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NEW YORK UNIVERSITY

INSTITUTE OF !V!ATHE^TAT1CAL SCIENCES

25 Waverly Pi.ci :''.-. 3, N. "'.'

f^ ^ -rx T^ NEW YORK UNIVERSITY

CO

ua

AFCRC-TN-60-464

^E ET PR^^

m

^^cccxx+

InsNtute of Mathematical Sciences

Division of Electromagnetic Research

RESEARCH REPORT No. CX-49

Bounds on Scattering Phase Shifts:

Static Central Potentials

LEONARD ROSENBERG and LARRY SPRUCH

Contract Nos. A F 1 9 ( 6 04 ) 4 5 5 5

DA 30-06 9-ORD-2581, Project No. 2360

JUNE, 1960

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. CX-U9

Boxinds on Scattering Phase Shifts:

Static Central Potentials

LEONARD ROSENBERG AND LARRY SPRUCH

Leonard Rosenberg

Larry Spruch

Morris Kline, Director

The research reported in this paper

has been jointly sponsored by the

Geophysics Research Directorate of

the Air Force Cambridge Research

Center, Air Research and Development

Command, under Contract No. AF 19(60U)

h555, and ths Office of Ordnance

Research under Contract No. DA-30-

069-ORD-2581, Project No. 2360.

Jime I960

Requests for additional copies by Agencies of the Department

of Defense, their contractors, and other Government agencies

should be directed to the:

AHMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

Depai^anent of Defense contractors must be established for

ASTIA sei^ices or have their 'need-to-know' certified by the

cognizant military agency of their project or contract. All

other i)ersons and organizations should apply to the:

U.S. DEPAR1MENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

i.

Table of Contents

Page

Abstract 1

1. Introduction 2

2. Upper bound on cot(Ti - 0) using energy eigenfunctions 6

A - Potentials vhich vanish identically beyond r = R 6

B - Potentials which are solvable beyond r = R 10

5. Lower bound on t] 11

A - Use of the conditional inequality 12

B - Comparison with Risberg-Percival result 15

h. Upper bound on tj 18

A - Use of Wigner inequality 19

B - Use of energy eigenfunction expansion 19

C - Lower boxind on energy eigenvalue 21

5. Upper bound on cot(Tj - 0) using potential eigenfunctions 22

References 29

1.

Abstract

It has recently been shovn that rigorous upper bounds on scattering

lengths cBJi be obtained by adding to the Kohn variational expression cer-

tain Integrals involving approximate wave functions for each of the nega-

tive energy states. For potentials which vanish Identically beyond a cer-

tain point, it is possible to extend the method to positive energy scat-

tering; one obtains upper bounds on (- k cot T[) , where t| is the phase

shift. In addition to the negative energy states one must now take into

account a finite number of states with positive energies lying below the

8catteil.ng energy. The states in this associated energy eigenvalue prob-

lem are defined by the imposition of certain boundary conditions on the

vave functions. A second approach, involving an associated potential

strength eigenvalue problem, is also used. The second method includes

the first as a special case and, more significantly, can be extended to

scattering by compound systems. If some states are not accounted for, a

bound on cot tj is not obtained; nevertheless it is still possible to ob-

tain a rigorous lower bound on T). Upper bounds on tj may also be obtain-

ed, but in a way which is probably not too useful for many body scatter-

ing problems.

2.

1. IHTRODUCTION

A variational expression vMch provides a rigoroiia upper bovmd on

the scattering length has recently been obtained for the scattering of

one compoxmd system by another where only one channel is open and vhere

1-4

a finite number of composite bound states exist. The method consisted

of expressing the scattering length as a variational estimate plus an

error term which is of second order, and then bounding the error term.

This term is of the form / w(H - E )wdT, where H is the Hamlltonian,

E Is the sum of the unpertiirbed groxmd state energies of the two colli-

g

ding systems, and w is the difference between the exact and the trial

vave function, each appropriately normalized. It is first recognized

that if there are no composite bound states, the integral is non-negative,

1 2

I.e., one has a bound immediately. The extension to the case where

there are N composite bound states, represented by the functions with

eigenvalues E. , proceeds with the observation that the function

N

1=1

has no components with an energy less than E . It follows that the expec-

tation value of H - E with respect to the above function is non-negative.

Evaluating this expectation value, one then immediately finds that

|v(H - E^)wdx . g (E^ I E^) [/?^i(H - E^)x^,dx] (l.l)

vhere u Is the trial scattering function. Now of coiirse the are not

t i

3.

generally knovn. It proved to be possible, hovever, to show that the

bovmd Is rigorous even though the bound state functions employed In the

calculation are not exact solutions of the Schroedlnger equation; the

only reqvilrement, essentially. Is that they be sufficiently accurate to

give binding. If then one can find N orthonormal trial functions, ,

vhlch have the property that the N eigenvalues of the Hamiltonlan matrix

formed from them all lie belov E , the Inequality given by Eq. (l.l) re-

mains valid vhen the are replaced by the 0. . .

