Geodesic
On a~problem of Bochner and von Neumann
Matematičeskie zametki, Tome 3 (1968) no. 5, pp. 529-538.

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The following theorem is proved. If there exists an everywhere dense set $\Gamma$ such that every solution of the equation $v_t'=-iA^*v$ satisfying the condition $v(0)\in\Gamma$ is defined on the entire axis and is bounded, then every compact solution of the equation $u'_t=iAu$ is an almost-periodic function. 
@article{MZM_1968_3_5_a4,
     author = {V. V. Zhikov},
     title = {On a~problem of {Bochner} and von {Neumann}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {529--538},
     publisher = {mathdoc},
     volume = {3},
     number = {5},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_5_a4/}
}
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V. V. Zhikov. On a~problem of Bochner and von Neumann. Matematičeskie zametki, Tome 3 (1968) no. 5, pp. 529-538. http://geodesic.mathdoc.fr/item/MZM_1968_3_5_a4/