ratio between the temporal and spatial discretization step sizes. The entrywise non-negativity of the matrix $M$ -- which is equivalent to the positivity preservation of the fully discrete scheme -- is investigated via discrete Fourier analysis and also by solving some low-order parametric linear recursions. We find that positivity preservation of the fully discrete system is impossible if the number of spatial grid points $m$ is even. However, it turns out that positivity preservation of the fully discrete system is recovered for \emph{odd} values of $m$ provided that $\theta\ge 1/2$ and $\nu$ are chosen suitably. These results are interesting since the systems of ordinary differential equations obtained via the spatial semi-discretizations studied are \emph{not} positivity preserving.

Let $Ω\subset\mathbb{R}^3$ be an open set, we study the spectral properties of the free Dirac operator $\mathcal{H}$ coupled with the singular potential $V_κ=(εI_4 +μβ+η(α\cdot N))δ_{\partialΩ}$. The open set $Ω$ can be either a $\mathcal{C}^2$-bounded domain or a locally deformed half-space. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of $\mathcal{H}+V_κ$ for the so-called critical combinations of coupling constants, when $Ω$ is a locally deformed half-space. Finally, we introduce a new model of Dirac operators with $δ$-interactions and deals with its spectral properties. More precisely, we study the coupling $\mathcal{H}_{\upsilon}=\mathcal{H}+i\upsilonβ(α\cdot N)δ_{\partialΩ}$. In particular, we show that $\mathcal{H}_{\pm2}$ is essentially self-adjoint and generates confinement., This article corresponds to the first part of the article arXiv:2102.10207. In this version we corrected many typos and errors

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of $(\mathbb{N}-\tfrac12)\times(\mathbb{N}-\tfrac12)$ inside triangles with vertices $(0, 0), (0, \lambda \sqrt{\beta})$ and $(\lambda/{\sqrt{\beta}}, 0)$ with respect to $\beta>0$, for fixed $\lambda\geq 0$. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice $(\mathbb{N}+\sigma)\times(\mathbb{N}+\tau)$, for $\sigma, \tau >-1$. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, $\sigma=\tau=-\tfrac12$, is a critical case., Comment: Accepted and final version. 24 pages

In this paper, we study the existence of nontrivial solutions to a class fractional Schrödinger equations$$ {( - \Delta )^s}u + V(x)u = \lambda f(x,u)\,\,{\rm in}\,\,{\mathbb{R}^N}, $$where$ {( - \Delta )^s}u(x) = 2\lim\limits_{\varepsilon \to 0} \int_ {{\mathbb{R}^N}\backslash {B_\varepsilon }(X)} {{u(x) - u(y)} \over {|x - y{|^{N + 2s}}}}dy,\,\,x \in {\mathbb{R}^N} $is a fractional operator ands∈ (0, 1). By using variational methods, we prove this problem has at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

Given two planar, conformal, smooth open sets $\Omega$ and $\omega$, we prove the existence of a sequence of smooth sets $\Omega_n$ which geometrically converges to $\Omega$ and such that the (perimeter normalized) Steklov eigenvalues of $\Omega_n$ converge to the ones of $\omega$. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur., Comment: typos corrected in section 2 and 4, statement of main result was precised

The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure., Comment: 16 pages, 1 figure

In this paper, we show that for all triangles in the plane, the equilateral triangle maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This is an extension of work by Siudeja, who proved the inequality in the case of acute triangles. The proof utilizes inequalities due to Siudeja and Freitas, together with improved variational bounds., 16 pages, 4 figures

International audience; In this paper, hypocoercivity methods are applied to linear kinetic equations with mass conservation and without confinement, in order to prove that the solutions have an algebraic decay rate in the long-time range, which the same as the rate of the heat equation. Two alternative approaches are developed: an analysis based on decoupled Fourier modes and a direct approach where, instead of the Poincar\'e inequality for the Dirichlet form, Nash's inequality is employed. The first approach is also used to provide a simple proof of exponential decay to equilibrium on the flat torus. The results are obtained on a space with exponential weights and then extended to larger function spaces by a factorization method. The optimality of the rates is discussed. Algebraic rates of decay on the whole space are improved when the initial datum has moment cancellations.

