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compute.m
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% Hi,
% This is basically Professor Zworski's code (that I got a long time ago.)
% We represent our operators in the fourier basis.
% The code is separated into sections- in the Matlab editor, type Ctrl-Enter to run a section.
% First let's set up some constants:
om = exp(2i*pi/3);
K = 4*pi/3;
zS = 1i/sqrt(3);
e1 = om^2; % basis of Lambda
e2 = -om;
f1 = (4i*pi/sqrt(3))*om; % dual basis of Lambda^* (<e_i, f_j> = 2pi 1_{i==j})
f2 = (4i*pi/sqrt(3))*om^2;
% Now, for example, we can make the Hamiltonian for non-scalar model.
% (Using the functions at the end of this file.)
N = 50;
% Dbar
Db_1 = Dbar(N, -K, f1, f2);
Db_2 = Dbar(N, +K, f1, f2);
% Potential
Up = sym_potential(N,0,0);
Um = Up.';
%for f_{-2}, use Up2 = sym_potential(N,1,-1); Um2 = Up2.';
D = [2*Db_1, 0*Up; 0*Up, 2*Db_2];
U = [0*Up, Up; Um, 0*Up];
%% Retrieve magic angles
save_angles=load('.\angles\1_-2_0.0002_circle.mat', 'save_angles').('save_angles');
%% Calculate magic angles with T_k
% for non-scalar (original) model
% Ak = Inv(2*Db_1) * Up * Inv(2*Db_2) * Um;
% for scalar model
Ak = Inv(4*Dbar(N,K,f1,f2)^2) * Up * Um;
Alphas = 1./sqrt(eigs(Ak, 200));
figure
hold on
scattermult([real(Alphas), imag(Alphas)], 5)
axis equal
%% Protected state for scalar model
[~,~,V] = svds(4*Dbar(N,0,f1,f2)^2 - 10^2*Up*Um, 1, 'smallest');
[z,v] = K2X3(V,0,600,e1,e2);
v = v ./ exp(1i*angle(v(1,1)));
figure
tiledlayout(1,2,'TileSpacing','compact')
minlevel=-16;
levels=linspace(minlevel,1,28);
% Plot log(abs(u_0))
nexttile
hold on
contourf(real(z), imag(z), max(log(abs(v)), minlevel), levels)
hex(zS); axis equal; colorbar
xlim([-0.63,0.63]), ylim([-0.63,0.63]); hold off
% Plot angle(u_0)
nexttile; hold on
surf(real(z), imag(z), angle(v), 'EdgeColor', 'none');
clim([-pi,pi]); colormap(gca,wheelmap);
hex(zS,10);
view(2); axis equal;
xlim([-0.63,0.63]), ylim([-0.63,0.63]); hold off;
colorbar;
%% Eigenvalues of H: single alpha
H = [0*D, (D+5*U)'; (D+5*U), 0*D];
E = svds(D+5*U, 200, 'smallest');
figure hold on
plot(E, (1:200)');
plot(E, sqrt(3)/(4*pi) * E.^2);
% Compare with Weyl's law
figure hold on
plot(E, (sqrt(3)/(4*pi) * E.^2 - (1:200)' ) ./ E);
yline(0);
%% Eigenvalues of H for range of alpha
alphas = 0:0.1:15;
EE = zeros(length(alphas),40);
for id=1:length(alphas)
disp(alphas(id))
EE(id,:) = svds(D+alphas(id)*U, 40, 'smallest');
end
figure hold on
plot(alphas,EE,'Color','b');
xlabel("\alpha")
ylabel("Positive eigenvalues of $H_0(\alpha)$",'Interpreter','latex')
%% Graph potential function V
V = Up*Um*const_fun(N);
[z,v] = K2X3(V,0,300,e1,e2);
% v = sqrt(v);
fig=figure;
set(fig, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1])
tiledlayout(1,2,'TileSpacing','compact')
% |V|
nexttile; hold on
contourf(real(z), imag(z), abs(v), 32)
hex(zS); axis equal; colorbar
xlim([-0.63,0.63]), ylim([-0.63,0.63])
% angle(V)
nexttile; hold on
surf(real(z), imag(z), angle(v), 'EdgeColor', 'none');
clim([-pi,pi]); colormap(gca,wheelmap);
hex(zS,10); view(2); axis equal; colorbar
xlim([-0.63,0.63]), ylim([-0.63,0.63])
%% Sample use of code for finding bracket area
figure
hold on
[boundary,unsure,areas,count,zz,bra]=bracket_area([0 0 1],600,3,4);
disp([areas, count]);
title('$|\{q, \bar q\}|$','Interpreter','latex');
surf(real(zz),imag(zz),abs(bra),'EdgeColor','none');
scatter3(real(boundary), imag(boundary), 1+ 0*boundary, 3,'red','filled');
scatter3(real(unsure), imag(unsure), 1+ 0*unsure, 4,'red','filled');
