**Solutions of certain
Schroedinger equations are sought via certain power
series. **

**Firstly: The action of H is
assumed to create a non-numerical Krylov-Lanczos basis,
triangularizing Hamiltonian matrix H and making the
wavefuctions easily defined power-by-power, recurrently.**

**Secondly: Matrices H-E are
assumed to acquire Hesssenberg form, trivializing thereby
the determination of energies.**

**We construct examples
satisfying these two assumptions. In a way generalizing
certain solvable (sc. shape-invariant) models, we get new
exactly solvable models with bound states quantized via
their smoothness at zero.**