**On the inverse Sturm - Liouville problem
on a finite interval.**

In this paper the following boundary problem
is being considered:

-y¢¢ + q(x)y = ly, x Î
(0,p), q Î L _{p}(0,p), (p = 1;2),(1)

y¢(0) - hy(0) = 0, y¢(p) + Hy(p) = 0, (2)

h and H are real
numbers.

The following question is solved:
Under which conditions the sequences l_{0} <
l_{1} < l_{2} < ... and 0 < r_{0} < r_{1}
< ... are the eigenvalues and the normalizing numbers for some boundary
problem (1), (2). Tne answer is given in terms of an asymptotical behaviour of
these sequences.

It is also studied the
problem of unique restoration of the boundary problem (1) , (2) by spectrum only
and numbers h and H.