axis equal;
%% Animation for log |u_K| / alpha as alpha grows
vid=VideoWriter('.\results\log_norm_asymptotics_4.mp4','MPEG-4'); open(vid)
fig=figure;
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
alphas = 3:0.1:20;
for id=1:length(alphas)
clf(fig);
tl=tiledlayout(2,2,'TileSpacing','compact');
title(tl, ['$\alpha = ' sprintf('%.1f', alphas(id)) '$'], 'Interpreter', 'latex')
% Get protected state
[~,~,V] = svds(D + alphas(id)*U + K*speye(size(D)), 1, 'smallest');
[z,v1,~] = K2X4(V,K,300,e1,e2);
% Plot in hexagon
nexttile(1, [2 1]);
hold on
title('$\alpha^{-1} \log |u_K|$', 'Interpreter', 'latex')
minlevel=-1;
levels=linspace(minlevel,0.25,40);
contourf(real(z), imag(z), max(log(abs(v1)) ./ alphas(id), minlevel), levels)
hex(zS); axis equal; colorbar
xlim([-0.63,0.63]), ylim([-0.63,0.63]), clim([minlevel 0.25])
% Plot on two lines
M = 600;
[~,v1,~] = K2X2(V,K,M,e1,e2);
nexttile;
hold on
title('restricted to line from $0$ to $-\sqrt 3 i$', 'Interpreter', 'latex')
plot((0:M-1)'./M, log(abs(diag(v1))) ./ alphas(id))
xlim([0 1]), ylim([-1 0.2])
xline(1/3), xline(2/3), yline(0, 'Color', 'red')
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
nexttile;
hold on
title('restricted to line from $0$ to $\omega$', 'Interpreter', 'latex')
plot((M-1:-1:0)'./M, log(abs(v1(1,:))) ./ alphas(id))
xlim([0 1]), ylim([-1 1])
xline(1/2), yline(0, 'Color', 'red')
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
drawnow
frame=getframe(fig);
writeVideo(vid,frame)
end
close(vid)
%% Animation of u_K restricted to edge of hexagon, and fourier transform
vid=VideoWriter('.\results\edge_fourier_asymptotics_3.mp4','MPEG-4'); open(vid);
fig=figure;
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
alphas = 5:0.1:20;
for id=1:length(alphas)
clf(fig);
tl=tiledlayout(1,2,'TileSpacing','compact');
title(tl, ['$\alpha = ' sprintf('%.1f', alphas(id)) '$'], 'Interpreter', 'latex');
[~,~,V] = svds(D + alphas(id)*U + K*speye(size(D)), 1, 'smallest');
MM = 600; M = 3*MM;
[~,v1,~] = K2X2(V,K,M,e1,e2);
v0 = diag(v1); v0 = v0(MM: 2*MM-1);
v0 = v0 / norm(v0); v0 = v0 / sign(v0(1));
v0 = [v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0; 0*v0]; MM = 13*MM;
% fourier transform
V0 = fft(v0);
V0 = V0 * sqrt(alphas(id)/13);
V0 = circshift(V0, round(MM/2));
nexttile;
hold on
title('protected state on line from $-\frac{\sqrt{3}i}{3}$ to $-\frac{2\sqrt{3}i}{3}$', 'Interpreter', 'latex');
plot((0:(MM/13)-1)' ./(MM/13) .*(2/sqrt(3)), v0(1:MM/13));
xlim([0 2/sqrt(3)]);
ylim([-3 3]);
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 1 1]; % [width, height, depth]
nexttile;
hold on
title('scaled fourier transform', 'Interpreter', 'latex');
xlabel('$\xi / \alpha$', 'Interpreter', 'latex');
plot(((0:MM-1)' - round(MM/2)) .* (2*pi*sqrt(3)/(13*2*alphas(id))), abs(V0));
xlim([-10 10]);
ylim([-10 20]);
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 1 1]; % [width, height, depth]
drawnow;
frame=getframe(fig);
writeVideo(vid,frame);
end
close(vid);
%% Animation for log |u_K| / alpha as potential varies
vid=VideoWriter('.\results\log_norm_varying_potential_3.mp4','MPEG-4'); open(vid);
fig=figure;
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
Up2 = sym_potential(N,1,-1); Um2 = Up2.';
U2 = [0*Up, Up2; Um2, 0*Up];
alphas = @(id) 10;
thetas = 0:0.002:1;
theta = thetas * pi;
for id=1:length(theta)
clf(fig)
tl=tiledlayout(3,2,'TileSpacing','compact');
title(tl, ['$U = \cos\theta f_1 + \sin\theta f_{-2},\,\,\theta = ' ...
