Operators with spectral density increasing as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∼</mml:

Motivated by a putative model of black holes as quantum objects, we consider what types of operators would have a corresponding spectrum. We find that there are indeed such operators, but of a rather unusual type, and for which the wave functions are only barely localized. We point out a tension between such almost delocalized states and black holes as compact objects.