Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

Abstract

It was recently shown that the nodal deficiency of an eigenfunction is encoded in the spectrum of the Dirichlet-to-Neumann operators for the eigenfunction's positive and negative nodal domains. While originally derived using symplectic methods, this result can also be understood through the spectral flow for a family of boundary conditions imposed on the nodal set, or, equivalently, a family of operators with delta function potentials supported on the nodal set. In this paper we explicitly describe this flow for a Schrödinger operator with separable potential on a rectangular domain, and determine a mechanism by which lower energy eigenfunctions do or do not contribute to the nodal deficiency.

Spectral flow for pair compatible equipartitions

“…Our main goal is to extend the construction and analysis of spectral flow and Dirichlet-to-Neumann operators, which was done for nodal partitions in [1], to spectral equipartitions that satisfy the PCC. We describe briefly the construction for nodal domains first, and return to our setting in Section 1.3.…”

“…To state the main result of [1], we need to introduce Dirichlet-to-Neumann operators. We only do this at an intuitive level at this point, and refer the reader to [2] for more details.…”

“…considered as an unbounded operator in L 2 ðCÞ: This is (for a suitable k, see below) the operator K þ ðeÞ þ K À ðeÞ introduced in [1] but we have preferred this presentation which is easier to generalize. Formally, the operator is defined via the quadratic form with form domain H 1=2 ðCÞ: 1,3]). If e > 0 is sufficiently small, then…”

“…The number k à À lðu Ã Þ in the L.H.S above is non-negative due to Courant's nodal theorem. It is usually called the nodal deficiency of the eigenfunction u à (see for example [1]). If k à is a simple eigenvalue of S X then the R.H.S.…”

“…We follow the approach of [1] but to be able to treat non-admissible partitions we introduce the AB Hamiltonian attached to the partition. Thus, let D be a k-partition in X.…”