sprintf('%.3f', theta0(id)) '\pi$'], 'Interpreter', 'latex')
% get protected state
[~,~,V] = svds(D + alphas(id)*(cos(theta(id))*U + sin(theta(id))*U2) + K*speye(size(D)), 1, 'smallest');
[z,v1,~] = K2X4(V,K,300,e1,e2);
% points where max is achieved
v1_max = max(abs(v1), [], 'all');
max_inds = find(abs(abs(v1) - v1_max) < 0.0001);
% graph in hexagon
nexttile(1, [3 1])
hold on
title(['$\alpha^{-1} \log |u_K|,\quad\quad \alpha = ' sprintf('%.1f', alphas(id)) '$'], 'Interpreter', 'latex')
minlevel=-1;
levels=linspace(minlevel,0.25,40);
contourf(real(z), imag(z), max(log(abs(v1)) ./ alphas(id), minlevel), levels)
hex(zS); axis equal; colorbar
plot(real(z(max_inds)), imag(z(max_inds)), '.', 'color', 'red', 'MarkerSize', 10)
xlim([-0.63,0.63]), ylim([-0.63,0.63]), clim([minlevel 0.25])
% graph on two lines
M = 600;
[~,v1,~] = K2X2(V,K,M,e1,e2);
nexttile;
hold on
title('restricted to line from $0$ to $-\sqrt 3 i$', 'Interpreter', 'latex')
plot((0:M-1)'./M, log(abs(diag(v1))) ./ alphas(id))
xlim([0 1]), ylim([-1 0.2])
xline(1/3), xline(2/3), yline(0, 'Color', 'red')
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
nexttile;
hold on
title('restricted to line from $0$ to $\omega$', 'Interpreter', 'latex')
plot((M-1:-1:0)'./M, log(abs(v1(1,:))) ./ alphas(id))
xlim([0 1]), ylim([-1 0.5]);
xline(1/2), yline(0, 'Color', 'red')
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
% graph magic angles
nexttile;
Alphas = save_angles(:, 1+10*(id-1));
hold on
scattermult([real(Alphas), imag(Alphas)], 13)
xlim([-0.1 11]), ylim([-8 8]), yline(0)
drawnow;
frame=getframe(fig);
writeVideo(vid,frame)
end
close(vid)
%% Graph log bands / alpha as alpha grows
alphas=3:0.05:20;
E = zeros(20, length(alphas));
for id=1:length(alphas)
disp(alphas(id))
E(:,id) = log( svds(D + alphas(id)*U, 20, 'smallest') ) ./ alphas(id);
end
fig=figure; hold on
tl=tiledlayout(2,1,'TileSpacing', 'compact');
title(tl, '$k=0$', 'Interpreter', 'latex');
nexttile; hold on
title('$\alpha^{-1}\log E_j$ for $j=1,2,3$', 'Interpreter', 'latex');
plot(alphas, E(1:3, :));
xlabel('$\alpha$', 'Interpreter', 'latex');
ylim([-1, 0.5]); xlim([3, 20]);
nexttile; hold on
title('$\alpha^{-1}\log E_j$ for $j=1,\dots, 20$', 'Interpreter', 'latex');
plot(alphas, E(1:20, :));
xlabel('$\alpha$', 'Interpreter', 'latex');
ylim([-1, 0]); xlim([3, 20]);
saveas(fig,'./results/scaled_log_eigenvalues_k0_2.png')
%% Animation overlaying scalar and non-scalar log |u| / alpha as alpha grows
vid=VideoWriter('.\results\log_norm_asymptotics_overlay2_1.mp4','MPEG-4'); open(vid);
fig=figure;
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
alphas = 10:0.1:20;
for id=1:length(alphas)
clf(fig);
tl=tiledlayout(2,2,'TileSpacing','compact');
title(tl, ['$\alpha = ' sprintf('%.1f', alphas(id)) '$'], 'Interpreter', 'latex');
[~,~,V] = svds(D + alphas(id)*U + K*speye(size(D)), 1, 'smallest');
V = V(1:length(V)/2); V = V ./norm(V);
[z,v1] = K2X3(V,0,300,e1,e2);
