We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R² for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a smaller concentric disk in function of their size.

Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic χ of metric graphs to the spectrum of their standard Laplacian. These ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured. In the present work we analyse sufficient hypotheses which guarantee the successful recovery of χ. We also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required. Finally, we compare our findings with numerical examples-surprisingly, just a few dozens of eigenvalues can be enough.

We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet–Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion’s leading term. This allows inferring some remarkable consequences for Aharonov–Bohm eigenvalues when the singular part of the operator has two coalescing poles.

We show that equality in Courant’s nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in an open, bounded, and connected subset of Rn with a C1,1 boundary, when n≥2. This result is analogous to the theorem proved by Pleijel in 1956 for the Dirichlet Laplacian. We also show that the argument and the result extend to a class of Robin boundary conditions.

We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov–Bohm operators with two colliding poles moving on an axis of symmetry of the domain.