Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics

In paper approach of double complex SUSY-transformations with not coincident complex energies of transformation is developed, allowing to deform given real potential $V_{1}$ with obtaining exact solutions. The explicit solutions of the deformation of shape of the potential, its wave function at arbitrary energy, not coincident with energies of transformation, wave functions at the energies of transformation are obtained, condition of keeping of continuity of the solutions and isospectral condition are determined. Using a rectangular well of finite width with infinitely high walls as the starting $V_{1}$ with discrete energy spectrum, by the proposed approach new types of deformation of this potential with deformation of the energy spectrum as a whole have been obtained. The new potential contains the rectangular well as own partial case (with simultaneous transformation of the shape of this new potential, energy spectrum, wave functions of all bound states, wave function at arbitrary energy into corresponding characteristics of the rectangular well at needed choice of parameters). Using null potential as the starting $V_{1}$ with continuous energy spectrum, new form of reflectionless real potential has been constructed. This potential generalizes well-known reflectionless potential of the type $V_{\rm ref}(x) = A^{2}(1-2 {\rm sech}^{2}{Ax})$, allowing: to pull down tails of the potential $V_{\rm ref}$ in the asymptotic regions up to zero (with keeping of nonzero depth); to pull down continuously the depth of the hole; to displace arbitrary along axis $x$ the hole with its passing through zero; to create and to increase the second hole, transforming $V_{\rm ref}$ into double-well potential; to control continuously and simply the asymmetry of the shape of such reflectionless potential.