Nodal domains and spectral critical partitions on graphs
Seminar Room 1, Newton Institute
AbstractThe $k$-th eigenfunction of a Schrodinger operator on a bounded regular domain $\Omega$ with Dirichlet boundary conditions defines a partition of $\Omega$ into $n$ nodal subdomains. A famous result by Courant establishes that $n \leq k$; the number $k-n$ will be referred to as the nodal deficiency. The nodal subdomains, when endowed with Dirichlet boundary conditions, have equal first eigenvalue, which coincides with the $k$-th eigenvalue of the original Schrodinger problem. Additionally, the partition is bipartite, i.e. it consists of positive and negative subdomains (corresponding to the sign of the eigenfunction), with two domains of the same sign not sharing a boundary. Conversely, for a given partition, define the energy of the partition to be the largest of the first Dirichlet eigenvalues of its subdomains. An $n$-partition with the minimal energy is called the minimal $n$-partition. It is interesting to relate the extremal properties of the partitions to the eigenstates of the operator on $\Omega$. Recently, Helffer, Hoffmann-Ostenhof and Terracini proved that $n$-th minimal partition is bipartite if and only if it corresponds to a Courant-sharp eigenfunction (an eigenfunction with nodal deficiency zero). We study partitions on quantum graphs and discover a complete characterization of eigenfunctions as critical equipartitions. More precisely, equipartitions are partitions with all first eigenvalues equal. We parameterize the manifold of all equipartitions and consider the energy of an $n$-equipartition as a function on this manifold. For a generic graph and large enough $n$ we establish the following theorem: a critical point of the energy function with $b$ unstable directions is a bipartite equipartition if and only if it corresponds to an eigenfunction with nodal deficiency $b$. Since by constructions it has $n$ nodal domains it is therefore the $n+b$-th eigenfunction in the spectral sequence. Since at a minimum the number of unstable directions is $b=0$, our results include the quantum graph analogue of the results of Helffer et al. They also provide a new formulation of known bounds on the number of nodal domains on generic graphs. This is joint work with Rami Band, Hillel Raz and Uzy Smilansky.
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