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This paper is a continuation and an extension of our recent work [3] on thegeometric structures of Laplacian eigenfunctions and their applications toinverse scattering problems. In [3], the analytic behaviour of the Laplacianeigenfunctions is investigated at a point where two nodal or generalisedsingular lines intersect. It reveals a certain intriguing property that thevanishing order of the eigenfunction at the intersecting point is related tothe rationality of the intersecting angle. In this paper, we consider the 3Dcounterpart of such a study by studying the analytic behaviours of theLaplacian eigenfunctions at places where nodal or generalised singular planesare intersect. Compared to the 2D case, the geometric situation is morecomplicated: the intersection of two planes generates an edge corner, whereasthe intersection of more than three planes generates a vertex corner. Weprovide a comprehensive characterisation for all of those cases. Moreover, weapply the spectral results to establish some novel unique identifiabilityresults for the geometric inverse problems of recovering the shape as well asthe (possible) surface impedance parameter by the associated scatteringfar-field measurements.

On nodal and generalized singular structures of Laplacian eigenfunctions and applications in $\mathbb{R}^3$