He then was a doctoral student in the Leningrad University's theoretical physics group supervised by Vladimir Fock, and wrote thesis on the foundations of statistical mechanics entitled Mixing processes in phase space awarded by for the degree of Candidate of Science in 1941.
[1][4] During the German invasion of the Soviet Union and the Siege of Leningrad, Krylov was assigned to the air defense of the city.
He continued research work at Kazan for the Physical-Technical Institute of the Academy of Sciences of the Soviet Union, when the Institute was relocated due to the siege, while on active duty and defended dissertation The processes of relaxation of statistical systems and the criterion of mechanical instability awarded by the degree of Doctor of Science the following year.
He then worked at the various Soviet Union's academic institutes, in 1947 together with his supervisor coauthored the Fock-Krylov theorem[5][6] on quasi-stationary state decay in quantum mechanics, returned to Leningrad, but fell ill in 1946 and died due to sepsis caused by a streptococcus.
Thanks to this effort, Krylov's research results had received a permanent place in modern theoretical physics and have laid the foundations of dynamical systems theory and quantum mechanics.