I describe a way to modify the multipliers of a-priori estimates, so as to include potential perturbations of the Laplacian., Comment: 13 pages

In 1954, G. P��lya conjectured that the counting function of the eigenvalues of the Laplace operator of Dirichlet (resp. Neumann) boundary value problem in a bounded set $��\subset{\mathbb R}^d$ is lesser (resp. greater) than $C_W |��| ��^{d/2}$. Here $��$ is the spectral parameter, and $C_W$ is the constant in the Weyl asymptotics. In 1961, P��lya proved this conjecture for tiling sets in the Dirichlet case, and for tiling sets under some additional restrictions for the Neumann case. We prove the P��lya conjecture in the Neumann case for all tiling sets.

We analize the limit problem of the anisotropic $p$-Laplacian as $p\rightarrow\infty$ with the mean of the viscosity solution. We also prove some geometric properties of eigenvalues and eigenfunctions. In particular, we show the validity of a Szeg\"o-Weinberger type inequality., Comment: 24 pages

International audience; This paper deals with semiclassical asymptotics of the three-dimensional magnetic Laplacian in presence of magnetic confinement. Using generic assumptions on the geometry of the confinement, we exhibit three semiclassical scales and their corresponding effective quantum Hamiltonians, by means of three microlocal normal forms à la Birkhoff. As a consequence, when the magnetic field admits a unique and non degenerate minimum, we are able to reduce the spectral analysis of the low-lying eigenvalues to a one-dimensional pseudo-differential operator whose Weyl's symbol admits an asymptotic expansion in powers of the square root of the semiclassical parameter.

Given a bounded Euclidean domain $��$, we consider the sequence of optimisers of the $k^{\rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $��$ with fixed total volume. We show that this sequence encodes information yielding conditions for $��$ to satisfy P��lya's conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether P��lya's conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of P��lya's original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa's bound in terms of packing density., 35 pages, 3 figures

International audience; This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.

In this paper analogues of Sobolev inequalities for compact and connected metric graphs are derived. As a consequence of these inequalities, a lower bound, commonly known as Cheeger inequality, on the first non-zero eigenvalue of the Laplace operator with standard vertex conditions is recovered.

For an $(n-1)$-dimensional compact orientable smooth metric measure space $\big(M,g,e^{-f}dv_{g}\big)$ embedded in an $n$-dimensional compact orientable Riemannian manifold $N$, we successfully give a lower bound for the first nonzero eigenvalue of the drifting Laplacian on $M$, provided the Ricci curvature of $N$ is bounded from below by a positive constant and the weighted function $f$ on $M$ satisfies two constraints.

The goal of the present paper is to calculate the limit spectrum of the Hodge-de Rham operator under the perturbation of collapsing one part of a manifold obtained by gluing together two manifolds with the same boundary. It appears to take place in the general problem of blowing up conical singularities as introduced in \cite{Maz} and \cite{Row1,Row2}.

We generalize to the case of the $p-$Laplacian an old result by Hersch and Protter. Namely, we show that it is possible to estimate from below the first eigenvalue of the Dirichlet $p-$Laplacian of a convex set in terms of its inradius. We also prove a lower bound in terms of isoperimetric ratios and we briefly discuss the more general case of Poincar\'e-Sobolev embedding constants. Eventually, we highlight an open problem., Comment: 20 pages, 3 figures

To M. Tsfasman and S. Vladut.; International audience; The present paper is a follow up of our paper \cite{nS}. We investigate here the maximization of higher order eigenvalues in a conformal class on a smooth compact boundaryless Riemannian surface. Contrary to the case of the first nontrivial eigenvalue as shown in \cite{nS}, bubbling phenomena appear.

43 pages; International audience; We consider a twisted magnetic Laplacian with Neumann condition on a smooth and bounded domain of $\R^2$ in the semiclassical limit $h\to 0$. Under generic assumptions, we prove that the eigenvalues admit complete asymptotic expansions in powers of $h^{1/4}$.

