Spectral problems on triangles and disks: Extremizers and ground states

Abstract

In this dissertation we study how the energy levels of the Laplacian depend upon the shape of the domain, and identify the ground state energy level of the magnetic Laplacian on the disk. Although we consider these different questions we can find some kind of unity in the sense that we are looking at the eigenvalues (energy levels) of the operator. In the first case, with no magnetic field, we can say a lot about how the eigenvalues depend on the shape of the triangular domain. On the other hand, in the second case, with magnetic fi eld, it is much harder to prove properties of the energy levels. Even identifying what the ground state is for the disk is a challenge.

In the first part of the thesis, we prove that among all triangles of given diameter, the equilateral triangle minimizes the sum of the first n eigenvalues of the Neumann Laplacian, when n is greater than or equal to 3. The result fails for n = 2, because the second eigenvalue is known to be minimal for the degenerate acute isosceles triangle (rather than for the equilateral), while the first eigenvalue (n=1) is 0 for every triangle. We show the third eigenvalue is minimal for the equilateral triangle.

The second part of the thesis is concerned with the properties of Dirichlet eigenvalues for the magnetic Laplacian on the unit disk. We find an orthonormal basis of eigenfunctions for the magnetic Laplacian on a disk with Dirichlet boundary condition and explicitly identify the ground state of the magnetic Laplacian. Then we have the symmetry of eigenfunctions and eigenvalues with respect to angular momentum and magnetic field strength. Lastly we prove that positive angular momentum gives lower energy and establish the properties of magnetic eigenvalues; As angular momentum increases the energy level goes up, the ground state is radial and positive, and we find asymptotic behavior of the magnetic eigenvalues as the fi eld strength tends to infinity.