V[r_,a_,b_,c_]=a*((1-Exp[-Sqrt[b/(2a)](r-c)])^2-1)
DV[r_,a_,b_,c_]=D[V[r,a,b,c],{r,1}]
DV2[r_,a_,b_,c_]=D[DV[r,a,b,c],{r,1}]

Assuming[a>0&&b>0&&c>0&&r>0,Solve[DV[r,a,b,c]==0,r]]

DV2[c,a,b,c]

Manipulate[Plot[V[r,a,b,c],{r,0,6}],{a,0.1,5},{b,0.1,5},{c,0.1,5}]

Limit[V[r,1,1,1],r->∞]

Plot[V[r,1,1,2],{r,0,10}]

V[c,a,b,c]

VL[x_]=-2x;
VR[x_,b_,c_]=1/2*x^2+b*x+c;

V2[x_,b_,c_]=VL[x]*HeavisideTheta[-x]+VR[x,b,c]*HeavisideTheta[x]
DVR[x_,b_,c_]=D[VR[x,b,c],{x,1}]
D2VR[x_,b_,c_]=D[VR[x,b,c],{x,2}]

Plot[V2[x,1,1],{x,-2,2}]

row=Solve[{VR[0,b,c]==0,DVR[0,b,c]==-2},{b,c}]

V3[x_]=V2[x,b,c]/.row[[1]]

Plot[V3[x],{x,-15,15}]

ϕ[x_,i_]=1/Sqrt[i!*2^i*Sqrt[π]]HermiteH[i,x-2]*Exp[-(x-2)^2/2]

TVij[i_,j_]:=-1/2NIntegrate[ϕ[x,i]D[ϕ[x,j],{x,2}],{x,-∞,∞}]+NIntegrate[ϕ[x,i]V3[x]ϕ[x,j],{x,-∞,∞}]
Sij[i_,j_]:=NIntegrate[ϕ[x,i]ϕ[x,j],{x,-∞,∞}]

Hij[i_,j_]:=TVij[i,j]

H=Table[Hij[i-1,j-1],{i,1,21},{j,1,21}];
S=Table[Sij[i-1,j-1],{i,1,21},{j,1,21}];
v=Table[ϕ[x,i-1],{i,1,21}];

{ee,ev}=Eigensystem[{H,S}]//Transpose//Sort//Transpose

Psi[x_,i_]:=ev[[i]].v;

1/Sqrt[NIntegrate[Psi[x,1]^2,{x,-∞,∞}]]

Plot[{V3[x],25Psi[x,19]^2+ee[[19]]},{x,-15,15}]