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a K\"ahler manifold whose Ricci curvature satisfies $\Ric \ge k$ for some constant $k>0$. When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field. We show that a ball of sufficiently large radius in complex projective space provides an example of a strongly pseudoconvex domain which is not convex, and for which the Lichnerowicz estimate fails., Comment: 9 pages

We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in R n when n ≥ 3 . This is in contrast to the situation when n = 2 , where a result of Weinstock from 1954 shows that the disk uniquely maximizes the first Steklov eigenvalue among all simply connected domains in the plane having the same boundary length. When n ≥ 3 , we show that increasing the number of boundary components does not increase the normalized (by boundary volume) first Steklov eigenvalue. This is in contrast to recent results which have been obtained for surfaces and for convex domains.

We study the behavior of eigenvalues of a magnetic Aharonov-Bohm operator with non-half-integer circulation and Dirichlet boundary conditions in a planar domain. As the pole is moving in the interior of the domain, we estimate the rate of the eigenvalue variation in terms of the vanishing order of the limit eigenfunction at the limit pole. We also provide an accurate blow-up analysis for scaled eigenfunctions and prove a sharp estimate for their rate of convergence., Comment: 35 pages

International audience; Let $\Omega$ be a smooth, convex, unbounded domain of $\mathbb{R}^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$% -dimensional Gaussian measure.

The spectral problem of anisotropic elasticity with traction-free boundary conditions is considered in a bounded domain with a spatial cusp having its vertex at the origin and given by triples $(x_1,x_2,x_3)$ such that $x_3^{-2}(x_1,x_2) \in \omega$, where $\omega$ is a two-dimensional Lipschitz domain with a compact closure. We show that there exists a threshold $\lambda_\dagger>0$ expressed explicitly in terms of the elasticity constants and the area of $\omega$ such that the continuous spectrum coincides with the half-line $[\lambda_\dagger,\infty)$, whereas the interval $[0,\lambda_\dagger)$ contains only the discrete spectrum. The asymptotic formula for solutions to this spectral problem near cusp's vertex is also derived. A principle feature of this asymptotic formula is the dependence of the leading term on the spectral parameter.

In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are different and mainly probabilistic relying on coupling methods adapted to the special situation under investigation. Moreover, we answer a question raised by Ben-Ari and Pinsky concerning the dependence of the spectral gap on the jump distribution in a multi-dimensional setting., 18 pages

Schrödinger operators $T_0 = -\Delta + q(x)$ with rapidly oscillating complex-valued potentials $q(x)$ are considered. Each of such operators is sectorial and hence has Friedrichs extension. We prove that $T_0$ is essentially m-sectorial in the sense that the closure of $T_0$ coincides with its Friedrichs extension $T$. In particular, $T_0$ is essentially self-adjoint if the rapidly oscillating potential $q(x)$ is realvalued. Further, we prove $\sigma_{ess} (T) = [0, \infty)$ under somewhat stricter condition on the potentials $q(x)$.

We prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations, 15 pages, 1 figure

International audience; Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be build, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy-Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities.

In this paper the linear stability properties of the steady states of a no-slip lubrication equation are studied. In the physical context, these steady states correspond to configurations of droplets that arise during the late-phase dewetting process under the influence of both destabilizing van der Waals and stabilizing Born intermolecular forces, which in turn give rise to the minimum thickness $\varepsilon$ of the remaining film connecting the droplets. The goal of this paper is to give an asymptotic description of the eigenvalues and eigenfunctions of the problem, linearized about the one-droplet solutions, as $\varepsilon\to0$. For this purpose, corresponding asymptotic eigenvalue problems with piecewise constant coefficients are constructed such that their eigenvalue asymptotics can be determined analytically. A comparison with numerically computed eigenvalues and eigenfunctions shows good agreement with the asymptotic results and the existence of a spectrum gap for sufficiently small $\varepsilon$.