[~,~,V_scalar] = svds((4*Dbar(N,0,f1,f2)^2 -alphas(id)^2 * Up*Um - K^2), 1, 'smallest');
[~,u0] = K2X3(V_scalar,0,300,e1,e2);
nexttile(1);
hold on
title('$\alpha^{-1} \log |u_{K,1}|$, where $D(\alpha)u_K = 0$', 'Interpreter', 'latex');
minlevel=-1;
levels=linspace(minlevel,0.25,40);
contourf(real(z), imag(z), max(log(abs(v1)) ./ alphas(id), minlevel), levels);
hex(zS); axis equal; colorbar;
xlim([-0.63,0.63]), ylim([-0.63,0.63]); clim([minlevel 0.25]);
nexttile(3);
hold on
title('$\alpha^{-1} \log |v_0|$, where $v_0$ is state for smallest singular value of $Q(\alpha,0)-K^2$', 'Interpreter', 'latex');
minlevel=-1;
levels=linspace(minlevel,0.25,40);
contourf(real(z), imag(z), max(log(abs(u0)) ./ alphas(id), minlevel), levels);
hex(zS); axis equal; colorbar;
xlim([-0.63,0.63]), ylim([-0.63,0.63]); clim([minlevel 0.25]);
M = 600;
[~,v1] = K2X(V,K,M,e1,e2);
[~,u0] = K2X(V_scalar,0,M,e1,e2);
nexttile
hold on
title('restricted to line from $0$ to $-\sqrt 3 i$', 'Interpreter', 'latex');
plot((0:M-1)'./M, log(abs(diag(v1))) ./ alphas(id), 'Color', 'blue');
plot((0:M-1)'./M, log(abs(diag(u0))) ./ alphas(id), 'Color', 'red');
xlim([0 1])
ylim([-1 0.2])
xline(1/3); xline(2/3); yline(0, 'Color', 'black')
legend('nonscalar', 'scalar')
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
nexttile
hold on
title('restricted to line from $0$ to $\omega$', 'Interpreter', 'latex');
plot((M-1:-1:0)'./M, log(abs(v1(1,:))) ./ alphas(id), 'Color', 'blue');
plot((M-1:-1:0)'./M, log(abs(u0(1,:))) ./ alphas(id), 'Color', 'red');
xlim([0 1])
ylim([-1 1])
xline(1/2); yline(0, 'Color', 'black');
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
drawnow;
frame=getframe(fig);
writeVideo(vid,frame);
end
close(vid);
%% Finite element for a hexagon dirichlet problem
model = createpde;
rho = exp(1i*pi/3);
hex_z = zS * rho.^(0:5);
g = decsg([2 6 real(hex_z) imag(hex_z)]');
geometryFromEdges(model,g);
applyBoundaryCondition(model,"dirichlet", ...
"Edge",1:model.Geometry.NumEdges, ...
"u",0);
% f= {q, bar q} / |xi|^2
V = const_fun(N);
V = Up*Um*V;
dzV = 1i* Dbar(N,0,f1,f2)' * V;
M=1000;
[~,v] = K2X(V,0,M,e1,e2);
[~,dzv] = K2X(dzV,0,M,e1,e2);
bracket = imag( sqrt(conj(v)) .* dzv );
f = abs(bracket) ./ abs(v);
f(1,1) = f(2,1); % lol
% % sanity checks
% figure
% hold on
% surf(real(z),imag(z),abs(bracket) ./ 1500, 'EdgeColor', 'none')
% view(2); axis equal; colorbar;
% figure
% hold on
% contourf(real(z),imag(z),f, 18)
% view(2); axis equal; colorbar;
ffun = @(z,etc) interpolate(e1,e2,f,M,z.x,z.y);
specifyCoefficients(model, "m",0,"d",0,"c",1,"a",0,"f",ffun);
generateMesh(model,"Hmax",0.01);
result=solvepde(model);
figure
hold on
pdeplot(result.Mesh, XYData=result.NodalSolution, Contour="on", ColorMap="default", Levels=18);
axis equal
title("$\Phi$", 'Interpreter', 'latex')
% saveas(gcf,'./Phi_solution.png')
%% Graph the first 10 eigenfunctions of H
p=10;
[~,s,V] = svds(D + 17*U + K*speye(size(D)), p, 'smallest');
