import sympy as sp

sp.init_printing()

o = sp.Symbol('omega', positive=True)
a = sp.Symbol('alpha', positive=True)
A = sp.Symbol('A', positive=True)
B = sp.Symbol('B', positive=True)
g = sp.Symbol('gamma', positive=True)
b = sp.Symbol('b', positive=True)
l = sp.Symbol('lambda')
z,x= sp.symbols('z x')

z**4 + b*z**2 - l

$$b z^{2} - \lambda + z^{4}$$

C1, C2, C3, C4 = sp.symbols('C_1 C_2 C_3 C_4')

Формулы для выражения корней через коэффициенты:

Случай $\pm i \omega$, $\pm i \alpha$

sp.collect(sp.expand((z-sp.I*a)*(z+sp.I*a)*(z-sp.I*o)*(z+sp.I*o)), z)

$$\alpha^{2} \omega^{2} + z^{4} + z^{2} \left(\alpha^{2} + \omega^{2}\right)$$

gen_sol_1 = C1*sp.cos(a*x)+C2*sp.sin(a*x)+C3*sp.cos(o*x)+C4*sp.sin(o*x)
lhs1 = sp.Matrix([sp.diff(gen_sol_1, x).subs([(x, 0)]),
          sp.diff(gen_sol_1, x, x, x).subs([(x, 0)]),
          sp.diff(gen_sol_1, x, x).subs([(x, 1)]),
          sp.diff(gen_sol_1, x, x, x).subs([(x, 1)]),])
chpoly1 = sp.det((lhs1.jacobian([C1, C2, C3, C4])))
(chpoly1)

$$- \alpha^{6} \omega^{3} \sin{\left (\alpha \right )} \cos{\left (\omega \right )} + \alpha^{5} \omega^{4} \sin{\left (\omega \right )} \cos{\left (\alpha \right )} + \alpha^{4} \omega^{5} \sin{\left (\alpha \right )} \cos{\left (\omega \right )} - \alpha^{3} \omega^{6} \sin{\left (\omega \right )} \cos{\left (\alpha \right )}$$

chpolyImaginaryPair = sp.lambdify([a, o], chpoly1)

print(sp.solve([a**2+o**2 - b,
          a**2*o**2+l],[a, o], dict=True)[7])

{omega: sqrt(b/2 - sqrt(b**2 + 4*lambda)/2), alpha: sqrt(b/2 + sqrt(b**2 + 4*lambda)/2)}

Эти формулы работают и дают положительные числа при $-\frac{b^2}{4} < \lambda < 0$

Случай $\pm i \alpha$, $\pm \gamma$

sp.collect(sp.expand((z-g)*(z+g)*(z-sp.I*a)*(z+sp.I*a)), z)

$$- \alpha^{2} \gamma^{2} + z^{4} + z^{2} \left(\alpha^{2} - \gamma^{2}\right)$$

gen_sol_2 = C1*sp.cos(a*x)+C2*sp.sin(a*x)+C3*sp.cosh(g*x)+C4*sp.sinh(g*x)
(gen_sol_2)

$$C_{1} \cos{\left (\alpha x \right )} + C_{2} \sin{\left (\alpha x \right )} + C_{3} \cosh{\left (\gamma x \right )} + C_{4} \sinh{\left (\gamma x \right )}$$

lhs2 = sp.Matrix([sp.diff(gen_sol_2, x, x).subs([(x, 1), (C2, 0), (C4, 0)]),
          sp.diff(gen_sol_2, x, x, x).subs([(x, 1)]),])


chpoly2 = sp.det((lhs2.jacobian([C1, C3])))
print(chpoly2)

-alpha**3*gamma**2*sin(alpha)*cosh(gamma) - alpha**2*gamma**3*cos(alpha)*sinh(gamma)

chpolyComplexReal = sp.lambdify((a, g), chpoly2)

sp.solve([a**2-g**2 - b,
          l - a**2*g**2], [a, g], dict=True)

$$\left [ \left \{ \alpha : - \sqrt{\frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : - \sqrt{- \frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}, \quad \left \{ \alpha : - \sqrt{\frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : \sqrt{- \frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}, \quad \left \{ \alpha : \sqrt{\frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : - \sqrt{- \frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}, \quad \left \{ \alpha : \sqrt{\frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : \sqrt{- \frac{b}{2} - \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}, \quad \left \{ \alpha : - \sqrt{\frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : - \sqrt{- \frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}, \quad \left \{ \alpha : - \sqrt{\frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : \sqrt{- \frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}, \quad \left \{ \alpha : \sqrt{\frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : - \sqrt{- \frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}, \quad \left \{ \alpha : \sqrt{\frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}, \quad \gamma : \sqrt{- \frac{b}{2} + \frac{1}{2} \sqrt{b^{2} + 4 \lambda}}\right \}\right ]$$

Эти формулы работают при $\lambda > 0$

print(sp.solve([a**2-g**2 - b,
          l - a**2*g**2], [a, g], dict=True)[7])

