For canonical systems, the start of the essential spectrum is at least half and at most about 2.62 times that of the diagonal system, with the 1/2 factor optimal; the diagonal value itself is pinned by limsup/liminf integrals of sin²φ.
Weidmann, Spectral Theory of Ordinary Differential Operato rs, Springer Lecture Notes 1258, Springer, Berlin, 1987
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The essential spectrum of canonical systems
For canonical systems, the start of the essential spectrum is at least half and at most about 2.62 times that of the diagonal system, with the 1/2 factor optimal; the diagonal value itself is pinned by limsup/liminf integrals of sin²φ.