On Diagonalizations of One-Parameter Unitary Groups Depending on Time Direction

The paper studies tridiagonal (Jacobi-type) realizations of unitary representations of the Lie algebra $$su(1,1)$$ in the space of square-summable sequences. Explicit similarity transformations reducing these operators to diagonal form are obtained; in the general case, these transformations are nonunitary and may be unbounded. For representations of elliptic elements of the Lie algebra $$su(1,1)$$ , a nonuniqueness of such a diagonalization is revealed, together with its fundamental distinction from spectral diagonalization. For the corresponding one-parameter unitary groups, this leads to diagonalizations that depend on the direction of time.