A Lecture Notes Convergence of Perturbation Theory Asymptotic Series An asymptotic series in a parameter ɛ of a function is given in a power series f(ɛ) = f n ɛ n () n=0 where the series actually does not converge. Instead, if you truncate the series at an order N N f N (ɛ) = f n ɛ n, () n=0 the difference between the true value f(ɛ) and and approximate expression f N (ɛ) goes to zero (f(ɛ) f N (ɛ))/ɛ N 0 as ɛ 0. In most examples, the perturbation theory gives an asymptotic series rather than a Taylor series. If it doesn t converge, why do we trust it? In the lecture notes on the steepest descent method, an asymptotic expansion of the Gamma function Γ(x) in inverse powers in x is discussed. The series is (obtained by Series[Γ[x], {x,,0}]), Γ(x) = πx x x e x ( + x + 88 x 39 5840 x 3 57 48830 x 4 (3) + 63879 0908880 x + 54689 5 7546796800 x 53470353 6 909656600 x 4483359 7 8668430993600 x 8 + 4369637 ) 54904800886784000 x + 635305089 9 865040065489797000 x + 0 O(x ). (4) It is an asymptotic series because its derivation ignores the exponentially suppressed tail when the integration region is changed from x [0, inf) to ( inf, inf). Therefore, it misses corrections that behave as e x or e x. In fact, if you attempt to expand e x in power series of /x, or equivalently, g(ɛ) = e /ɛ in power series of ɛ, you find that coefficient at each order vanishes. g(0) = lim ɛ 0 e /ɛ = 0, (5)
n sum 0 0.937 0.937 0.99898 3.0080 4 0.9997 5 0.999499 6.0000 7.00090 8 0.99974 9 0.999693 sum n sum 0.00047.00053 0.998768 3 0.99868 4.0045 5.0050 6 0.97779 7 0.975504 8.406 9.5495 n sum 0 0.8483 0.35955.88 3 3.54 4 3.44 5 43.57 6 878.3 7 047.4 8 34.9 9 3403.3 0.8 0.6 0.4 0. 5 0 5 0 5 30 n Table : The sum of the asymptotic series of Γ() up to the n-th order. g (0) = lim ɛ e /ɛ = 0, (6) ( g (0) = lim ɛ 0 ɛ ) e /ɛ = 0, (7) 4 ɛ 3 ɛ 0 and so on. At each finite order in Taylor expansion, the prefactor is always power-divergent while the exponential factor wins. Therefore the expansion of this function around ɛ = 0 never sees the function at all. In other words, there is no information about this function at the origin. The exponentially suppressed correction is nominally smaller than the power-suppressed correction in power series expansion and ignored, while it comes back and haunts you when you evaluate it for the finite value of the expansion parameter. On the other hand, this asymptotic series works quite well from the prac-
tical point of view. Even for x =, where an expansion in /x is not expected to be good at all, the series appears to converge to the true result Γ() =. The series approaches the true value up to the sixth order within the error of 0.000, quite remarkable. But beyond the sixth order, the series starts to deviate from the true result, and eventually goes completely wild. Convergence of the Perturbation Series. Infinite Radius of Convergence Some perturbation series is convergent with an infinite radius of convergence. The simplest example is probably the harmonic oscillator with a linear term as the perturbation, H = p + x x 0 x. (8) For simplicity, we take m = ω = h =. One can solve the system exactly, because H = p + (x x 0) x 0. (9) The ground state is only a shifted Gaussian, with the energy E 0 = x 0. The perturbative expansion gives the second-order shift () = 0(0) x 0 x (0) (0) x 0 x 0 (0) E (0) 0 E (0) = x 0/ = x 0, (0) reproducing the exact result. We used x = (a + a )/ to obtain this result. It is not obvious, but all the higher order corrections vanish. The ground-state wave function can be obtained by rewriting the Hamiltonian as H = a a + a + ( a x 0 = a x ) ( 0 a x ) 0 x 0. () The ground-state must satisfy ( a x ) 0 0 = 0, () and therefore it is a coherent state, 0 = e a x 0 / 0 (0) e x 0 /4. (3) 3
In the perturbation theory, the correction to the ground-state wave function is given by Eq. (5..44) in Sakurai, 0 = 0 (0) + (0) (0) x 0 x 0 (0) E (0) 0 E (0) + (0) (0) x 0 x (0) (0) x 0 x 0 (0) + O(x (E (0) 0 E (0) )(E (0) 0 E (0) 0 ) 3 ) = 0 (0) + x 0/ (0) + x 0/ (0) + O(x 0 ) 3. (4) This expression agrees with the Taylor series expansion of e a x 0 / 0 (0). However, this state is not properly normalized. It needs the wavefunction renormalization factor Eq. (5..48b) in Sakurai, Z 0 = (0) x 0 x 0 (0) = x 0 ( ) (5) The normalization factor e x 0 /4 agrees with Z / 0 up to this order. Even if x 0 is large, the perturbation series converges.. Finite Radius of Convergence Another simple example is again the harmonic oscillator with a quadratic term as the perturbation, H = p + x + ɛ x. (6) This is the example in Sakurai, pp. 94 96. The exact ground-state energy is E 0 = ( + ɛ = + ɛ ) 8 ɛ + O(ɛ 3 ), (7) which agrees with the perturbative result. However, we know that the radius convergence of the Taylor series expansion of + ɛ is ɛ <. In particular, there is a branch cut for ɛ < and it is clear that the series does not converge there. There is a physical intuitive picture why the perturbation series does not converge for ɛ >. For ɛ <, the potential is actually upside down and there is no stable ground state. Therefore, the perturbation series should not converge! 4
