arXiv subjects

B. A. Siudeja

Publications and source records attributed to B. A. Siudeja.

3 recordsLinked to original sources

arXiv2014

Steklov eigenvalues and quasiconformal maps of simply connected planar domains

We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains, for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings.

math.SP↗

arXiv2014

Magnetic spectral bounds on starlike plane domains

We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that $\sum_{j=1}^n Φ\big( λ_j A/G \big)$ is maximal for a disk whenever $Φ$ is concave increasing, $n \geq 1$, the domain has area $A$, and $λ_j$ is the $j$-th Dirichlet eigenvalue of the magnetic Laplacian $\big( i\nabla+ \fracβ{2A}(-x_2,x_1) \big)^2$. Here the flux $β$ is constant, and the scale invariant factor $G$ penalizes deviations from roundness, meaning $G \geq 1$ for all domains and $G=1$ for disks.

math-ph↗

arXiv2010

Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane

The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio $\text{(area)}^3/\text{(moment of inertia)}$ for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.

math.SP↗