Question: finding the root of a complicated funtion

I have quite a complicated function given by the de solve part of this equation, it gives the correct plot. I now need to find the points where it crosses the axis. From the plot i can see this is around E=0.25, 2.2.

ii have no idea how to solve this using the fsolve function. I would be very greatfull for any help any one could provide.
 
Thank you in advance.

 A:= 50: 
VB:= Vector(A):
VE:=Vector(A) :
  for i from 1 by 1 to A do 
  j:=i-1:
  E:=((3*j)/(A))-3:
  DE := (-1/(2)*diff(diff(F(x),x),x) +(x^(4)-3)(*exp((-x^(2))/(2)))^*F(x))=E*F(x):
ini := F(-10)=exp(-10*(-2E)^(0.5)), D(F)(-10)=exp(-10*(-2E)^(0.5))*(-2E)^(0.5): 
dsol :=dsolve({DE, ini}, F(x),numeric, range = -10..10):  
J:= eval(F(x),dsol(10));
K:=eval(diff(F(x),x),dsol(10));
Bx:= ((((-2*E)^0.5)*J+K)*exp(-10*(-2*E)^0.5 ))/(2*(-2*E)^0.5); 
B := Bx; 
VE(i):=E:
  if B>1 
then VB(i):= 1 
else VB(i):=B 
end if:  
end do:
plot(VE,VB) ;