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The spectral form factor is a dynamical probe for level statistics of quantumsystems. The early-time behaviour is commonly interpreted as a characterizationof two-point correlations at large separation. We argue that thisinterpretation can be too restrictive by indicating that the self-correlationimposes a constraint on the spectral form factor integrated over time. Moregenerally, we indicate that each expansion coefficient of the two-pointcorrelation function imposes a constraint on the properly weightedtime-integrated spectral form factor. We discuss how these constraints canaffect the interpretation of the spectral form factor as a probe forergodicity. We propose a new probe, which eliminates the effect of theconstraint imposed by the self-correlation. The use of this probe isdemonstrated for a model of randomly incomplete spectra and a Floquet modelsupporting many-body localization.

Sensitivity of the spectral form factor to short-range level statistics