Local-scaling point transformations

 

Suppose $\{\varphi_{\alpha}({\mbox{{\boldmath {$r$ }}}})\}$ represents a complete set of orthonormal single-particle wave functions depending on the spatial coordinate ${\mbox{{\boldmath {$r$ }}}}$. (To simplify the presentation, we suppress the spin and isospin labels here.) Then, one can introduce a local-scaling point transformation (LST) of the three dimensional vector space, which is a generalization of the analogous spherically-symmetric LST [12,13,14], namely

$$\begin{array}{ll} x\longrightarrow & x^{\prime}\equiv x^{\pri... ...ime}\equiv z^{\prime}(x,y,z)=\frac{z}{{r}}f_z({r}), \end{array}$$ (1)

where r= $\sqrt{x^2+y^2+z^2}$.

The LST functions f_k(r), k=x, y, or z, should have mathematical properties ensuring that (1) is a valid invertible transformation of the three-dimensional space. In particular, f_k(r) should be monotonic functions of r such that

$$\begin{array}{lll} f_k(0)=0 & \mbox{~~and~~} & f_k(\infty )=\infty \end{array} ,$$ (2)

and should lead to a non-vanishing Jacobian of the LST (1), i.e.,

 

$$\equiv \frac{\partial(x^{\prime},y^{\prime},z^{\prime})}{\partial(x,y,z)}$$ D

$$\frac{x^2f^{\prime}_xf_yf_z+y^2f_xf^{\prime}_yf_z +z^2f_xf_yf^{\prime}_z}{{r}^{4}}\neq 0,$$ = (3)

where primes denote derivatives with respect to r.

When we apply the LST ( 1) to the set of wave functions $\varphi_{\alpha}({\mbox{{\boldmath {$r$ }}}})$, we obtain another set of single-particle wave functions

$$\psi _{\alpha}(x,y,z)=D^{1/2}\varphi_{\alpha} \textstyle{ \le... ...({r}), \frac{y}{{r}}f_y({r}), \frac{z}{{r}}f_z({r}) \right)} .$$ (4)

Due to the factor D^1/2 entering Eq.(4), the LST of wave functions is unitary and the new wave functions $\psi _{\alpha}({\mbox{{\boldmath {$r$ }}}})$ are automatically orthonormal, i.e., $\langle\psi_{\alpha}\vert\psi_{\beta} \rangle$= $\langle\varphi_{\alpha}\vert\varphi_{\beta}\rangle$= $\delta_{\alpha\beta}$.

Summarizing, the LST (1) generates from a given complete set of orthonormal single-particle wave functions another orthonormal and complete set of single-particle wave functions (4) depending on three almost-arbitrary scalar LST functions f_k(r). The freedom in the choice of f_k(r) provides great flexibility in the new set $\{\psi _{\alpha}({\mbox{{\boldmath {$r$ }}}})\}$, and this opens up the possibility of improving on undesired properties of the initial set. This is the motivation for the present study in which we use the LST to modify the incorrect asymptotic properties of deformed HO wave functions.