We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity
f
(
ρ
)
=
ρ
σ
f(\rho ) = \rho ^\sigma
, where
ρ
=
|
ψ
|
2
\rho =|\psi |^2
is the density with
ψ
\psi
the wave function and
σ
>
0
\sigma >0
is the exponent of the semi-smooth nonlinearity. Under the assumption of
H
2
H^2
-solution of the NLSE, we prove error bounds at
O
(
τ
1
2
+
σ
+
h
1
+
2
σ
)
O(\tau ^{\frac {1}{2}+\sigma } + h^{1+2\sigma })
and
O
(
τ
+
h
2
)
O(\tau + h^{2})
in
L
2
L^2
-norm for
0
>
σ
≤
1
2
0>\sigma \leq \frac {1}{2}
and
σ
≥
1
2
\sigma \geq \frac {1}{2}
, respectively, and an error bound at
O
(
τ
1
2
+
h
)
O(\tau ^\frac {1}{2} + h)
in
H
1
H^1
-norm for
σ
≥
1
2
\sigma \geq \frac {1}{2}
, where
h
h
and
τ
\tau
are the mesh size and time step size, respectively. In addition, when
1
2
>
σ
>
1
\frac {1}{2}>\sigma >1
and under the assumption of
H
3
H^3
-solution of the NLSE, we show an error bound at
O
(
τ
σ
+
h
2
σ
)
O(\tau ^{\sigma } + h^{2\sigma })
in
H
1
H^1
-norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional
L
2
L^2
-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of
0
>
σ
≤
1
2
0 > \sigma \leq \frac {1}{2}
, and to establish an
l
∞
l^\infty
-conditional
H
1
H^1
-stability to obtain the
l
∞
l^\infty
-bound of the numerical solution by using the mathematical induction and the error estimates for the case of
σ
≥
1
2
\sigma \ge \frac {1}{2}
; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.