Dirichlet, one may think of the solution for $\beta=0$ as the zero-energy limit of a scattering state, while that for $\beta=1$ as the zero-energy limit of a bound state. For the case (ii), the attractive potential problem is ill-defined, where the energy of the single-particle ground state (in the sense of the potential problem) collapses to $-\infty$ [@Landau; @Sutherland]. Thus, the SSD problem does not have a proper single-particle excited state with positive energy, unlike the SSD for the lattice systems. We conclude that the SSD problem in continuous space is well defined only below the Fermi level.

Geometric factor and lattice SSD
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In this section, we discuss the relation between the above continuous-space SSD and its lattice version, in the former of which a geometric factor associated with the inner product of the SSD eigenfunctions is of particular importance. So far, we have not defined the inner product of the SSD eigenfunctions. This is because we should pay a particular attention to the geometric factor in the continuous SSD, unlike the SSD for lattice systems. For instance, the standard definition of the inner product for the plane waves fails to satisfy the orthogonality relation among $\psi^{N-\beta}_\beta(x)$ ($\beta=1,2, ..., 