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Miren Zubeldia

Publications and source records attributed to Miren Zubeldia.

6 recordsLinked to original sources
arXiv2015

Discreteness of Transmission Eigenvalues for Higher-Order Main Terms and Perturbations

In this paper we extend Sylvester's approach via upper triangular compact operators to establish the discreteness of transmission eigenvalues for higher-order main terms and higher-order perturbations. The coefficients of the perturbations must be sufficiently smooth and the coefficients of the higher-order terms of the perturbation must vanish in a neighbourhood of the boundary of the underlying domain. The zeroeth order term must satisfy a suitable coercivity condition in a neighbourhood of the boundary.

math.SP↗
arXiv2014

The inverse Robin boundary value problem in a half-space

We study the inverse Robin problem for the Schrödinger equation in a half-space. The potential is assumed to be compactly supported. We first solve the direct problem for dimensions two and three. We then show that the Robin-to-Robin map uniquely determines the potential q.

math.AP↗
arXiv2013

Limiting absorption principle for the electromagnetic Helmholtz equation with singular potentials

We study the following Helmholtz equation $$ (\nabla +iA(x))^{2} u+ V_{1}(x) u + V_{2}(x) u + λu = f(x) $$ in $\mathbb{R}^d$ with magnetic and electric potentials that are singular at the origin and decay at infinity. We prove the existence of a unique solution satisfying a suitable Sommerfeld radiation condition, together with some a priori estimates. We use the limiting absorption method and a multiplier technique of Morawetz type.

math.AP↗
arXiv2012

Energy concentration and explicit Sommerfeld radiation condition for the electromagnetic Helmholtz equation

We study the electromagnetic Helmholtz equation \notag (\nabla + ib(x))^{2}u(x) + n(x)u(x) = f(x), \quad x\in\Rd with the magnetic vector potential $b(x)$ and $n(x)$ a variable index of refraction that does not necessarily converge to a constant at infinity, but can have an angular dependency like $n(x) \to n_{\infty}(\frac{x}{|x|})$ as $|x|\to\infty$. We prove an explicit Sommerfeld radiation condition \notag \int_{\Rd} |\D u - in_{\infty}^{1/2}\frac{x}{|x|}u|^{2} \frac{dx}{1+|x)} < + \infty for solutions obtained from the limiting absorption principle and we also give a new energy estimate \notag \int_{\Rd}| \nabla_ωn_{\infty}(\frac{x}{|x|})|^{2}\frac{|u|^{2}}{1+|x|} dx < +\infty, which explains the main physical effect of the angular dependence of $n$ at infinity and deduces that the energy concentrates in the directions given by the critical points of the potential.

math.AP↗