From e1ddd40e512f7cfad58eaf7e63ae36a27f9fb971 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <jauslin@ias.edu>
Date: Mon, 1 Oct 2018 22:59:41 +0000
Subject: As presented at Rutgers on 2018-09-27

---
 figs/contour.fig/contour.py | 82 +++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 82 insertions(+)
 create mode 100644 figs/contour.fig/contour.py

(limited to 'figs/contour.fig/contour.py')

diff --git a/figs/contour.fig/contour.py b/figs/contour.fig/contour.py
new file mode 100644
index 0000000..102df86
--- /dev/null
+++ b/figs/contour.fig/contour.py
@@ -0,0 +1,82 @@
+#!/usr/bin/env python3
+
+import cmath
+import math
+import scipy.special as sp
+from scipy import optimize as op
+import random
+import sys
+
+# number of roots
+nr_roots=4
+
+# size of plot
+plotsize_x=3
+plotsize_y=3
+# rescale plot (so roots are not too close together)
+plotsize_scale_x=1
+plotsize_scale_y=6
+
+# numerical values
+hbar=6.58e-16 # eV.s
+m=9.11e-31 # kg
+Vn=9 # eV
+En=20e9 # V/m
+
+V=1
+E=En*hbar/(Vn**1.5*m**0.5)*math.sqrt(1.60e-19)
+
+# sqrt with branch cut along iR_+
+def sqrt_p(x):
+    r,p=cmath.polar(x)
+    if(p<cmath.pi/2):
+        return(cmath.rect(math.sqrt(r),p/2))
+    else:
+        return(cmath.rect(math.sqrt(r),(p-2*math.pi)/2))
+
+# solution of (-\Delta+V-ip)phi=0
+def phi(p,x,E,V):
+    return(sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[0])
+# its derivative
+def dphi(p,x,E,V):
+    return(cmath.exp(-1j*math.pi/3)*E**(1/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1])
+
+
+# the function whose roots are to be computed and its derivative
+def f(p):
+    return(sqrt_p(-1j*p)*phi(p,0,E,V)-dphi(p,0,E,V))
+def df(p):
+    return(-1j/(2*sqrt_p(-1j*p))*phi(p,0,E,V)+sqrt_p(-1j*p)*dp_phi(p,0,E,V)-dp_dphi(p,0,E,V))
+
+# derivatives of phi with respect to p
+def dp_phi(p,x,E,V):
+    return(1j*cmath.exp(-1j*math.pi/3)*E**(-2/3)*sp.airy(cmath.exp(-1j*math.pi/3)*(E**(1/3)*x-E**(-2/3)*(V-1j*p)))[1])
+def dp_dphi(p,x,E,V):
+    return(-1j*(x-(V-1j*p)/E)*phi(p,x,E,V))
+
+# check whether the root was already found
+def check(root,roots):
+    # reject if the root doesn't fit on the plot
+    if(plotsize_scale_x*root.real<-plotsize_x or abs(plotsize_scale_y*root.imag)>plotsize_y):
+        return(False)
+    for x in roots:
+        if(abs(root-x)<1e-12):
+            return(False)
+    return(True)
+
+# find roots
+roots=[]
+while len(roots)<nr_roots:
+    try:
+        root=op.newton(f, -random.random()*plotsize_x/plotsize_scale_x+(2*random.random()-1)*plotsize_y/plotsize_scale_y*1j, fprime=df)
+        if(check(root,roots)):
+            roots.append(root)
+            print("found "+str(len(roots))+" roots of "+str(nr_roots), file=sys.stderr)
+    except RuntimeError:
+        root=0
+    except:
+        break
+
+# print result
+for root in roots:
+    print("\\pole{(% .3f,% .3f)}" % (plotsize_scale_x*root.real,plotsize_scale_y*root.imag))
-- 
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