1 Xw

It vould be extremely useful to generalize the above result to the

case for which the initial relative kinetic energy of the systems is

greater than zero. More precisely, ve seek bounds on tan t\, vhere t^ is

one of the phase shifts. (Actually, it will prove convenient as well as

\iseful to seek bounds on cot(r) - 0), where ^ 9 < jt.) For purposes of

simplicity, the present paper is confined to consideration of the zero

angular momentum scattering of a splnless particle by a short range cen-

tral potential. (The case of a short range central potential plus a

Coulomb potential will also be treated. See Sec. 2B.) As in the zero

energy case, many of the results are applicable to a wider class of prob-

lems, and in fact throughout the present paper stress is placed upon

those methods which do Indeed allow extensions. The wider class of

problems, which are always restricted to those for which only one channel

is open. Include the scattering of one system by another, with the effects

of the Paull principle taken fully into account . The results can also be

extended to the case of an arbitrary anguleir momentum. On the other hand,

in contrast to the zero energy case, the inclusion of tensor forces wovild

require a major modification . The origin of this difference is that the

k.

mixing parameter, the parameter vhlch characterizes the relative admix-

tures of states of different orbital angular momentum, vanishes In general

only at zero energy.

Henceforth, then, we restrict our attention to the one body problem,

for zero angular momentum. Since the Initial kinetic energy, to be de-

noted by E, Is greater than 0, there are nov an Infinite number of solu-

tions of the Hamlltonlan with energies less the energy of the system under

consideration, the N bound states and the continuum below E. If, however,

ve restrict ourselves to potentials which vanish Identically beyond r = R,

it will prove possible by the Imposition of appropriate boundary conditions,

to require the subtraction of only a finite number of states with energies

less than E. The problem Is then almost Identical In form with that at

zero energy.

The technique Involved utilizes a connection between the scattering

pi^blem at energy E for a potential, V(r), which cuts off beyond r = R and

the bound state problem for the potential V(r) for ^ r < R followed by

an Infinitely rep\il8lve potential for r ^ R. The spectrum of the bound

state problem Is of course discrete, and a boirnd on / w(H - E)wdr may be

obtained by peirformlng a subtraction that Involves the N negative energy

7

states and the finite niomber of positive energy elgenstates of the bound

state problem which lie below the energy E. This Is shown In Sees. 2 and 5-

The reduction of the positive energy problem to a form similar to

that of the zero energy problem previously studied has of course been

achieved only at a price, namely, the restriction on the potential. One

would expect, however, that In practice, for low energy scattering, R can

be chosen to be sufficiently large so that the neglect of the potential

tall vhlch exists beyond r = R gives rise to a negligible error in q; it

may be possible. In fact, to make reasonably reliable estimates of the

error Inciirred. Of course at higher scattering energies the labor involved

in the calculation is Increased, due to the larger number of states to be

8\ibtracted off, for R fixed. At higher scattering energies, therefore, one

vould have to strike a balance in the choice of R; it would have to be suf-

ficiently large so as to give reasonable accuracy, but not so large as to

unduly increase the necessary labor.

The results of Sees. 2 and 3 lead to a lover bound on t]. Two methods

of obtaining an upper boiind on t] are discussed in Sec. k.

The method of Sees. 2 and 3 involves the introduction of an associated

bound state energy eigenvalue problem. It is shown in Sec. 5 that a more

general result (which includes the above as a special case) can be obtain-

ed by the introduction of an associated eigenvalue problem in which the

eigenvalues are certain appropriate strengths of an auxiliary arbitrarily

chosen positive definite potential, p(r); this will be referred to as the

associated potential strength eigenvalue problem. The significant feature

of this alternate approach is not simply that its greater generality leads

to an Inrproved bovmd on cot(yi - 0) and on t^. Rather, it is that it makes

possible the extension of the determination of a bound on tj to the scat-

tering of one system by another, with the Pauli prtnciple fully accoimted

for ^ .