{gamma: sqrt(-b/2 + sqrt(b**2 + 4*lambda)/2), alpha: sqrt(b/2 + sqrt(b**2 + 4*lambda)/2)}

Случай $\pm a \pm b i$

sp.collect(sp.expand((z-(A+sp.I*B))*(z-(A-sp.I*B))*(z-(-A+sp.I*B))*(z-(-A-sp.I*B))), z)

$$A^{4} + 2 A^{2} B^{2} + B^{4} + z^{4} + z^{2} \left(- 2 A^{2} + 2 B^{2}\right)$$

При $\lambda <-\frac{b^2}{4}$

gen_sol_3 = C1*sp.exp(A*x)*sp.cos(B*x)+C2*sp.exp(A*x)*sp.sin(B*x)+C3*sp.exp(-A*x)*sp.cos(B*x)+C4*sp.exp(-A*x)*sp.sin(B*x)
(gen_sol_3)

$$C_{1} e^{A x} \cos{\left (B x \right )} + C_{2} e^{A x} \sin{\left (B x \right )} + C_{3} e^{- A x} \cos{\left (B x \right )} + C_{4} e^{- A x} \sin{\left (B x \right )}$$

lhs3 = sp.Matrix([sp.diff(gen_sol_3, x).subs([(x, 0)]),
          sp.diff(gen_sol_3, x, x, x).subs([(x, 0)]),
          sp.diff(gen_sol_3, x, x).subs([(x, 1)]),
          sp.diff(gen_sol_3, x, x, x).subs([(x, 1)]),])
(lhs3)

$$\left[\begin{matrix}A C_{1} - A C_{3} + B C_{2} + B C_{4}\\A^{3} C_{1} - A^{3} C_{3} + 3 A^{2} B C_{2} + 3 A^{2} B C_{4} - 3 A B^{2} C_{1} + 3 A B^{2} C_{3} - B^{3} C_{2} - B^{3} C_{4}\\A^{2} C_{1} e^{A} \cos{\left (B \right )} + A^{2} C_{2} e^{A} \sin{\left (B \right )} + \frac{A^{2} C_{3}}{e^{A}} \cos{\left (B \right )} + \frac{A^{2} C_{4}}{e^{A}} \sin{\left (B \right )} - 2 A B C_{1} e^{A} \sin{\left (B \right )} + 2 A B C_{2} e^{A} \cos{\left (B \right )} + \frac{2 C_{3}}{e^{A}} A B \sin{\left (B \right )} - \frac{2 C_{4}}{e^{A}} A B \cos{\left (B \right )} - B^{2} C_{1} e^{A} \cos{\left (B \right )} - B^{2} C_{2} e^{A} \sin{\left (B \right )} - \frac{B^{2} C_{3}}{e^{A}} \cos{\left (B \right )} - \frac{B^{2} C_{4}}{e^{A}} \sin{\left (B \right )}\\A^{3} C_{1} e^{A} \cos{\left (B \right )} + A^{3} C_{2} e^{A} \sin{\left (B \right )} - \frac{A^{3} C_{3}}{e^{A}} \cos{\left (B \right )} - \frac{A^{3} C_{4}}{e^{A}} \sin{\left (B \right )} - 3 A^{2} B C_{1} e^{A} \sin{\left (B \right )} + 3 A^{2} B C_{2} e^{A} \cos{\left (B \right )} - \frac{3 C_{3}}{e^{A}} A^{2} B \sin{\left (B \right )} + \frac{3 C_{4}}{e^{A}} A^{2} B \cos{\left (B \right )} - 3 A B^{2} C_{1} e^{A} \cos{\left (B \right )} - 3 A B^{2} C_{2} e^{A} \sin{\left (B \right )} + \frac{3 C_{3}}{e^{A}} A B^{2} \cos{\left (B \right )} + \frac{3 C_{4}}{e^{A}} A B^{2} \sin{\left (B \right )} + B^{3} C_{1} e^{A} \sin{\left (B \right )} - B^{3} C_{2} e^{A} \cos{\left (B \right )} + \frac{B^{3} C_{3}}{e^{A}} \sin{\left (B \right )} - \frac{B^{3} C_{4}}{e^{A}} \cos{\left (B \right )}\end{matrix}\right]$$

chpoly3 = sp.det((lhs3.jacobian([C1, C2, C3, C4])))
print(chpoly3)

