We use the factorization method to find the exact eigenvalues and eigenfunctions for a particle in a box with the delta function potential V (x) = λδ(x − x 0 ). We show that the presence of the potential results in the discontinuity of the corresponding ladder operators. The presence of the delta function potential allows us to obtain the full spectrum in the first step of the factorization procedure even in the weak coupling limit λ → 0.

The upward motion and deformation of a single large-scale two-dimensional planar gas bubble rising in an otherwise quiescent viscoelastic liquid obeying the Oldroyd-B rheological model is numerically investigated using the weakly-compressible smoothed particle hydrodynamics (WC-SPH) method. It is shown that unlike the incompressible version of this mesh-less method (I-SPH), the WC-SPH method combined with the continuum surface force (CSF) idea for modelling surface tension is well capable of capturing the cusped-shape trailing edge during its rise in viscoelastic liquids. The failure of the I-SPH method for predicting the cusp is attributed to the several stages of weighted arithmetic and harmonic interpolations used for smoothing the transport parameters of the constituents. The WC-SPH code developed in the present work has enabled us to obtain results at high density and viscosity ratios not reported in similar works. The results suggest that WC-SPH is better suited than I-SPH for handling bubble rise problem in viscoelastic liquids even at large density and viscosity ratios typical of underwater explosions.

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