disp([(1:p)', diag(s)])
fig=figure;
set(fig, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
tiledlayout(2,5,'TileSpacing','tight');
minlevel=-1;
levels=linspace(minlevel,0.25,40);
for id=1:10
disp(id)
V2 = V(:,id);
[U0,U1,U2] = PROJC(N,V2,1); % at least look symmetric bruh
disp([id, norm(U0), norm(U1), norm(U2)])
if norm(U0) > 10^(-5)
V2 = U0;
elseif norm(U1) > 10^(-5)
V2 = U1;
else
V2 = U2;
end
V2 = V2 ./ norm(V2);
[z,v1,v2] = K2X4(V2,K,200,e1,e2);
nexttile; hold on
contourf(real(z), imag(z), max(log(abs(v1)) ./ 17,minlevel), levels);
hex(zS); axis equal; colorbar;
xlim([-0.63,0.63]), ylim([-0.63,0.63]); clim([minlevel 0.25]);
end
%% Log eigenfunction / alpha smoothened, then take laplacian
alpha = 20;
[~,~,V] = svds(D + alpha*U + K*speye(size(D)), 1, 'smallest');
V = V(1:length(V)/2); V = V / norm(V);
[z,v] = K2X(V,K,300,e1,e2);
rad = 8;
chi = ones(rad,rad); chi(300,300) = 0; %chi=circshift(chi, [-rad/2, -rad/2]);
v_conv = ifft2(fft2(abs(v)) .* fft2(chi)) ./ rad^2;
v_conv = log(abs(v_conv)) ./ alpha;
% v_conv = max(log(abs(v)) ./ alpha, -0.5);
V_conv =X2K(v_conv,0,N,e1,e2);
[~,Lap_v_conv]=K2X(Dbar(N,0,f1,f2) * Dbar(N,0,f1,f2)' * V_conv, 0, 300, e1,e2);
figure
tiledlayout(1,4,'TileSpacing','compact');
minlevel=-1;
levels=linspace(minlevel,0.25,40);
nexttile;
hold on
title('$\chi$ where $\int \chi = 1$', 'Interpreter', 'latex');
contourf(real(z), imag(z), -chi ./ 2, levels);
axis equal; clim([minlevel 0.25]);
nexttile;
hold on
title('$\alpha^{-1} \log |u_K|$', 'Interpreter', 'latex');
contourf(real(z), imag(z), max(log(abs(v)) ./ alpha, minlevel), levels);
clim([minlevel 0.25]); axis equal;
nexttile;
hold on
title('$\alpha^{-1} \log (|u_K| \ast \chi)$', 'Interpreter', 'latex');
contourf(real(z), imag(z), max(v_conv, minlevel), levels);
axis equal; clim([minlevel 0.25]); colorbar;
% nexttile;
% hold on
% title('$\Delta(\alpha^{-1} \log (|u_K| \ast \chi))$', 'Interpreter', 'latex');
% surf(real(z), imag(z), real(Lap_v_conv),'EdgeColor', 'None');
% axis equal;
% view(2);
% % hex(zS); axis equal; colorbar;
% ylim([-1.721,0]), xlim([-0.5,0.5]);
% colorbar;
nexttile;
hold on
title('max$(-3,\Delta(\alpha^{-1} \log (|u_K| \ast \chi)) )$', 'Interpreter', 'latex');
contourf(real(z), imag(z), min(max(real(Lap_v_conv),-3), 18),15);
axis equal;
view(2);
% hex(zS); axis equal; colorbar;
ylim([-1.721,0]), xlim([-0.5,0.5]);
colorbar;
% saveas(gcf,'./results/Laplacian_of_phi_1.png')
%% Conjugate by multiplication with e^(phi/h)
M = 300;
figure
tiledlayout(1,2)
% Sine wave in one direction (real function)
Phi = (fourier_shift(N,1,0) + fourier_shift(N,-1,0))*const_fun(N) .* 0.1;
[z, phi] = K2X(Phi, 0, M, e1, e2);
phi = real(phi);
alpha = 10;
ephi = exp(phi .* alpha);
% protected state times e^(phi/h)
alpha = 19;
[~,~,U0] = svds(4*Dbar(N,0,f1,f2)^2 - alpha^2 * Up*Um, 1, "smallest");
[~,u0] = K2X(U0, 0, M, e1, e2);
nexttile
hold on
minlevel=-1.5;
levels=linspace(minlevel,1,80);