.3 Zero Radius of Convergence The most famous example is the anharmonic oscillator, H = p + x + λx 4. (8) The ground-state energy is expanded in the power series, E 0 = ( ) + λ n A n. (9) n= The coefficients A n can be calculated using the recursion formula given by Carl M. Bender and Tai Tsun Wu, Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order, Phys. Rev. D 7, 60 636 (973), Section VI. They are given in Table. In order to test the perturbative result, I also computed the ground-state energy numerically. I give examples for λ = 0.0 in Table 3 and 0.05 in Table 4. For the former, the series converges to the true result and appears stable up to the 40th order. For the latter, the series comes close to the true result at the sixth and seventh order, and then it starts to diverge. We actually expect the perturbation series to have the zero radius of convergence. This is because a negative λ, no matter how small in the magnitude, leads to a potential unbounded from below and does not give a stable ground state. See Fig.. The ground state would tunnel through the potential barrier. The tunneling amplitude can be estimated using the WKB method if the barrier is sufficiently high, or equivalently, if λ is sufficiently small. Neglecting the energy relative to the height of the potential barrier, the WKB amplitude of the tunneling is exp / λ Changing the variable x = ξ/ λ, it becomes 0 ( ) x λ x 4 dx. (0) exp / ( ) λ 0 ξ ξ 4 dξ = e /6 λ. () Clearly, this effect cannot be obtained in the perturbation theory. 5
n A n 0.75000000.6500000 3.085000 0 4.48906 0 5 3.58098047 0 3 6 6.398835 0 4 7.3973373 0 6 8 3.44847 0 7 9 8.3354603 0 8 0.44789407 0 0 7.8933336 0.77387770 0 3 3.05564666 0 5 4 4.368068 0 6 5.900870 0 8 6 8.9075 0 9 7 4.4455089 0 8.3464643 0 3 9.309506 0 5 0 7.69399985 0 6 n A n 4.754077 0 8 3.0757995 0 30 3.0830009 0 3 4.479049 0 34 5.085596 0 36 6 8.348368 0 37 7 6.633937 0 39 8 5.483943 0 4 9 4.69784 0 43 30 4.650700 0 45 3 3.8734896 0 47 3 3.699554 0 49 33 3.577800 0 5 34 3.5503394 0 53 35 3.6797476 0 55 36 3.930600 0 57 37 4.30055097 0 59 38 4.843778 0 6 39 5.59966 0 63 40 6.6486378 0 65 Table : The coefficients A n in the perturbative expansion of the groundstate energy of the anharmonic oscillator. Becker and Wu showed that the coefficients of the power series behaves as A n = ( )n+ 6 3 n Γ(n + ( π 3/ ) 95 ) 7 n + O(n ), () and hence grow faster than n!. No matter how small λ is, A n λ n grows very fast for large n and makes the series non-convergent. In general, it is typical that the perturbation theory has zero radius of convergence, and the perturbative series is only an asymptotic series. Nonetheless, as long as the expansion parameter is small, the series approaches the true result quite well at a finite order. 6
.0500000.0447500 3.04566 4.04580 5.0455 6.04539 7.0454 8.0454 9.0454 0.0454.0454.0454 3.0454 4.0454 5.0454 6.0454 7.0454 8.0454 9.0454 0.0454.0454.0454 3.0454 4.0454 5.0454 6.0454 7.0454 8.0454 9.0454 30.0454 3.0454 3.0454 33.0454 34.0454 35.0454 36.0454 37.0454 38.0454 39.0454 40.0454 Table 3: The good apparent convergence of the series for λ = 0.0. The true result is.0454 obtained by solving the Schrödinger equation numerically..07500000.0687500 3.0670783 4.064060 5.0663003 6.06430066 7.06637837 8.063948 9.0677750 0.0639646.0700479.05656047 3.0833309 4.095557 5.45537 6 0.87355644 7.5543608 8 0.3877459 9 4.7553874 0 9.9990573 3.539474 0.7384 0 3 3.85354663 0 4.37048546 0 3 5 5.0997380 0 3 6.9707 0 4 7 7.933605 0 4 8 3.9450770 0 5 9.406588 0 6 30 6.3373449 0 6 3.94504 0 7 3.390879 0 8 33 6.8346905 0 8 34 3.4506745 0 9 35.79649554 0 0 36 9.6098984 0 0 37 5.960979 0 38.99434377 0 39.73769365 0 3 40.03437934 0 4 Table 4: The apparent convergence and then divergence of the series for λ = 0.05. The true result is.065855 obtained by solving the Schrödinger equation numerically. 7
V, λ=0.05 0.5-4 - 4 x -0.5 - -.5 Figure : The potential of the anharmonic oscillator V = x + λx 4 with λ = 0.05. 8