.
and 
.
.
solution for 
is 
with 
so 
, 
.
.
On first excited state, you have:
and 
This gives eigenvalues: 
. So eigenstates are 
.
If symmetry under x,y interchange then diagonal basis is 
.
Parity (interchange of x,-x), then parity of perturbation splits parity odd systems.
diagonal in parity eigenstates. Ex: 
goes to 
parity odd or 
for
parity even.