## Abstract

It has been 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.

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## Acknowledgements

The authors would like to thank Ram Band for interesting discussions and helpful suggestions regarding the manuscript. G.B. acknowledges partial support from the NSF under Grant DMS-1410657. G.C. acknowledges the support of NSERC Grant RGPIN-2017-04259. J.L.M. was supported in part by NSF Applied Math Grant DMS-1312874 and NSF CAREER Grant DMS-1352353.

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## Appendix A. Example for a Rectangle

### Appendix A. Example for a Rectangle

Let us consider first the one-dimensional eigenvalue problem for the case \(q(x) = 0\) from Sect. 3. Namely, we wish to compute the eigenvalues \(\{ \lambda _n (\sigma ) \}\) for \(\sigma \ge 0\).

### A.1. \(\{ Z_k \} = \{ \frac{1}{2} \ell \}\)

The second Dirichlet eigenfunction for the Laplacian the interval \([0,\ell ]\) has a zero at \(\ell /2\). Using this nodal point to define the boundary conditions in \(\sigma \), as in Sect. 3, we look for the eigenvalues \(\lambda _n (\sigma )\). We will use the notation \(\lambda _n (\sigma ; 2)\) to denote the *n*th eigenvalue that arises from the spectral flow in \(\sigma \) set at the nodal point of the second Dirichlet eigenfunction. Symmetry considerations guarantee that the corresponding lowest eigenfunction, \(u_1 (x)= u_1 (x, \sigma ; 2)\), is symmetric with respect to \(\ell /2\). The eigenvalues \(\lambda _n (\sigma ; 2)\) in this case can be found by taking \(u_1 (x) = \sin (\kappa x)\) on \([0, \ell /2]\) for \(\kappa ^2 = \lambda _n\). Condition (17) gives

and hence

Thus, \(\lambda _1 (\sigma ; 2) = \kappa ^2\) is given as the implicit solution to (21) for finding the lowest eigenvalue.

### A.2. \(\{ Z_k \} = \{ \frac{1}{3} \ell , \frac{2}{3} \ell \}\)

Now, let us consider the next excited state, or the case of the nodal set given by 2 zeros equidistributed throughout the interval. As before, the lowest eigenfunction of \(L_\sigma \), denoted \(u_1 (x)= u_1 (x, \sigma ; 3)\), is symmetric with respect to \(\ell /2\) and we can write

Hence, conditions (16) and (17) at \(\ell /3\) imply

for \(c = u_1 (\ell /3)\). Solving out for *c*, we arrive at

which can be solved implicitly for \(\lambda _1 (\sigma ; 3) = \kappa ^2\).

A similar approach applies to find the second eigenfunction \(u_2 (x) = u_2 (x,\sigma ; 3)\), which is anti-symmetric with respect to \(\ell /2\). Following the same logic, we arrive at

which can be solved implicitly for \(\lambda _2 (\sigma ; 3) = \kappa ^2\).

### A.3. An example with nodal deficiency 3 on the rectangle

Let us now consider a rectangle of the form \([0,\pi ] \times [0, \alpha \pi ]\) with \(\alpha < 1\) but such that \(1-\alpha \ll 1\). We observe in this case that for the Laplacian with Dirichlet boundary conditions,

Therefore, the sixth eigenvalue \(\lambda _6 = \lambda _{1,3}\) has 3 nodal domains and therefore nodal deficiency 3, see Fig. 6.

Setting \(\lambda _* = \lambda _6 = \lambda _{1,3}\), we obtain the spectral flow

which was the flow depicted in Fig. 1(right). The above equations can be analyzed to show that \(\gamma _2, \gamma _4, \gamma _5\) all cross \(\gamma _6\) as \(\sigma \rightarrow \infty \), whereas \(\gamma _1\) and \(\gamma _3\) do not.

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Berkolaiko, G., Cox, G. & Marzuola, J.L. Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map.
*Lett Math Phys* **109, **1611–1623 (2019). https://doi.org/10.1007/s11005-019-01159-x

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### Keywords

- Nodal domains
- Nodal sets
- Dirichlet-to-Neumann map
- Spectral indices

### Mathematics Subject Classification

- 58J50
- 35B05
- 35P05
- 35J10