Analytic interpolation of the eigenvalues of an asymmetric rotator

Nuclear Physics A161 (1911) 481-491; Not to be reproduced

ANALYTIC

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INTERPOLATION

Co., Amsterdam from the publisher

OF THE EIGENVALUES

OF AN ASYMMETRIC

ROTATOR

J. D. TALMAN University

of Western

Ontario, London, Ontario, Canada t and The Niels Bohr Institute, University of Copenhagen, DenmaFk

Received 1G September 1970 Abstract: The behavior of the eigenvalucs of the asymmetric rotator as functions of a continuous variable z .y 1(1-‘1) is considered. The eigenvaluc problem for the Hamiltonian matrix in a basis of cigcnstates of I2 and 4, where I3 is parallel to one of the principal moments of inertia, is studied by transforming it to a Sturm-Liouville problem. It is found that a proper cigenvalue problem is obtained only if I3 is parallel to the largest or smallest moment of inertia. It is shown in this case that an eigenvaluc 3, is an analytic function for .z in the neighborhood of the real axis, but may have branch points off the real axis, and that 1 is real for t real. The behavior of the eigenvalues as the intermediate moment of inertia approaches one of the others is considered in some detail.

1. Introduction The current great interest in the analytic properties of the scattering amplitude as a function of the total angular momentum stems originally from the study by Regge of two-particle potential scattering. One of the directions in which one could hope to extend this analysis is to three or more particle systems. The extension from two to three-particle systems is, however, faced with serious difficulties of convergence as described by Omnb and Alessandrini ‘). It might be worthwhile, therefore, to discuss some aspects of a more complicated system in the limit of very strong binding. One of the results of the analysis of the two-body problem is that the poles of the scattering amplitude, as a function of I, provide an interpolation of the bound state energies of the system. In the case of a strongly bound system, the bound state energies are just those of a rigid rotator. In the case of the two-particle system, the strongbinding limit is that of an axially symmetric rotator, the Hamiltonian is of the form k(1: +f:) and the energies are /cl@+ 1). (The system cannot have angular momentum parallel to the 3-axis, so that I3 = 0.) In order to go beyond these linear trajectories of the strongly bound system, it may be of interest to consider the trajectories of an asymmetric rotator which might be considered as a model of a strongly bound system of several particles. In this article, some properties of an interpolation scheme for the eigenvalues of the rigid asymmetric rotator will be described. t Present address. 481

J. D. TALMAN

482

The Hamiltonian of a rigid rotator can be written A,Z: +AzZ; +AJZi. However, by a suitable scaling and inclusion of a term in Z2, the Hamiltonian can be written in the one-parameter form HP = p(Zf-Z;)+Z,2 = jrp(z”++z2_)+Zg.

(1)

For the sake of comparison, the Hamiltonian usually used in molecular spectroscopy is H: = Z;+K+-I: = 3(1 +4Z2+(3+4HB,

/l=-.

1-k 3+lc

The Hamiltonian can be expressed as a matrix in the basis of eigenstates of I2 and 1s. It will be found, though, that an interpolation of the physical eigenvalues is obtained only if the quantization axis is chosen to be parallel either to the largest or smallest of the moments of inertia; this is equivalent to the condition - 1 < ~1< 1. There are then two different interpolations, corresponding to the choice of the 3-axis parallel to the largest, or smallest moment of inertia. It may be of interest to note that only the physical points lie on both types of trajectories. The non-vanishing matrix elements of ZZP,in the basis of eigenstates of Z2 and I,, are hwl = k2,

3~(Cz-k(k+1)1Cz-(k+l)(k+2)1}~ = &[(1-k)(Z+k+l)(Z-k-l)(Z+k+2)]*,

hk,k+z =

(3)

kc,ik--2 = kk-z,lr, where z = Z(1+ 1) and -1 5 k 5 I. The interpolation for the eigenvalues is to be performed by considering the eigenvalues of H,, for z other than Z(Z+l), I an integer. It is seen that the matrix is the sum of two commuting matrices corresponding to even and odd values of k, and that the eigenvectors must have non-vanishing components only for k even or k odd. This corresponds to the symmetry of the Hamiltonian under the transformation Zr + -Z1, I2 --, -I,. The symmetry under the transformation I, -+ -I, requires that the components a, of an eigenvector must satisfy a, = +a_,. This is reflected by the symmetry h_,,_,_, = h,,,,, in the matrix elements. At first glance, the matrix elements (3) are ambiguous because of the square root. If, however, the matrix is transformed with the diagonal matrix with elements %I” = 1,n 5 k,u,,, = - 1, n > k, the sign of h,,,n+2 is changed, so that the definition of the square root is irrelevant. We assume then, that a definite sign for (I+k)‘:‘, k = 0, fl, . . . has been fixed initially and that this sign will be used consistently each time such a factor occurs.