The technique of the associated potential strength eigenvalue problem

8,9

was Introduced into scattering theory some time ago by Kato, who obtain-

ed some rather striking results for zero angular momentum scattering by a

static central potential. He was able to get both bounds on cot(Ti - 0)>

6.

and he did not need to cut off the potentleil. The differences between the

present approach and that of the original work of Kato, for the one bound

that ve obtain, are however essential, both as matters of practice and of

principle, if results are to be obtained for compound systems. The matter

of practice is that we need only calculate matrix elements of H while the

2

Kato method requires the calculation of matrix elements of H ; the former

is difficult enough -vrtille the latter is all but impossible for even the

slurplest compoxind systems. The matter of principle is that the result de-

duced by Kato will be less accurate, for the same choice of trial scatter-

ing function, than the result obtained in Sec. 5Â«

2. UPPER BOUND ON COT(ti - O) USING ENERGY EIGENFUNCTIONS

A. Potentials which vanish identically beyond r = R

The scattering problem is defined by the differential equation

(H - E)u(r) = (- 1^ -^ + V(r) - ^ju(r) = 0,

with boundary conditions

u(0) =

u(r) = co8(kr + 9) + cot(Ti - 0)sin(kr +9), r * R.

The last result follows from the restriction, in line with the previous

dlsciisslon, to potentials V(r) which vanish identically for r > R.

We Introduce the trial function u (r) vMch satisfies

u^(0) - 0,

(2.1)

u^(r) Â» co8(kr + 9) + cot(Ti^- 0)sln(kr + e), r S R .

We then have the Identity

R

k cot(Ti - 9) x= k cot(Ti^- 9) + (t^/2n) I u^(H - E)u^dr

o

R

- (t^/2n) I v(H - E)vdr, (2.2)

o

vhere the difference function

v(r) = u^(r) - u(r),

has the properties

v(0) Â« 0,

v(r) = [cot(i]^- 9) - cot(T] - 9)]sln(kr + 9)^ r * R (2.5)

(H - E)u (r) Â» (H - E)w(r), for all r,

and where ve have made use of the fact that

(H - E)u^(r) = (H - E)v(r) =0, r > R.

The Identity Is due to Kato, but Is here specialized to the case of a cut-

off potential. The last term in Eq.. (2.2) is of second order; if dropped

ve remain with a one parameter family of variational principles. It is

8.

our present purpose, however, to boiind this term.

One procedxire for doing so is to adjust and R such that

kR + = (P + l)rt,

where P is an arbitrary non-negative integer . It then follows from

Eqs. (2.5) that w(r) vanishes at r = R as well as at r = 0. These bound-

ary conditions are precisely those that would be placed on an energy

eigenfunction for a particle in a spherical box, with a rigid wall at a

radius R, within which a potential may exist. The desired bound on

R

w(H - E)wdr

o

may then be obtained by considering the associated energy eigenvalue prob-

lem in which the potential Is V(r) for r < R and is + Â» for r > R. Sup-

pose this potential supports M states with energies below E. (These M

states Include N negative energy states.) Denote the normalized eigen-

functlons by 0. (r) with corresponding eigenvalues E . We now assume that

we have been able to find M trial functions 0. . (r) with the properties

0it(O) = ?f,^(R) = 0,

(^It'^st) = ^is' (^-^^

(^it'^^st^ = Vis' ^it N), and write

Ej - â€¢fe^j^/2n . (5.4)

The sequence then Is E,,...E ,...E.,... . The bound states are charac-

1 n' J'

terlzed by the requirement that the function vanish at r = R. Now consider

that function which Is Identical with the J'th bound state solution for

r S R and which Is a solution of the free wave equation for the energy E

for r> R, with contlnuoiis value and slope at r = R. From \inlqueness, this

amst be Identical with the scattering solution, which Is a multiple of

BlnTk r + Tj(k.)3 In the external region. Since this must vanish at r = R,

It follows that k R + Ti(k ) must be a multiple of n.

J J

To prove the equivalence of the two different definitions of k., Eqs.

(5.2) and (5.4), it must still be shown that the multiple of Â« is in fact

J itself. To see this, we note firstly that by Levlnson's theorem.

13a

Figure 1

{N> Z)ir

N-fl

N+2

A schematic plot of ^(k) a kR + T](k) versus k; ^(k) need be

defined only for non-negative energies. Since there are N negative

energy states, Le-vlnson's theorem gives ^(O) = Nrt. By Wigner's causal

Inequality, d5(k)/dk > for any interval of k vhlch satisfies

Jx ^ C(l^) - (J + 2)jt> where J is an integer, which, from the above dis-

cussion, must be greater than N. By continuity the slope will be posi-

tive for ^(k) less than but sufficiently close to (j + l)jt. The eigen-

values k are defined by C(k.) = Jrt- Once ^(k) has passed through the

u J

value Jjt, which occurs at k = k., it can never retiim to that value.

J

The k. as defined above determine the positive energy eigenstates, E.,

J J

of the particle constrained to the interval to R in the presence of

the potential, V(r), by the relationship E. = "fe ic /2|a.

Ik,

5(0) = Tj(0) Â« Nrt. Further, it follows from Eq.8. (5.2) and (5.5) that once

5(k) has passed through a given multiple of Â«, it can never return to that

15

value -'. Thus ^(k) starts, at k = 0, at Njt; at the first positive energy

boimd state, vhlch is the state labelled by N + 1, it has the value (N + l)n.

