Usually, the reflection probability R ( E ) of a particle of zero energy incident on a potential
which converges to zero asymptotically is found to be 1: ##IMG##
[http://ej.iop.org/images/0143-0807/38/2/025401/ejpaa4f96ieqn1.gif] {$R(0)=1$} . But earlier, a
paradoxical phenomenon of zero reflection at zero energy ( ##IMG##
[http://ej.iop.org/images/0143-0807/38/2/025401/ejpaa4f96ieqn2.gif] {$R(0)=0$} ) has been revealed
as a threshold anomaly. Extending the concept of half-bound state (HBS) of 3D, here we show that in
1D when a symmetric (asymmetric) attractive potential well possesses a zero-energy HBS, ##IMG##
[http://ej.iop.org/images/0143-0807/38/2/025401/ejpaa4f96ieqn3.gif] {$R(0)=0$} ##IMG##
[http://ej.iop.org/images/0143-0807/38/2/025401/ejpaa4f96ieqn4.gif] {$(R(0)\ll 1)$} . This can
happen only at some critical values q c of an effective parameter q of …