Perturbation theory with a tridiagonal zeroth-order hamiltonian

Volume 120, number 7

PHYSICS LETTERS A

9 March 1987

PERTURBATION THEORY WITH A TRIDIAGONAL ZEROTH-ORDER HAMILTONIAN M. ZNOJIL Institute ofNuclear Physics, Czechoslovak AcademyofSciences, 25068 1~e~ Czechoslovakia Received 17 November 1986; revised manuscript received 4 January 1987; accepted forpublication 13 January 1987

We generalize the Rayleigh—Schrodinger perturbationformalism to the hamiltonians H=H

0+2H1 where the correction AN1 is small and the unperturbed 2E operator H0 is representedby 21W2)an +... infinite in terms tridiagonal ofthe analytic matrix.continued This enables fractions. us to construct the solutions E=E0+AE~+A 2+... and ~‘>= IWo> +AIw~> +2

A knowledge of the unperturbed states I n> and of the unperturbed hamiltonians

Ho=>~In>a 0
(1)

is needed in most perturbation theories [1]. Usually, we assume that the operator H0 is a good approximation to the exact hamiltonian H and solve the full Schrodinger equation ~HIy/>=EIy,>

(2)

by the power series (Rayleigh—Schrodinger) ansatz 2E 21w E=E0+AE1+A 2+..., Iw>=Iwo>+21w1>+A 2>+....

(3)

Here,the parameter A characterizes formally the smallnessof the perturbation H—H0—ill1. The Rayleigh—Schrodinger (RS) approach to (2) may fail to work even for the simplest anharmonic oscillator 2+vr4, 11>0. (4)

H=—~+jir

Indeed, when we interpret yr4 as a perturbation with A = y, we find that the RS series (3) diverges [2]. A number of papers (cf., e.g., their list in ref. [3]) has been devoted to a resummation and meaningful re-interpretation of this result. Also, a natural source ofconvergence has been sought in a repartitioning ofHand found in an improved choice ofH 2)2In>[4],a 0 (1) with a0= [5], etc. In all these cases, a guarantee of further acceleration of the slow rate of convergence would still be needed. In practice, a change ofthe basis n> is rarely desirable. For example, the harmonic-oscillator states are easily generalized to the many-body case, their polynomial structure simplifies an evaluation of the matrix elements of H1, etc. Hence, we encounter a challenge to generalize H0 (1) to a more general class of matrices which are not entirely diagonal. In the present letter, we shall consider the tridiagonal unperturbed hamiltonians H0=

n~0

ln>a0b0
~ In+1>b0_1
(5)

We shall describe the corresponding extension ofthe RS formalism and show that a lot of its merits may still be preserved. Mathematically our use of the analytic continued fractions [6] will in effect enable us to compress the redundant set ofparameters a0 and b0 into a single auxiliary sequence again. 0375-9601/87/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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PHYSICS LETTERS A

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The easiest way to proceed is to parallel simply the textbook presentation of the RS theory. Thus, inserting (3) into (2), [H0_E0+A(H1—E1)—A2E2—...](g~0>+Ai~1>+...)=0 (to be satisfied for each 2) we arrive at a sequence of the independent requirements H01w0>=EoIwo>

(6)

and k ~ i= I

(Ho—Eo)Iy/k>=—HI IWk—l >+

E1~ V/k_i>,

k=l,2

(7)

When we multiply (7) by
/

k—I

(\—



~ Ek_J)~ k=l,2

(8)

j=I

Thus, we may treat Ek as mere abbreviations and re-interpret (6) and (7) as an implicit recurrent definition of the wavefunction components. Immediately, we may notice that the overlaps a~=+ ~

m= 1

, k=l,2,...

(9)

remain unspecified by (6) and (7). We shall choose here the free parameters a,,, in such a way that =

l

0>

~

(ak_

) =0,

k=1,2


This differs from the RS theory where a ~,,RS) = <~I w~’~ > = 0, so that we have now I akIWO>. In the next step, multiplication of (7) by
,n~I

(10)

,n=l V/k>

I y4RS) > +

=~,

gives fk

I1lJk>Ro(EO)(~ ~ El Wk_1>—HlIWk_I> i= I

R 0(E0)=Q[Q(H0—E0)Q]~Q,

k=l,2,...,

(11)

where Q= 1 — 10> <01. This is the final explicit formula for the wavefunctions. In (6 )—( 11), an arbitrary matrix H0 may be used in principle. Although the RS theory seems to be the only formalism inverting Q( H0 E0 ) Q in a non-numerical manner, an alternative construction of the unperturbed propagator R0(E0) = [Q(H0 — E0) Q] I is possible also by means of analytic continued fractions [7]. This is our central idea. Its four essential non-RS technical features may be characterized as follows. (1) The continuedfractionalformula for R0 ( E0). We shall transform the parameters E0, a0 and b0 into a new auxiliary sequence of quantitiesf0 ( Es). They bare defined by the recurrences —

—

fk=(ak—Eo—b~J,,+I)’

,

318

k=l,2,...N,

fN+I=0,

(12)

Volume 120, number 7

PHYSICS LETTERSA

9 March 1987

in the limit N—~co, i.e., as the analytic continued fractions [61. Then, with the abbreviation k

