where primes denote derivatives with respect to x and We now follow the well-known point-canonical-transformation (PCT) recipe (cf. e.g., [19]) through the substitution ψ ( x ) = m ( x ) 1 / $^{4}$φ ( q ( x )) in (3) to imply (with q $^{′}$( x ) = √ m ( x )) and obtain a Schr¨ odinger equation of the form where Which would in turn (with m $^{′}$( x ) = ∂$_{x}$m ( x ) , and f $^{′}$( h ( x )) = ∂ h ( x $_{)}$f ( h ( x )) ) yield with Where δ ( x ) denotes Dirac delta function and δ $^{′}$( x ) = ∂$_{x}$δ ( x ) is the derivative of the Dirac delta function. Hereby, the terms associated with δ $^{′}$( x ) and δ ( x ) 2 V$_{eff}$ ( q ( x )) = G$_{1}$ m $^{′′}$( x ) m ( x ) 2 − G$_{2}$ m $^{′}$( x ) 2 m ( x ) 3 = G$_{1}$ δ $^{′}$( x ) f $^{′}$( h ( x )) f ( h ( x )) 2 + δ ( x ) 2 f ( h ( x )) 2 [ G$_{1}$f $^{′′}$( h ( x )) − G$_{2}$ f $^{′}$( h ( x )) 2 f ( h ( x )) ] , (9) G$_{1}$ = 1 8 (1 + 2 β ) ; G$_{2}$ = 1 2 [ α ( α + β + 1) + β + 9 16 ] (10) V$_{eff}$ ( q ) = ˜ V ( x ) + 1 2 ( 7 m $^{′}$( x ) 2 32 m ( x ) 3 − m $^{′′}$( x ) 8 m ( x ) 2 ) . (8) [ − 1 2 ∂ 2 q + V$_{eff}$ ( q ) ] φ ( q ) = Eφ ( q ) , (7) q ( x ) = ∫ x √ m ( t ) dt = ∫ x √ f ( h ( t )) dt = x √ f ( h ( x )) , (6) 5 g$_{1}$ = 1 4 (1 + β ) ; g$_{2}$ = 1 2 [ α ( α + β + 1) + β + 1] . (5)