fNTERPOLATION

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OF EIGENVALUES

If 2 is an integer, h,, 1+2 = h_I_-2, -I = 0. In this case the Hamiltonian is the sum of three commuting pieces, one of finite dimension 21+ 1, and two of infinite dimension. It is, of course, the finite-dimensional piece that provides the physical eigenvalues of the rotator. If p = 0, the eigenvalues of H are m2, m = 0, & 1, . . . . The states for ,U# 0 can be labelled by m as an approximate quantum number that describes the dominant state in the wave function, together with the symmetry type in k. If I is not an integer, the Ha~ltonian defined by (3) is symmetric rather than Hermitian. This is reasonable if we wish to consider analyticity properties; the eigenvalues of the Hermitian matrix cannot have any analyticity property as a function of z since the Hermitian matrix involves complex conjugation. In the following section we will reformulate the eigenvalue problem for the matrix (3) as a Sturm-Liouville problem. We will then be able to discuss the analytic behavior of the eigenvalues for 1~1< 1 as functions of z on the basis of this formulation. We will also consider the qualitatively different case l,u] > 1 corresponding to qusntization about the axis of the intermediate moment of inertia; at first glance it might be expected that one problem could be transformed into the other by a rotation of the coordinate axes, and it will be shown why this is not the case. We will also consider in detail the limit /A-+ I- to see how the change in behavior at p = 1 develops. 2. Formulation as a Sturm-Liouville problem The eigenvalue problem h,,~-zap-2+(h,--l)ak~hk,k+2aA+*

= 0,

(4)

is to be studied by transforming it to one of the familiar Sturm-Liouville type. The first step is to remove the square roots in the matrix elements by making the substitution a, = [(l-k)!]-*[(Z+k)!]-tbk,

(9

to obtain the eigenvalue equation ~,~(E+k)(l-k+l)b,_,+(k~-;L)b,+-),u(f-k)(l-k-l)b,+,

= 0.

(6)

The coefficients in (5) are again ambiguous: the square roots of the factorials must be defined by [(Z+k)!]*

= (l+k)“[(l+k-l)!]”

,

where the sign of (i+ k)” is determined by the definition of the Hamiltonian, and one coefficient, say (E!)+, can be defined arbitrarily. There is a further ambiguity in changing variables from z to I. We define I = -$+(z+$)* so that Re(f+$) < 0. Since the problem is symmetric in 1-l-4, this is no real restriction. Stirling’s formula can be applied to show that, for k + co, aR = O(k-“b,), a = Re(I-f-3)

< 0.

(7)

J. D. TALMAN

484

The inverse of the transformation defined by (5) is not defined for all vectors in the Hilbert space but only those for which Clk-“b,l’ converges. If an eigenvector of the transformed problem is found, it should therefore be checked whether it can be transformed back to the original problem. The eigenvalue problem (6) is no longer symmetric, in keeping with the fact that the transformation defined by (5) is not unitary. The eigenvalue problem (6) can be transformed to a differential equation by defining a “wave function” by S(4) =,=$ mb,ei’9. The wave functionf -(l+~

(8)

then satisfies the eigenvalue problem

cos 24)f”($)+(21+3)~

sin 247(4)+(l+1)(1+2)~

cos 2$f($)

= 3$(4), (9)

with periodic boundary conditions on (0,2n). This equation is not self-adjoint. It can be transformed, however, to a form that depends on I only through f(I+ 1) and is self-adjoint, for z = /(I+ 1) real. We write f(P) = [1+~cos2~]-*~2’+1)g(~),

(10)

and obtain the eigenvalue problem

The transformation (10) does not affect the periodic boundary conditions. Eq. (11) is the desired expression as a Sturm-Liouville problem. If 1,nl< 1, the eigenfunctions g of (I l), and f of (9), are arbitrarily differentiable functions. It can then be shown that the Fourier coefficients b, off approach zero more rapidly than k-” for any n, so that the solutions of (11) must correspond to eigenvectors of the original problem. The eigenvalues of (11) interpolate the lowest eigenvalues of the physical problem; that is, the ground state of (11) as a function of z passes through the smallest eigenvalues of (3) for integral 1. The Hamiltonian (1) can also be written