contourf(real(z), imag(z), max(phi+ (log(abs(u0)) ./ alpha), minlevel), levels)
axis equal
colorbar
% {q, bar q}
Lap_Phi = -Dbar(N,0,f1,f2) * Dbar(N,0,f1,f2)' * Phi;
[~, lap_phi] = K2X(Lap_Phi, 0, M, e1, e2);
V = const_fun(N);
V = Up*Um*V;
dzV = 1i* Dbar(N,0,f1,f2)' * V;
[~,v] = K2X(V,0,M,e1,e2);
[~,dzv] = K2X(dzV,0,M,e1,e2);
bracket = imag( sqrt(conj(v)) .* dzv );
bracket = abs(bracket)-8 .* abs(v) .* real(lap_phi);
nexttile
hold on
contourf(real(z), imag(z), bracket, 30)
axis equal
colorbar
%% Animation for log |u|/alpha and log |2hDbar u / sqrt(V)|/alpha as alpha grows
vid=VideoWriter('.\results\log_norm_scalar_overlay_4.mp4','MPEG-4'); open(vid);
fig=figure;
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
V0 = Up * Um * const_fun(N);
alphas = 10:0.1:20;
for id=1:length(alphas)
clf(fig);
tl=tiledlayout(2,2,'TileSpacing','compact');
title(tl, ['$\alpha = ' sprintf('%.1f', alphas(id)) '$'], 'Interpreter', 'latex');
[~,~,V] = svds((4*Dbar(N,0,f1,f2)^2 -alphas(id)^2 * Up*Um), 1, 'smallest');
hdbarV = 2*Dbar(N,0,f1,f2) * V ./ alphas(id);
[z,v] = K2X3(V,0,300,e1,e2);
[~,dv] = K2X3(hdbarV,0,300,e1,e2);
[~,v0] = K2X3(V0,0,300,e1,e2);
dv = dv ./ sqrt(v0);
nexttile(1);
hold on
title('$h\log |u_{0}|$, where $P(\alpha)u_0 = 0$', 'Interpreter', 'latex');
minlevel=-1;
levels=linspace(minlevel,0.5,40);
contourf(real(z), imag(z), max(log(abs(v)) ./ alphas(id), minlevel), levels);
hex(zS); axis equal; colorbar;
xlim([-0.63,0.63]), ylim([-0.63,0.63]); clim([minlevel 0.25]);
nexttile(3);
hold on
title('$h\log |2hD_{\bar z} u_{0} / \sqrt{V}|$', 'Interpreter', 'latex');
minlevel=-1;
levels=linspace(minlevel,0.5,40);
contourf(real(z), imag(z), max(log(abs(dv)) ./ alphas(id), minlevel), levels);
hex(zS); axis equal; colorbar;
xlim([-0.63,0.63]), ylim([-0.63,0.63]); clim([minlevel 0.25]);
M = 600;
[~,v] = K2X(V,K,M,e1,e2);
[~,dv] = K2X(hdbarV,0,M,e1,e2);
[~,v0] = K2X(V0,0,M,e1,e2);
dv = dv ./ sqrt(v0);
dv(201,201) = dv(202,202); dv(401,401)=dv(402,402);
nexttile
hold on
title('restricted to line from $0$ to $-\sqrt 3 i$', 'Interpreter', 'latex');
plot((0:M-1)'./M, log(abs(diag(v))) ./ alphas(id), 'Color', 'blue');
plot((0:M-1)'./M, log(abs(diag(dv))) ./ alphas(id), 'Color', 'red');
xlim([0 1])
ylim([-1 0.2])
xline(1/3); xline(2/3); yline(0, 'Color', 'black')
legend('h log u', 'h log (2hDbar u / sqrt V)')
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
nexttile
hold on
title('restricted to line from $0$ to $\omega$', 'Interpreter', 'latex');
plot((M-1:-1:0)'./M, log(abs(v(1,:))) ./ alphas(id), 'Color', 'blue');
plot((M-1:-1:0)'./M, log(abs(dv(1,:))) ./ alphas(id), 'Color', 'red');
xlim([0 1])
ylim([-1 1])
xline(1/2); yline(0, 'Color', 'black');
ax = gca; % Get current axes
ax.PlotBoxAspectRatio = [1 0.5 1]; % [width, height, depth]
drawnow;
frame=getframe(fig);
writeVideo(vid,frame);
end
close(vid);
%% Does restriction of u_0 to edge of hexagon, approximately solve semiclassical ode?
