arXiv subjects

Demetrios A. Pliakis

Publications and source records attributed to Demetrios A. Pliakis.

5 recordsLinked to original sources

arXiv2022

Small time heat expansion of the laplacian on an analytic hypersurface with an isolated singularity

I prove the existence of small time heat expansion for the Laplace operator on an analytic hypersurface with an isolated singularity. First we obtain a local parametrization of the hypersurface near the singularity. We introduce the notion of quasihomogeneous tangent cone. Then perturb the parametrization of the cone employing a Newton scheme and obtain a parametrization with functions of specific form. These allow us to obtain local models for the Laplace operator near the singularity. These are operators with irregular singularities. We derive the estimates required by singular asymptotics.

math.AP↗

arXiv2014

Eigenvalue gaps for the Laplacian on hypersurfaces of the sphere

We provide lower estimates for the eigenvalues of the laplacian for hypersurfaces of the round sphere.

math.AP↗

arXiv2013

The size of the nodal sets for the eigenfunctions of the smooth laplacian

We prove estimates for eigenfunctions on a manifold equipped with a smooth metric. We use these estimates in order estimate the size of their nodal sets.

math.AP↗

arXiv2013

Spectrally determined singularities in a potential with an inverse square initial term

We study the inverse spectral problem for Bessel type operators with potential (v(x)): (H_κ=-\partial_x^2+\frac{k}{x^2}+v(x)). The potential is assumed smooth in ((0,R)) and with an asymptotic expansion in powers and logarithms as (x\rightarrow 0^+, v(x)=O(x^α), α>-2). Specifically we show that the coefficients of the asymptotic expansion of the potential are spectrally determined. This is achieved by computing the expansion of the trace of the resolvent of this operator which is spectrally determined and elaborating the relation of the expansion of the resolvent with that of the potential, through the singular asymptotics lemma.

math-ph↗

arXiv2010

On certain generalized Hardy's inequalities and applications

We prove certain generalization of Hardy's inequality where the "boundary defining function" is replaced by a polynomial defining a singular algebraic variety. An application is given on the existence of a small time heat trace expansion for a Schrödinger operator with mild singularities along this algebraic set.

math.AP↗