For p > 0, this expression, when quantized about the 2-axis, provides an interpolation of the largest eigenvalues of the physical problem. It is of interest to consider the symmetries of eq. (11) corresponding to those of the original problem. If k is even, g satisfies g(4) = g(n--&) and hence g’&) = 0. On the other hand, if k is odd, g satisfies g(4) = -g(n-4) and g&r) = 0. If a, = a-p, g(q) is even and satisfies g’(0) = 0, and if a, = -a+, g(q) is odd and g(0) = 0. Therefore the original problem is equivalent to solving eq. (11) on (0, &r) with

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OF EIGENVALUES

485

boundary conditions g’(0) = g’(_lix) = 0, etc. Changing the sign of JL is equivalent to the substitution cp -+ 3x-4 so that the solutions for -,u are those for ,u with the boundary conditions at 0 and $n interchanged. It is somewhat surprising that although the variables in the S&r&linger equation for the asymmetric rotator cannot be separated, it is still possible to formulate the problem as an ordinary differential equation. 3. Analyticity of the interpolation for lpj c 1 The analyticity properties of an eigenvalue 1 can be obtained from eq. (11). It is seen immediately that 3, is real if z is real. Furthermore, in this case, g, andfmay be assumed real, and it follows that the bk may be assumed to be real. The consequence of this for the a, is that if I is real and positive, uk may be chosen to be real for jkj < 1. In this case uk,,, k,- 1 -C I 1. The eigenvectors may be transformed to be real; the four imaginary matrix elements in (3) then become real, but antisymmetric. The Sturm-Liouville problem corresponding to eq. (11) may be formulated as follows. For arbitrary 2, form the solution of (11) satisfying g(0) = 0 (g’(0) = 0) and g’(0) = 1 (g(0) = 1). F orm the function q(d, z) = g(&r) (g’@). This is an entire function of A and z. The eigenvalues A are the roots of the equation

tt(5 4 = 0,

(12)

and arc analytic functions of z provided the derivative ~~(1, z) # 0. In the case r(A z) = s&), ttAcan be obtained from the derivative g1 which satisfies the inhomogeneous equation -(~g~)‘+zqg~~~g~

= Ag,+g.

(13)

For convenience, eq. (11) has been written as - (pg’)‘+zqg +rg = Lg. The function gA satisfies also gl(0) = g;(O) = 0 since g(0) and g’(0) are fixed. The solution of (13) with these boundary conditions is g&4 = Ag(~)~~g,(~)s(~lp(~)-

id~-ag,t~)~~g(g)2~(~~-id~y

where g1 is any second solution of eq. (11) and A is the constant (gg; -gig’)/p. In the case g&c) = 0, g,(in) = 0 only if

the same condition can be seen to apply in the case ?(A.,z) = g’&).

(14) value of

J. D. TALMAN

486

Eq. (15) is certainly not valid for z real since then g can be assumed to be real. If z is complex, however, eq. (15) can be satisfied and n(z) has a singularity. This singularity is a branch point corresponding to the merging of two roots of (13). This is readily seen by expanding the analytic function q(A, z) about the singularity: ?(A, z) = a(Z-Z,)+~(n-&)z+2c(n-lz,)(z-z,)+d(z-z,)2.

. ..

The solution of (12) then shows two roots that approach A, as z approaches ze. The eigenvalues 1 lie on different sheets of a many-sheeted function A(z). The various sheets meet at the branch points and it may be possible to pass continuously from one eigenvalue’to another by varying z in the complex plane. Degeneracy of eigenvalues corresponding to different symmetries does not, of course, give rise to branch points. Unfortunately, it appears to be diflicult to make definite statements about the location of the branch points. They can at least exist; in the case g’(0) = g’&) = 0, p = 0.2, one has been found by a numerical calculation near z = 0.362- 14.5 i, corresponding to a value of Iz of 2.06-0.104 i. It is, however, unknown whether the branch points are Unite or infinite in number, or how they depend on p. They probably tend to infinity for p + 0 since they are absent for p = 0. Since they arise from the degeneracy of two eigenvalues, they can probably approach the real axis only if the eigenvalues are closely spaced for z on the nearby real axis. It will be seen that for P--+ 1-9 the spacing of the eigenvalues decreases, so that the branch points may approach the real axis in this limit, and may be associated with the change in behavior atp = *I. It is possible to obtain an indication of the location of some of the branch points by truncating the Hamiltonian to a 2 x 2 matrix. This gives the secular equation (k2-,l)[(k+2)2-rl]

= &u”[z-k(k+

l)][z-(k+

l)(k+2)].