alpha = 20;
[~,~,V] = svds(4*Dbar(N,0,f1,f2)^2 - alpha^2 * Up * Um, 1, 'smallest');
[~,v] = K2X(V,0,300,e1,e2);
phase = exp(1i*angle(v(1,1)));
V = V ./ phase;
Pot = Up*Um*const_fun(N);
DxV = Dx(N,0,f1,f2) * V;
DyV = Dy(N,0,f1,f2) * V;
DxxV = Dx(N,0,f1,f2) * DxV;
DxyV = Dy(N,0,f1,f2) * DxV;
DyyV = Dy(N,0,f1,f2) * DyV;
M = 500;
[~,v] = K2X(V,0,3*M,e1,e2);
[~,dxv] = K2X(DxV,0,3*M,e1,e2);
[~,dyv] = K2X(DyV,0,3*M,e1,e2);
[~,dxxv] = K2X(DxxV,0,3*M,e1,e2);
[~,dxyv] = K2X(DxyV,0,3*M,e1,e2);
[~,dyyv] = K2X(DyyV,0,3*M,e1,e2);
[~,pot] = K2X(Pot,0,3*M,e1,e2);
v = diag(v); dxv = diag(dxv).* (2 / alpha); dyv = diag(dyv).* (2 / alpha);
dxxv = diag(dxxv).* (2 / alpha)^2; dyyv = diag(dyyv).* (2 / alpha)^2;
dxyv = diag(dxyv).* (2 / alpha)^2; pot = diag(pot);
%1 v real
%2 Dxv real (Dx = -i dx)
%3 Dyv imag
%4 DxDxv real
%5 DyDyv real
%6 DxDyv imag
A = [real(v), real(dxv), imag(dyv), real(dxxv), real(dyyv), imag(dxyv), pot].';
names = ["u", "dxu", "dyu", "dxxu", "dyyu", "dxyu", "potential V"];
mm = 20;
A = A(:,M+1+mm:2*M-mm);
% B = normalize(A,2, "norm");
% ind = [1 2 4];
% [c,s,~] = svd(B(ind,:), "econ");
%
% disp(names(ind))
% disp("singular values")
% disp(s)
% disp("coefficients")
% disp(c)
%
%
% figure
% hold on
% plot((mm:M-1-mm)'./M, abs(B(ind,:)))
% yline(0)
% xlim([0 1])
% legend(names(ind))
figure
hold on
title("$V$ vs second derivatives of $u$ on vertical edge of hexagon", 'Interpreter', 'Latex')
plot((mm:M-1-mm)'./M, A(4,:) ./ A(1,:))
plot((mm:M-1-mm)'./M, A(5,:) ./ A(1,:))
plot((mm:M-1-mm)'./M, A(6,:) ./ A(1,:))
plot((mm:M-1-mm)'./M, real(A(7,:)))
ylim([-100, 100])
legend(["$(4h^2 \partial_x^2 u) / u$", "$(4h^2 \partial_y^2 u) / u$", "$\Im (4h^2 \partial_x\partial_y u) / u$", "V"], 'Interpreter', 'Latex')
% disp( (imag(dyv)' * dxv) / (norm(dxv) * norm(dyv)) )
% disp( -norm(dxv) / norm(dyv) )
%% (related to previous) calculate c (coefficient in ode) for some values of alpha
alphas=5:0.1:20;
xx_yy = alphas * 0;
xx_xy = alphas * 0;
x_y = alphas * 0;
for id=1:length(alphas)
alpha = alphas(id);
[~,~,V] = svds(4*Dbar(N,0,f1,f2)^2 - alpha^2 * Up * Um, 1, 'smallest');
[~,v] = K2X(V,0,300,e1,e2);
phase = exp(1i*angle(v(1,1)));
V = V ./ phase;
Pot = Up*Um*const_fun(N);
DxV = Dx(N,0,f1,f2) * V;
DyV = Dy(N,0,f1,f2) * V;
DxxV = Dx(N,0,f1,f2) * DxV;
DxyV = Dy(N,0,f1,f2) * DxV;
DyyV = Dy(N,0,f1,f2) * DyV;
M = 500;
[~,v] = K2X(V,0,3*M,e1,e2);
[~,dxv] = K2X(DxV,0,3*M,e1,e2);
[~,dyv] = K2X(DyV,0,3*M,e1,e2);
[~,dxxv] = K2X(DxxV,0,3*M,e1,e2);
[~,dxyv] = K2X(DxyV,0,3*M,e1,e2);
[~,dyyv] = K2X(DyyV,0,3*M,e1,e2);
[~,pot] = K2X(Pot,0,3*M,e1,e2);
v = diag(v); dxv = diag(dxv).* (2 / alpha); dyv = diag(dyv).* (2 / alpha);
dxxv = diag(dxxv).* (2 / alpha)^2; dyyv = diag(dyyv).* (2 / alpha)^2;
dxyv = diag(dxyv).* (2 / alpha)^2; pot = diag(pot);
A = [real(v), real(dxv), imag(dyv), real(dxxv), real(dyyv), imag(dxyv), pot].';
names = ["u", "dxu", "dyu", "dxxu", "dyyu", "dxyu", "potential V"];
mm = 20;
A = A(:,M+1+mm:2*M-mm);
ind = [2 3];
[c,~,~] = svd(A(ind,:), "econ");
x_y(id) = c(3)/c(4);
ind = [4 5];
[c,~,~] = svd(A(ind,:), "econ");
xx_yy(id) = c(3)/c(4);
ind = [4 6];
[c,~,~] = svd(A(ind,:), "econ");
xx_xy(id) = c(3)/c(4);
end
figure
hold on
plot(alphas, xx_yy)
plot(alphas, xx_xy)
plot(alphas, x_y)
legend(["$\sqrt{\partial_{yy} u / \partial_{xx} u}$",...
