In 1977 the mathematicians Landau and Widom showed that for a rectangularly band‑limited signal observed through a finite length window the eigenvalues of the resulting concentration matrix form two tight clusters, one near one and one near zero. The narrow band of values that lies between these clusters – the so‑called “plunge” – has a width that does not stay fixed but grows logarithmically with the number of samples N. The growth rate is universal: it equals 1/π² times log N, independent of the bandwidth or of the threshold chosen to separate the two clusters.

The result can be written as : for any threshold α between zero and one, the number of eigenvalues exceeding α is 2 W N + (1/π²) log (1−α)/α · log N + o(log N). The symmetric plunge count (eigenvalues between α and 1−α) therefore equals 2 / π² · log (1−α)/α · log N.

This behaviour has a few consequences. First, when a band is split into M disjoint sub‑bands the plunge width simply multiplies by M – every band edge contributes the same 1/π² factor. Second, if the symbol of the concentration operator is smooth rather than piecewise constant, the classic logarithmic plunge does not appear; the spectrum becomes continuous and no sharp two‑cluster structure emerges. In that smooth case the useful information is instead encoded in a power‑law decay governed by a spectral‑edge exponent.

These findings come from a careful application of Szegő’s theorem to Toeplitz concentration operators. The theorem tells how the symbol’s shape controls the fine structure of the eigenvalues: for an elementwise p‑th Hadamard power of the kernel, one obtains a power‑law edge with exponent p−1.

Together the results set a clear theoretical benchmark for interpreting the spectral profile of time‑frequency concentration operators. They also explain why the plunge phenomenon is universal for rectangular band limits, while smooth band shapes lead to a different, continuously decaying spectrum.