U

dk

(bpnfrn±i), d01,

rn=I

it is easy to showthat R0(E0) is a simple matrix, 1

fI

1 QHOQ—EOQ

—

d1 d2

f2

1 d1

f3

1

d1 1

d2 d1 I

d3 d2 d1

13

(

Its insertion into (11) leads to the generalized RS formula rn-I

00

~ d,,,_1_1f÷1 ~

d0_1_i

n=i-i-I

i=I

1IW~—~ >—a

/

x(\_bfl_I
k 0
>—b0+ ~ EJ j=l

(14) containing the continued fractions. It is reducible precisely to the standard RS prescription when we put brn_1 0, i.e.,frn=(am—E0Y’ and dm=0 for all m~1. (2) The continuedfractionalformulafor I Wo>. The unperturbed equation (6) is closely related to the preceding construction. In accord with ref. [8], we may write =g~_——gobJjdrn_i,m=l,2,...,

(15)

and convert also the RS energy (8) into the continued fractional formula /oo

00

~

,n,n=0

Ek=,K

gmgn,

00

00

~

~ grn
2 \~

,4t( \ 1=0 ~g,

00

1In>_X ~ n=I rn=0

k—I

~ E,,,_1g0 k=2,3 ,

(16)

n=I j=l

Again, it degenerates to the ordinary RS prescription for brn_i = 0 since g,,,= 0, m ~ 1. (3) The continued-fractional determination ofE0 or a0. The unperturbed energy E0 should be an eigenvalue ofthe unperturbed matrix H0. Thus, the secular equation det(H0 —E0) = 0 determines the value ofE0 as a pole off0 [8], a0—E0—b~Jj(Eo)_—0.

(17)

This may be interpreted either (simply) as a definition ofthe parameter a0 or (numerically) as a transcendental equation for E0. The latter case represents merely a continued-fractional reformulation ofthe Lanczos standard numerical eigenvalue method [9]. In both cases, the evaluation of a0 orE0 generates alsothe auxiliary sequence f1(E0),f2(E0), ...asabyproduct. (4) An example of analysis of the continued-fractional convergence. We have the choice of H0 under our control. Hence, a convergence ofthe functions defined by (12) in the limit N—pon may be easily guaranteed by the simple criterion [6] 319

Volume 120, number 7

PHYSICS LETTERS A

a~>4b~, n>N0.

9 March 1987

2

(18) r2/p of the “spring constant” in the

—~

This may be best illustrated on[5]. our The example (4) with requirement 1 I n> = 0 fixes then the matrix elements, ~ 112(n+1+~)’’2 2=v/p~ ,

,

n, 1=0,1,...,

(19)

y~=2n+1+~ /3~=(n+l) and we may choose also ,

—_=I3~I3~÷i, =_-=I3~y~±i, b~=(—p+~üp+Ay,,)fl~, m>N0.

(20)

Such a choice ofH0 (5) will imply that [10] ~

vn>>1,

(21)

i.e., the continued fractions converge and decrease quickly enough to suppress also the asymptotics of the auxiliary factors d~and g~for vn 1. In conclusion, we may notice that the present formalism may be also combined with the ordinary RS expansions. For example, we may choose the particular scaling p=p(M) in (20) such that bM—O, i.e., 3(M)—/.co(M)—vyM=0. (22) p Then, the auxiliary continued fractions will terminate, and the low-order formulae acquire a finite-dimensional character resembling the RS formalism with linear combinations of the oscillator states I n>. In the simplest cases, we even get the compact prescriptions ~‘

E=a

2), M=0, 0+O(v E=~{a 2+4b~]”2}+O(v2) M=l (23) 0+a1 [(a1—a0) reproducing and improving the first-order RS ground state energies, respectively. In the higher orders, a more detailed account of applications of the present formalism will be published in the near future [11]. ,

,

—

Note added I am obliged to the referee for noticing that the condition bM= 0 is closely related to some recent analyses of optimality of the diagonal RS operators H 0. Indeed, if we treat the scaling p as a free parameter and try to improve the RS convergence properties by its appropriate choice (P ~Poptimai), then the value OfPoptimal is believed tobe order dependent [12]. In this context, our explicit and order-dependent prescription (22) might represent a very natural tentative candidate for POPtirnaI.

References [1] P.M. Morse and H. Feshbach, Methods of theoretical physics, Vol. 2 (McGraw-Hill, New York, 1953). 12] B. Simon~Ann. Phys. (NY) 58 (1970) 76. [3] G,A. Arteca, F.M. Fernández and E.A. Castro, J. Math. Phys. 25 (1984) 2377. [4]1.G. Halliday and P. Suranyi, Phys. Rev. D21 (1980) 1529.

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PHYSICS LETTERS A

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I.D. Feranchuk and L.I. Komarov, Phys. Lett. A 88 (1982) 211. H.S. Wall, Analytic theory ofthe continued fractions (Van Nostrand, London, 1948). R. Haydock, J. Phys. Al (1974) 2120. M. Znojil,J. Math. Phys. 21(1980)1629. J.H. Wilkinson, The algebraic eigenvalue problem (Clarendon, Oxford, 1965). M. Znojil, J. Phys. A 13 (1980) 2375, Appendix. M. Znojil, Phys. Rev. A., to bepublished. G.A. Arteca, F.M. Fernández and E.A. Castro, J. Math. Phys. 25 (1984) 3492; F.M. Fernández, A.M. Meson and E.A. Castro, Phys. Lett. A 104 (1984) 401; 111(1985)104.

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