The Band Structure of the Periodic Delta Potential

Code

RHS[q_,W_]:=Cos[q]+(W/q)Sin[q];
KofE[EE_,W_]:=Module[{k},
If[Abs[RHS[(Sqrt[EE]),W]]≤1,k=ArcCos[RHS[(Sqrt[EE]),W]]//N,k=None];
Return[k]
];

BandQ[EE_,W_]:=If[Abs[RHS[(Sqrt[EE]),W]]≤1,1,-1];

EMin = -2;
EMax=160;
RMax=4;

PlotBandContour[W_] :=ContourPlot[((RHS[(Sqrt[EE]),W])^2)==1,{k,-π,π},{EE,EMin,EMax},MaxRecursion->2];

PlotBand[W_]:=RegionPlot[BandQ[EE,W]>0,{k,-π,π},{EE,EMin,EMax}(*,MaxRecursion->2*),Mesh->None,PlotPoints->120(*,BoundaryStyle->None*),PlotStyle->Directive[Green,Opacity[0.2]]];

PlotEnergy[W_]:=ParametricPlot[{{KofE[EE,0],EE},{-KofE[EE,0],EE},{KofE[EE,W],EE},{-KofE[EE,W],EE}(*,{BandQ[EE,W],EE}*)},{EE,EMin,EMax},AspectRatio->1,MaxRecursion->2RMax,Mesh->None,PlotStyle->{Directive[Red,Dashed],Directive[Red,Dashed],Directive[Blue,Thick],Directive[Blue,Thick](*,Directive[Green,Thickness[0.02],Opacity[1]]*)},PlotRange->{{-π,π},{0,EMax}}];

PlotRHS[W_]:=Plot[{Cos[(Sqrt[EE])]+W*((Sin[(Sqrt[EE])])/(Sqrt[EE])),1,-1},{EE,0,(((4π))^2)}, PlotRange->{-(π/2),(π/2)}, GridLines->{{(((π))^2),(((2π))^2),(((3π))^2),(((4π))^2)}, {}}];

dpi=72;

Scratch

(*
KofE[EE_,W_]:=Module[{q=(Sqrt[EE]),RHS,k},
RHS=Cos[q]+(W/(2q))Sin[q];
If[Abs[RHS]≤1,k=ArcCos[RHS]//N,k=None];
Return[k]
];
*)

(*
BandQ[EE_,W_]:=Module[{BandV},
If[KofE[EE,W]==None,BandV=None,BandV=1];
Return[BandV];
];
*)

(*

Plot[BandQ[EE,24],{EE,0,160}]

FindRoot[((RHS[(Sqrt[EE]),48])^2)==1,{EE,20}]

Manipulate[ContourPlot[((RHS[(Sqrt[EE]),W])^2)==1,{k,-π,π},{EE,0,160}],{W,0,50}]

FindRoot[{BandQ[EE,24]==0,EE>0,EE<160},{EE,0.1}]

*)

Manipulate[
Show[PlotBand[g+(g^3)],PlotEnergy[g+(g^3)],AspectRatio->2.4,FrameLabel->{"qL","E(q)"},Axes->False,Frame->True,ImageSize->{dpi*5/1.8,dpi*5}]
Show[PlotRHS[g+(g^3)],AxesLabel->{"E","Cos(qL)"},ImageSize->{dpi*4.6*GoldenRatio,dpi*4.6}],
{{g,0},0,4}]