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Kernel: Python 2 (SageMath)
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
bz2λ+z4b 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 ±iω\pm i \omega, ±iα\pm i \alpha

sp.collect(sp.expand((z-sp.I*a)*(z+sp.I*a)*(z-sp.I*o)*(z+sp.I*o)), z)
α2ω2+z4+z2(α2+ω2)\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)
α6ω3sin(α)cos(ω)+α5ω4sin(ω)cos(α)+α4ω5sin(α)cos(ω)α3ω6sin(ω)cos(α)- \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 b24<λ<0-\frac{b^2}{4} < \lambda < 0

Случай ±iα\pm i \alpha, ±γ\pm \gamma

sp.collect(sp.expand((z-g)*(z+g)*(z-sp.I*a)*(z+sp.I*a)), z)
α2γ2+z4+z2(α2γ2)- \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)
C1cos(αx)+C2sin(αx)+C3cosh(γx)+C4sinh(γx)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)
[{α:b212b2+4λ,γ:b212b2+4λ},{α:b212b2+4λ,γ:b212b2+4λ},{α:b212b2+4λ,γ:b212b2+4λ},{α:b212b2+4λ,γ:b212b2+4λ},{α:b2+12b2+4λ,γ:b2+12b2+4λ},{α:b2+12b2+4λ,γ:b2+12b2+4λ},{α:b2+12b2+4λ,γ:b2+12b2+4λ},{α:b2+12b2+4λ,γ:b2+12b2+4λ}]\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 λ>0\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 ±a±bi\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)
A4+2A2B2+B4+z4+z2(2A2+2B2)A^{4} + 2 A^{2} B^{2} + B^{4} + z^{4} + z^{2} \left(- 2 A^{2} + 2 B^{2}\right)

При λ<b24\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)
C1eAxcos(Bx)+C2eAxsin(Bx)+C3eAxcos(Bx)+C4eAxsin(Bx)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)
[AC1AC3+BC2+BC4A3C1A3C3+3A2BC2+3A2BC43AB2C1+3AB2C3B3C2B3C4A2C1eAcos(B)+A2C2eAsin(B)+A2C3eAcos(B)+A2C4eAsin(B)2ABC1eAsin(B)+2ABC2eAcos(B)+2C3eAABsin(B)2C4eAABcos(B)B2C1eAcos(B)B2C2eAsin(B)B2C3eAcos(B)B2C4eAsin(B)A3C1eAcos(B)+A3C2eAsin(B)A3C3eAcos(B)A3C4eAsin(B)3A2BC1eAsin(B)+3A2BC2eAcos(B)3C3eAA2Bsin(B)+3C4eAA2Bcos(B)3AB2C1eAcos(B)3AB2C2eAsin(B)+3C3eAAB2cos(B)+3C4eAAB2sin(B)+B3C1eAsin(B)B3C2eAcos(B)+B3C3eAsin(B)B3C4eAcos(B)]\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])
[(12b2λ,b4λ2),(12b2λ,b4λ2),(12b2λ,b4λ2),(12b2λ,b4λ2),(12b+2λ,b4+λ2),(12b+2λ,b4+λ2),(12b+2λ,b4+λ2),(12b+2λ,b4+λ2)]\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])
{A:12b+2λ,B:b4+λ2}\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, [-0.0001, 0.0001], colors='red') #plt.contourf(bs, ls, caseComplexRealDets, [-0.001, 0.001], colors='blue') plt.contour(bs, ls, caseImaginaryPairsDets, [-0.0001, 0.0001], colors='green') plt.xlabel('$b$') plt.ylabel('$\lambda$') _, xmax = plt.xlim(); ymin, ymax = plt.ylim(); plt.plot(bs, [-bb**2/4.0 for bb in bs], 'k--') plt.ylim([ymin, ymax]);

Брутфорсовая проверка разрешимости задачи на собственные числа с отрицательным λ\lambda при положительном bb

import numpy as np import matplotlib.pyplot as plt
bb = 0.1 detAllComplex = [] range1 = np.linspace(-25, -bb**2/4.0, 1000, endpoint=False) for ll in range1: BB, AA = np.sqrt(bb/4 + np.sqrt(-ll)/2), np.sqrt(-bb + 2*np.sqrt(-ll))/2 detAllComplex.append(chpolyAllComplex(AA, BB)) detImaginaryPair = [] range2 = np.linspace(-bb**2/4.0+0.001, 0.0, 1000, endpoint=False) for ll in range2: OO, AA = np.sqrt(bb/2 - np.sqrt(bb**2 + 4*ll)/2), np.sqrt(bb/2 + np.sqrt(bb**2 + 4*ll)/2) detImaginaryPair.append(chpolyImaginaryPair(AA,OO)) plt.plot(range1, detAllComplex, 'r')
[<matplotlib.lines.Line2D at 0x7f6229d9de90>]
Image in a Jupyter notebook
plt.plot(range2, detImaginaryPair, 'b')
[<matplotlib.lines.Line2D at 0x7f6229d30f50>]
Image in a Jupyter notebook