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.
Main novelty: We are able to deal with the centrally asymmetric cases.
Presented at the III-rd Int. Workshop on classical and quantum integrable systems held during June 29 - July 4, 1998 in Yerevan, Armenia.
Full text in preparation, to appear in proceedings (JINR Dubna 1999, G.S.Pogosyan, editor).