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Pedro Freitas

Publications and source records attributed to Pedro Freitas.

47 records · Page 3Linked to original sources
arXiv2007

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.

math.SP↗
arXiv2007

A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains

We show that as the ratio between the first Dirichlet eigenvalues of a convex domain and of the ball with the same volume becomes large, the same must happen to the corresponding ratio of isoperimetric constants. The proof is based on the generalization to arbitrary dimensions of Polya and Szego's 1951 upper bound for the first eigenvalue of the Dirichlet Laplacian on planar star-shaped domains which depends on the support function of the domain. As a by-product, we also obtain a sharp upper bound for the spectral gap of convex domains.

math.SP↗
arXiv2007

Location of the nodal set for thin curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in the limit when the volume of the cross-section diminishes. We show that spectral properties of the Laplacian are, in this limit, approximated well by those of the sum of the Dirichlet Laplacian in the cross-section and a one-dimensional Schrodinger operator whose potential is expressed solely in terms of the first curvature of the reference curve. In particular, we establish the convergence of eigenvalues, the uniform convergence of eigenfunctions and locate the nodal set of the Dirichlet Laplacian in the tube near nodal points of the one-dimensional Schrodinger operator. As a consequence, we prove the "nodal-line conjecture" for a class of non-convex and possibly multiply connected domains. The results are based on a perturbation theory developed for Schrodinger-type operators in a straight tube of diminishing cross-section.

math.SP↗
arXiv2007

Waveguides with combined Dirichlet and Robin boundary conditions

We consider the Laplacian in a curved two-dimensional strip of constant width squeezed between two curves, subject to Dirichlet boundary conditions on one of the curves and variable Robin boundary conditions on the other. We prove that, for certain types of Robin boundary conditions, the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Laplacian in a Dirichlet-Robin annulus determined by the geometry of the strip. Moreover, we show that an appropriate combination of the geometric setting and boundary conditions leads to a Hardy-type inequality in infinite strips. As an application, we derive certain stability of the spectrum for the Laplacian in Dirichlet-Neumann strips along a class of curves of sign-changing curvature, improving in this way an initial result of Dittrich and Kriz.

math-ph↗
arXiv2006

Hearing the weights of weighted projective planes

Which properties of an orbifold can we ``hear,'' i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kähler orbifolds: weighted projective planes $M:=\C P^2(N_1,N_2,N_3)$ with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on $M$ determine the weights $N_1$, $N_2$, and $N_3$. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of $N_1$, $N_2$, and $N_3$, we can hear whether $M$ is endowed with an extremal Kähler metric.

math.DG↗
arXiv2005

A Li-type criterion for zero-free half-planes of Riemann's zeta function

We define a sequence of real functions which coincide with Li's coefficients at one and which allow us to extend Li's criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip $0<\Re(z)<1$. We study some of the properties of these functions, including their oscillatory behaviour.

math.NT↗
arXiv2004

Instability results for the damped wave equation in unbounded domains

We extend some previous results for the damped wave equation in bounded domains in Euclidean spaces to the unbounded case. In particular, we show that if the damping term is of the form $αa$ with bounded $a$ taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive $α$. In order to prove these results, we generalize some existing results on the asymptotic behaviour of eigencurves of one-parameter families of Schrodinger operators to the unbounded case, which we believe to be of interest in their own right.

math.AP↗
arXiv2001

On the invariant spectrum of $S^1$-invariant metrics on $S^2$

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4π$, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard $S^1$-action on $S^2$, one can restrict the Laplace operator to the subspace of $S^1$-invariant functions and consider its spectrum there. The corresponding eigenvalues will be called invariant eigenvalues, and the purpose of this paper is to analyse its possible values. We first show that there is no general analogue of Hersch's theorem, by exhibiting explicit families of $S^1$-invariant metrics with total area $4π$ where the first invariant eigenvalue ranges through any value between 0 and $\infty$. We then restrict ourselves to $S^1$-invariant metrics that can be embedded in ${\bf R}^3$ as surfaces of revolution. For this subclass we are able to provide optimal upper bounds for all invariant eigenvalues. As a consequence, we obtain an analogue of Hersch's theorem with an optimal upper bound (greater than 2 and geometrically interesting). This subclass of metrics on $S^2$ includes all $S^1$-invariant metrics with non-negative Gauss curvature. One of the key ideas in the proofs of these results comes from symplectic geometry, and amounts to the use of the moment map of the $S^1$-action as a coordinate function on $S^2$.

math.SP↗
arXiv2001

A Heawood-type result for the algebraic connectivity of graphs on surfaces

We prove that the algebraic connectivity a(G) of a graph embedded on a nonplanar surface satisfies a Heawood-type result. More precisely, it is shown that the algebraic connectivity of a surface S, defined as the supremum of a(G) over all graphs that can be embedded in S, is equal to the chromatic number of S. Furthermore, and with the possible exception of the Klein bottle, we prove that this bound is attained only in the case of the maximal complete graph that can be embedded in S. In the planar case, we show that, at least for some classes of graphs which include the set of regular graphs, a(G) is less than or equal to four. As an application of these results and techniques, we obtain a lower bound for the genus of Ramanujan graphs. We also present some bounds for the asymptotic behaviour of a(G) for certain classes of graphs as the number of vertices goes to infinity.

math.CO↗
arXiv2000

On minimal eigenvalues of Schrodinger operators on manifolds

We consider the problem of minimizing the eigenvalues of the Schrödinger operator $H=-Δ+αF(\ka)$ ($α>0$) on a compact $n-$manifold subject to the restriction that $\ka$ has a given fixed average $\ka_{0}$. In the one-dimensional case our results imply in particular that for $F(\ka)=\ka^{2}$ the constant potential fails to minimize the principal eigenvalue for $α>α_{c}=μ_{1}/(4\ka_{0}^{2})$, where $μ_{1}$ is the first nonzero eigenvalue of $-Δ$. This complements a result by Exner, Harrell and Loss (math-ph/9901022), showing that the critical value where the circle stops being a minimizer for a class of Schrödinger operators penalized by curvature is given by $α_{c}$. Furthermore, we show that the value of $μ_{1}/4$ remains the infimum for all $α>α_{c}$. Using these results, we obtain a sharp lower bound for the principal eigenvalue for a general potential. In higher dimensions we prove a (weak) local version of these results for a general class of potentials $F(\ka)$, and then show that globally the infimum for the first and also for higher eigenvalues is actually given by the corresponding eigenvalues of the Laplace-Beltrami operator and is never attained.

math-ph↗