# Replacing a squared potential by a position-dependent mass

I'm studying the solutions of the Klein-Gordon and Dirac equations for a relativistic particle in a potential of the form $$V(x)=\left\lbrace\begin{array}{ll} 0, & x\in[0,L]\\V_0, & x\not\in[0,L]\end{array}\right.$$ where $V_0\to\infty$. In the papers i'm reading it says that, in order to avoid Klein paradox, we can replace this potential by a position-dependent mass such that $$m(x)=\left\lbrace\begin{array}{ll} m, & x\in[0,L]\\M\to\infty, & x\not\in[0,L]\end{array}\right.$$ Why are the two approximations to the problem equivalent?

The papers where I found that are here:

https://arxiv.org/abs/1711.06313

https://pdfs.semanticscholar.org/b4d4/9c62446394151bd2f437d449a17e96ed3eda.pdf

• Which two approximations? I see an attempt to modify the equation in order to obtain a desirable solution (no antiparticle appearing). The original equations produce Klein paradox. When $M\to\infty$, one may choose the limit $Mc^2\gg V_0$, sot the wall potential may not produce antiparticles due to lack of energy. – Vladimir Kalitvianski Aug 26 '18 at 16:43
• iopscience.iop.org/article/10.1088/0143-0807/17/1/004 – Qmechanic Aug 26 '18 at 16:49