"$(Im \partial_{xy} u) / \partial_{xx} u$", "$\partial_y u / Im(\partial_x u)$"],...
'Interpreter', 'Latex')
%% Animation of potential changing (between f_1 and f_-2) and corresponding protected state
vid=VideoWriter('.\results\varying_potential_2_1.mp4','MPEG-4'); open(vid);
fig=figure;
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
Up2 = sym_potential(N,1,-1); Um2 = Up2.';
alpha = @(id) 10;
theta = (0:0.01:1) * pi;
for id=1:length(theta)
thet = theta(id);
clf(fig);
tl=tiledlayout(1,3,'TileSpacing','compact');
V = (cos(thet)^2 * Up - sin(thet)^2 * Up2)* (cos(thet)^2 * Um - sin(thet)^2 * Um2);
[~,~,U] = svds((4*Dbar(N,0,f1,f2)^2 -alpha(id)^2 * V), 1, 'smallest');
[z,u0] = K2X3(U,0,300,e1,e2);
[~,v] = K2X3( V * const_fun(N), 0, 300, e1,e2 );
v = v ./ exp(1i*angle(v(150,150)));
title(tl, ['$V(z) = U(z)U(-z), U(z) = \cos^2(\theta)f_1 - \sin^2(\theta) f_{-2}, \theta = '...
sprintf('%.2f', id / length(theta)) '\pi$'], 'Interpreter', "latex");
nexttile % plot highest points? nah for now, just wanna see
hold on
title("$h \log |u_0|$", 'Interpreter', 'latex');
minlevel=-1;
levels=linspace(minlevel,0.25,40);
contourf(real(z), imag(z), max(log(abs(u0)) ./ alpha(id), minlevel), levels);
hex(zS)
axis equal
colorbar
xlim([-0.63,0.63]), ylim([-0.63,0.63])
clim([minlevel 0.25]);
nexttile
hold on
title("$|V|$", 'Interpreter', 'Latex')
levels=linspace(0,150,70);
contourf(real(z), imag(z), min(abs(v), 150), levels)
hex(zS,10)
axis equal
xlim([-0.63,0.63])
ylim([-0.63,0.63])
clim([0 60])
colorbar
nexttile
hold on
title("$angle(V)$", 'Interpreter', 'latex')
surf(real(z), imag(z), angle(v), 'EdgeColor', 'none')
clim([-pi,pi])
colormap(gca,wheelmap)
hex(zS,10)
view(2)
axis equal
xlim([-0.63,0.63])
ylim([-0.63,0.63])
colorbar
drawnow;
frame=getframe(fig);
writeVideo(vid,frame);
end
close(vid);
%% Comparison with Airy function in w coordinates
% prologue -------------
vid=VideoWriter('.\results\airy_holo_6.mp4','MPEG-4'); open(vid);
fig=figure;
set(fig, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
alphas = 0.1:0.1:19.5;
for id=1:length(alphas)
alpha = alphas(id);
% alpha = 17;
% ----------------------
disp(id)
clf(fig);
tl=tiledlayout(2,3,'TileSpacing','compact');
title(tl, ['$\alpha = ' sprintf('%.1f', real(alpha / 1i)) 'i$'], 'Interpreter', 'latex')
A = -1i*(8 * pi^3 * (alpha)^2 / 3)^(-1/3);
% Range of w
[x,y] = meshgrid(-1:0.005:1, -1:0.005:1);
w = x + 1i*y; w= 5*w;
z = zS + A * w;
% Get protected state
M=4000;
[~,~,V] = svds(4*Dbar(N,0,f1,f2)^2 - (alpha)^2*Up*Um, 1, 'smallest');
[~,v] = K2X(V,0,M,e1,e2);
apply = @(f) arrayfun(@(z) interpolate(e1,e2,f,M,real(z),imag(z)), z);