The condition for the equality of the roots of this equation gives the value z = (k+1)2+(k+1)L

(16)

with the corresponding degenerate eigenvalue at 1 = +[k2+(k+2)2]. The double root for k = - 1 does not produce a branch point since the corresponding eigenvectors belong to different symmetries. The estimate (16) is in reasonable agreement with the branch point found numerically; in particular the value of 1 at the branch point is close to midway between the eigenvalues 0 and 4 of the “unperturbed” problem. It is seen that the estimate (16) is in agreement with the qualitative remarks. The imaginary part of the branch points increases with the spacing of the unperturbed levels and with p-‘. Unfortunately, the estimate (16) is not exact in any limit. There are also, perhaps, more branch points further removed from the real axis, corresponding to degeneracy of unperturbed eigenvalues that are not adjacent.

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OF EIGENVALUES

A certain amount is known about the eigenvalue problem (1 I ) when z is complex so that the problem is not self-adjoint “). In particuIar, the eigenfunctions are complete, and, in the absence of degeneracy of the roots, the eigenfunction expansion

is valid, where c, is a normalization constant. It should be noted that gn rather than g: is used to calculate the Fourier coefficients. Eq. (17) implies a similar completeness relation for the eigenvectors of the original problem. 4. Qu~tizatio~

about the axis of the intermediate moment of inertia

The nature of the eigenvalue problem (11) is entirely different if 1~1> 1. This case arises when A, < A3 < A, or A, < A 3 < A,. There is then a regular singular point in (0,27x) at ‘pO= 4 cos-l( - 1,‘~). Analysis of the nature of the solution of eq. (11) in the neighborhood of the singular point shows that it is of the form wh ere m = &(21-t-1) and CIand p are analytic. A ,(~,)(~-~o)m+B(~)(~-~o)-m solution satisfying the appropriate boundary condition at the origin cannot, in general, be carried continuously through +e ; rather, the solution can be defined arbitrarily on the other side of$@, and, in particular, in such a way as to satisfy the boundary condition at _tn. This means that there is a solution of (1 l), albeit discontinuous, for all I, or that the spectrum of the eigenvalue problem consists of all complex numbers. It is necessary to ascertain the behavior of bk for k + 00 in order to determine whether the discontinuous functions g correspond to a normalizable eigenvector of the original problem. This asymptotic behavior is dictated by the nature off in the neighborhood of #O. From eq. (10) it is seen that f(p) is of the form a(q)+ (~-#~)-~-~g(~). Th c asymptotic behavior of bk is governed by

which, on changing variables to k#, is seen to be O(k’-“).

Therefore, from eq. (7),

the eigenvector corresponding to 1 is therefore normalizable. This argument depends on the choice Re(1+&) < 0 in order to assure that f is continuous at (lz, and that the integral for b, exists. The above conclusion in the case IpI > 1 has previously been drawn by Cutkosky “1 from the recurrence relation (4). In the case that I is an integer, there is also the discrete spectrum of the physical states, for which a, = 0, jk[ > 1. The structure of the physical spectrum must be independent of the magnitude of p, since two of the moments of inertia can be inter-

J. D. TALMAN

488

changed by a rotation about the third axis by $r. In other words, the Hamiltonian A,Zf + A,Zz +A3Zi can be transformed to AiZf +A3Zg +AzZi by the unitary matrix D'(R),R being a rotation about the l-axis by $7~. The infinite matrix for 1 not an integer can also be transformed by the infinitedimensional matrix D'(R),and, in fact, the angular momentum matrices satisfy D'(R)Z,D'(R)-' = (RI),,

(1%

provided the sum on the left hand side converges. This can be demonstrated from the group representation property of the D'(R),which is discussed by Omnb and Alessandrini ‘). It can be shown though ‘), that dk(p) = O(m”-+ tanm -@),

(20)

for n tied and rn+ co.From this it can be seen that the sum in (19) converges geometrically for j? < +rc, but may diverge at /3 = 3~. There is no contradiction, therefore, between the fundamentally different spectra for 1~1< 1 and 1~1> 1, and the possibility of permuting the moments of inertia in the Hamiltonian by means of rotation of the coordinate system.

5. Behavior of the interpolation for p + lIn view of the change of behavior at p = 1, it may be of interest to study the solutions of the eigenvalue problem (11) for /J -+ 1 to WCif there is an indication of the development of a continuous spectrum.The Hamiltonian in this case is Zf -Zi +Zz and corresponds to an axially symmetric rotator; this limit will therefore be called, for convenience, the axial limit, even though p = 0 is also axial. The physical eigenvalues are in this case Z(Z+ 1) - 2m2 and each level except m = 0 is doubly degenerate. It is convenient to convert eq. (9) to a Schrbdinger-like problem by defining a new independent variable t and wave function t(4) = s,“.[l. +P cos 2@J-*d$, x(4) = [1+~cos2$J*(1+1~(~).