International audience; In this paper, we consider an open tube of diameter $\varepsilon>0$, on the side of which a small hole of size $\varepsilon^2$ is pierced. The resonances of this tube correspond to the eigenvalues of the Laplacian operator with homogeneous Neumann condition on the inner surface of the tube and Dirichlet one the open parts of the tube. We show that this spectrum converges when $\varepsilon$ goes to $0$ to the spectrum of an explicit one-dimensional operator. At a first order of approximation, the limit spectrum describes the note produced by a flute, for which one of its holes is open.

We derive upper bounds for the trace of the heat kernel $Z(t)$ of the Dirichlet Laplace operator in an open set $\Omega \subset \R^d$, $d \geq 2$. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on $Z(t)$ in domains of infinite volume. For domains of finite volume the bound on $Z(t)$ decays exponentially as $t$ tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of $Z(t)$. To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means.

In this work we study the sequence of variational eigenvalues of a system of resonant type involving $p-$ and $q-$laplacians on $\Omega \subset \R^N$, with a coupling term depending on two parameters $\alpha$ and $\beta$ satisfying $\alpha/p + \beta/q = 1$. We show that the order of growth of the $k^{th}$ eigenvalue depends on $\alpha+\beta$, $\lam_k = O(k^{\frac{\alpha+\beta}{N}})$., Comment: Minor changes, Theorem 1.4 added

Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of P\'olya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-P\'olya-Weinberger conjecture proved by Ashbaugh and Benguria.

In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $\Gamma$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\eps) \to (M,g_\eps)$ such that the spectrum of the Laplacian $\Delta_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $\Gamma$ has a positive Kadison constant, then we can apply results by Br\"uning and Sunada to deduce that $\spec \Delta_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(\lambda)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,\lambda]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results., Comment: final version (26 pages, 2 figures). to appear in Rev. Math. Phys

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations., 16 pages; no figures; replaces math.SP/0403179; completely re-written

An eigenvalue problem is considered where the eigenvalue appears in the domain and on the boundary. This eigenvalue problem has a spectrum of discrete positive and negative eigenvalues. The smallest positive and the largest negative eigenvalue $\lambda_{\pm 1}$ can be characterized by a variational principle. We are mainly interested in obtaining non trivial upper bounds for $\lambda_{-1}$. We prove some domain monotonicity for certain special shapes using a kind of maximum principle derived by C. Bandle, J.v. Bellow and W. Reichel in [J. Eur. Math. Soc., 10 (2007), 73-104]. We then apply these bounds to the trace inequality.

International audience; By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the L2 norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrödinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators.

We study estimates involving the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold and L1-norms of exit time moments of Brownian motion. Our results generalize a classical inequality of Polya., Comment: 15 pages, added appendix on variational quotients, added lower bounds, improved exposition

In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In the particular case of a surface of nonnegative curvature, we prove that the surface is diffeomorphic to the Euclidean plane if the surface has tamed second fundamental form, and that the surface is isometric to the Euclidean plane if the surface has strongly tamed second fundamental form. In the last part of the paper we characterize the fundamental tone of any submanifold of tamed second fundamental form immersed in an ambient space with a pole and quadratic decay of the radial sectional curvatures., 19 pages. Title changed and several improvement of the main theorems are done. arXiv admin note: text overlap with arXiv:0805.0323 by other authors

This paper is concerned with non-selfadjoint elliptic problems having a principal part in divergence form and involving an indefinite weight. We study the asymptotic behavior of the principal eigenvalues when the zero order term becomes larger and larger. Use is made of a minimax formula for these principal eigenvalues.

This paper communicates recent results in theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schr\"odinger operators. In particular, we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schr\"odinger operators with spectral gaps. Second, we provide new ways of evaluating the resolvent norm for Schr\"odinger operators appearing in the complex scaling theory of resonances.

Let $B_1$ be a ball of radius $r_1$ in $S^n(\Hy^n)$, and let $B_0$ be a smaller ball of radius $r_0$ such that $\bar{B_0}\subset B_1$. For $S^n$ we consider $r_1< ��$. Let $u$ be a solution of the problem $-\La u =1$ in $\Om := B_1\setminus \bar{B_0}$ vanishing on the boundary. It is shown that the associated functional $J(\Om)$ is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on $\Om$ is maximal if and only if the balls are concentric., 10 pages