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We sharpen a classical result on the spectral asymptotics of the boundaryvalue problems for self-adjoint ordinary differential operator. Using thisresult we obtain the exact $L_2$-small ball asymptotics for a new class of zeromean Gaussian processes. This class includes, in particular, integratedgeneralized Slepian process, integrated centered Wiener process and integratedcentered Brownian bridge.

Exact L_2-small ball asymptotics of Gaussian processes and the spectrum of boundary value problems with "non-separated" boundary conditions