(21) (22)

This leads to the differential equation

-- d2x+Pz4t)X= k, dt2

(23

where the “potential” u(t) is given by u(t) =

p + cos 24(t) 1 +/c cos 24(t) -

(24)

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Eq. (24) is to be solved on the interval (to, 0)

(25)

with the appropriate boundary conditions, which are not affected by the change of variables in (21) and (22). For p > 0, the potential o(t) is a monotonic well with u(0) = - 1 and u(te) = 1. - 00. Further, ~(4) is close to 1, unless 4 is close to +a. A detailed Asp-, l-,t,-+ study of the behavior of u as a function of t shows that u rises from - 1 to 1 in an essentially Axed finite interval and that for ail but this finite interval o(t) = 1. If z is real and positive, the eigenvahtes of (23) consists of a fixed, finite number of “bound states” lying between -p.z and pz and an infinite number of unbound states of energy greater than z. In the axial limit, the energies of the unbound states decrease monotonically to z, and the density of states becomes infinite, giving an indication of the continuum of states that appears for p > 1. There is no indication, however, of the states in the continuum at complex energies or energies less than z. If z is real and negative, all the eigenvalues approach z in the axial limit. It is possible to show that the bound states are precisely the physical states of the axially symmetric rotator. They are doubly degenerate in the axial limit since they satisfy both x(to) = 0 and I’ = 0. The one non-degenerate state, corresponding to m = 0,is exactly at ‘“threshold”. To calculate the bound states of eq. (23), we make the change of variable x = cos 2J/ in (21) and obtain cos2IfP t=q

(1

-x2)-f(1f~x)-*ddx,

-1

Since the dominant part of the integral is for x close to - 1, we can replace 1 --x by 2 and obtain tz

---

2:2S~2~(l~x)-~(l+iu)-tdx t 1

We can introduce a new variable y defined by

J. 5. TALMAN

490 so

that t = - Lln(coshY+sinhY) P(J2 = --

1

Y*

PJ2 Also, ~(4) = -l+

i-?! P

tanh2 y.

The Schrijdinger equation now becomes, at p = 1, -2gyt

+z(-l+2tanh2y)X

= 22.

This can be transformed to the associated Legendre equation by putting x = tanh y: 1-z ___-

+zXf2(1-g)

3: =

0.

(27)

For integral I, the solutions are given by

x(x) = P;“(x)., = I(M)-2m2. For other values of 1, the eigcnvalue problem (27) c’dn be solved by the usual technique of looking for polynomial solutions about a singular point. It is then found that x(x) = (1 - x2)PP&), (28) j&= z-Q?=,

(29)

p = a[-2&l-(4z+l)“],

(30)

where k = 0, 1, . . . and Fk is a polynomial of degree k. The sign of the square root in (30) is determined by the condition Re(p) > 0. For real values of Z,the eigenvalues are given by A = I(I+-1)-2(k+f+l)2 and /z = 1(I+1)+2(1-k)2. There are two trajectories passing through each physical point except for nr = 0, because of the double degeneracy of the eigenvalues. Eq. (23) can be employed also to determine the behavior of the eigenvalues for z real and large, and arbitrary p. In this case the potential well ,u.zu(t) becomes very deep and can be approximated by a harmonic oscillator potential. If P > 0, the potential is a minimum at Cp= &c or t = 0. Since the wave function is only appreciable for 4 close to f~, one can make the approximation t x -$/(I -,u)*. It then follows that e(t) can be approximated by

INTERPOLATION OF EIGENVALUES

n(t) = -l+f(/L+l)P,

491

(31)

and the Schrijdinger eq. (23) becomes - 3

+/1z[-l++(~+l)]t2~

= I&

The eigenvalues are given therefore by an = -p1(2+1)+[2/4~+1)1(1+1)]~(2n+1),

(33)

inthelimitz-+ co. A result similar to this, but with [a(r+ l)]” replaced by 1 can be obtained by substituting the c-number I for I1 in the commutation relations for the angular momentum operators “). The motivation for this is that at large z, I must be oriented parallel to the largest moment of inertia. This work is a development of the study of this problem initiated by Bohr and Mottelson “). I am indebted to Dr. Mottelson for introducing me to the problem and for useful discussions. References 1) R. L. Om& and V. A. Alessandrini, Phys. Rev. 136 (1964) B1137 2) E. A. Coddington and N. Levinson, Theory of ordinary differential equations (McGraw-Hill, New York, 1955), ch. 12 3) R. Cutkosky, unpublished 4) A. Bohr and B. R. Mottelson, Nuclear structure,vol. 2 (Benjamin, New York) to be published