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.
Kac, A criterion for the discreteness of a singular canonical system, Funct
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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.