-8*A**8*B*sin(B)*cos(B) - 2*A**7*B**2*exp(2*A)*sin(B)**2 - 2*A**7*B**2*exp(2*A)*cos(B)**2 + 2*A**7*B**2*exp(-2*A)*sin(B)**2 + 2*A**7*B**2*exp(-2*A)*cos(B)**2 - 24*A**6*B**3*sin(B)*cos(B) - 6*A**5*B**4*exp(2*A)*sin(B)**2 - 6*A**5*B**4*exp(2*A)*cos(B)**2 + 6*A**5*B**4*exp(-2*A)*sin(B)**2 + 6*A**5*B**4*exp(-2*A)*cos(B)**2 - 24*A**4*B**5*sin(B)*cos(B) - 6*A**3*B**6*exp(2*A)*sin(B)**2 - 6*A**3*B**6*exp(2*A)*cos(B)**2 + 6*A**3*B**6*exp(-2*A)*sin(B)**2 + 6*A**3*B**6*exp(-2*A)*cos(B)**2 - 8*A**2*B**7*sin(B)*cos(B) - 2*A*B**8*exp(2*A)*sin(B)**2 - 2*A*B**8*exp(2*A)*cos(B)**2 + 2*A*B**8*exp(-2*A)*sin(B)**2 + 2*A*B**8*exp(-2*A)*cos(B)**2

chpolyAllComplex = sp.lambdify((A, B), chpoly3)

sp.solve([2*B**2-2*A**2 - b,
          (A**2+B**2)**2+l], [A, B])

$$\left [ \left ( - \frac{1}{2} \sqrt{- b - 2 \sqrt{- \lambda}}, \quad - \sqrt{\frac{b}{4} - \frac{\sqrt{- \lambda}}{2}}\right ), \quad \left ( - \frac{1}{2} \sqrt{- b - 2 \sqrt{- \lambda}}, \quad \sqrt{\frac{b}{4} - \frac{\sqrt{- \lambda}}{2}}\right ), \quad \left ( \frac{1}{2} \sqrt{- b - 2 \sqrt{- \lambda}}, \quad - \sqrt{\frac{b}{4} - \frac{\sqrt{- \lambda}}{2}}\right ), \quad \left ( \frac{1}{2} \sqrt{- b - 2 \sqrt{- \lambda}}, \quad \sqrt{\frac{b}{4} - \frac{\sqrt{- \lambda}}{2}}\right ), \quad \left ( - \frac{1}{2} \sqrt{- b + 2 \sqrt{- \lambda}}, \quad - \sqrt{\frac{b}{4} + \frac{\sqrt{- \lambda}}{2}}\right ), \quad \left ( - \frac{1}{2} \sqrt{- b + 2 \sqrt{- \lambda}}, \quad \sqrt{\frac{b}{4} + \frac{\sqrt{- \lambda}}{2}}\right ), \quad \left ( \frac{1}{2} \sqrt{- b + 2 \sqrt{- \lambda}}, \quad - \sqrt{\frac{b}{4} + \frac{\sqrt{- \lambda}}{2}}\right ), \quad \left ( \frac{1}{2} \sqrt{- b + 2 \sqrt{- \lambda}}, \quad \sqrt{\frac{b}{4} + \frac{\sqrt{- \lambda}}{2}}\right )\right ]$$

Формулы, описывающие удобное решение:

(sp.solve([2*B**2-2*A**2 - b,
          (A**2+B**2)**2+l], [A, B], dict=True)[7])

$$\left \{ A : \frac{1}{2} \sqrt{- b + 2 \sqrt{- \lambda}}, \quad B : \sqrt{\frac{b}{4} + \frac{\sqrt{- \lambda}}{2}}\right \}$$

Построение карты собственных значений

import numpy as np

N = 3001
M = 2001
bs = np.linspace(0.0 , 1.0, N)
ls = np.linspace(-0.3, 0.01, M)

caseImaginaryPairsDets = np.zeros((M, N))
caseComplexRealDets = np.zeros((M, N))
caseAllComplexDets = np.zeros((M, N))

for j, bb in enumerate(bs):
    for i, ll in enumerate(ls):
        caseImaginaryPairsDets[i, j] = np.nan
        caseComplexRealDets[i, j] = np.nan
        caseAllComplexDets[i, j] = np.nan
        if ll > 0:
            GG,AA = np.sqrt(-bb/2 + np.sqrt(bb**2 + 4*ll)/2), np.sqrt(bb/2 + np.sqrt(bb**2 + 4*ll)/2)
            caseComplexRealDets[i, j] = chpolyComplexReal(AA, GG)
        elif ll < - bb**2/4.0:
            BB, AA = np.sqrt(bb/4 + np.sqrt(-ll)/2), np.sqrt(-bb + 2*np.sqrt(-ll))/2
            caseAllComplexDets[i, j] = chpolyAllComplex(AA, BB)
        elif ((ll > -bb**2/4.0) and (ll < 0)):
            OO, AA = np.sqrt(bb/2 - np.sqrt(bb**2 + 4*ll)/2),  np.sqrt(bb/2 + np.sqrt(bb**2 + 4*ll)/2)
            caseImaginaryPairsDets[i, j] = chpolyImaginaryPair(AA,OO)

import matplotlib.pyplot as plt

plt.contour(bs, ls, caseAllComplexDets, [-0.001, 0.001], colors='red')
#plt.contourf(bs, ls, caseComplexRealDets, [-0.001, 0.001], colors='blue')
plt.contour(bs, ls, caseImaginaryPairsDets, [-0.001, 0.001], colors='green')
plt.xlabel('$b$')
plt.ylabel('$\lambda$')
#plt.plot([0.0, 1.0], [0.0, 0.0], 'k--')

<matplotlib.text.Text at 0x7faf931af2d0>