In general, then, we do in fact have that

k R + Ti(k ) = JÂ« , J = N+l,N+2, ...

and that E. = "h k. /3^ represents the J'th bound state. It follows that

J J

if Jjt < 5(k) < (j + l)rt, precisely J bound states of the associated energy

eigenvalue problem exist with energies less than "fe. k. /2\i,

The fact that the k. defined by Eq. (3.2) are identical with the k.

J

defined by Eq. (5.^) can also be proved , with the aid of continuity argu-

ments (the case E = must therefore be excluded in this parti ciilar proof),

by studying the phase shift as the potential strength is built up from zero

to its true strength.

We are now in a position to obtain a rigorous lower bouad on t). Thus,

suppose that M' bound stetes have been found with energies below the energy

E. If M' = M, where M is the true number of such states, then the inequa-

lity satisfied by k cot(Ti - 9), Eq. (2.5), is valid. If, however, M' < M,

the above mentioned inequality, in which M' trial bo'und state functions,

, are employed need not be valid. We may, however, simply assume that

M' = M, which Implies that

M'jt ^ kR + Ti(k) ^ (M' + l)rt.

If the aaeitmption is correct, Eq. (2.5) will provide a lower bound on t],

namely t) , where tj is a known number which is less than (M + l)jt - kR.

15.

It, on the other hand, the assiomption is Incorrect we have r\ > (M+l)jt-kR.

Therefore the Inequality t] > tu Is correct. Independent of the validity

of the original assumption, and we then have a rigorous lower bound on t].

In summary then, we have, if we have proved the existence of at least

M' eigenstates of our bound state problem with energies less than the scat-

tering energy E, that

n > Tjj^ / (3.5a)

where r\j Is defined by the relations

k cot(TV^ - e) = right hand side of Eq. (2.5), (5.5h)

and

M'k - kR S Ti^ s (M' + l)rt - kR. (5.5c)

The technique of treating Eq. (2.5) as a conditional Inequality which

leads nevertheless to a rigorous lower bound on the phase shift is closely

17

analogoiis to a procedure used earlier in a study of the Kato method.

It should be remeirked that while the lower bound on t\ is rigorous even

vhen one has not accounted for all of the bound states, a situation quite

different from that for the upper boxind on cot(Ti - 0), the lower bovmd on tj

deduced when M j^ M' is a very poor one, being too small by roughly (M - M')jt.

B. Comparison with Risberg Fercival Res\ilt

During the course of the last derivation, we arrived at the result

that 5(k) > Jjt if k >k . This inequality itself clearly provides a lower

J

bound on the phase shift for a scattering energy which lies above the appro-

18

prlate level. With the method of Hylleraas and Undheim upper bounds, E ,

16.

may be found, yielding a discrete set of scattering energies for vhlch

a lower bound on the phase shift Is determined. This method for getting

a boxond on Tj(k) is essentially that given earlier by Risberg and by

Of)

Perclval . The simileirlty betveen their result and that deduced in the

previous subsection, vhlch grew out of a quite different approach, is

rather striking and it may be of some interest to compare the two. We

INSTITUTE OF !V!ATHE^TAT1CAL SCIENCES

25 Waverly Pi.ci :''.-. 3, N. "'.'

f^ ^ -rx T^ NEW YORK UNIVERSITY

CO

ua

AFCRC-TN-60-464

^E ET PR^^

m

^^cccxx+

InsNtute of Mathematical Sciences

Division of Electromagnetic Research

RESEARCH REPORT No. CX-49

Bounds on Scattering Phase Shifts:

Static Central Potentials

LEONARD ROSENBERG and LARRY SPRUCH

Contract Nos. A F 1 9 ( 6 04 ) 4 5 5 5

DA 30-06 9-ORD-2581, Project No. 2360

JUNE, 1960

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. CX-U9

Boxinds on Scattering Phase Shifts:

Static Central Potentials

LEONARD ROSENBERG AND LARRY SPRUCH

Leonard Rosenberg

Larry Spruch

Morris Kline, Director

The research reported in this paper

has been jointly sponsored by the

Geophysics Research Directorate of

the Air Force Cambridge Research

Center, Air Research and Development

Command, under Contract No. AF 19(60U)

h555, and ths Office of Ordnance

Research under Contract No. DA-30-

069-ORD-2581, Project No. 2360.

Jime I960

Requests for additional copies by Agencies of the Department

of Defense, their contractors, and other Government agencies

should be directed to the:

AHMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

Depai^anent of Defense contractors must be established for

ASTIA sei^ices or have their 'need-to-know' certified by the

cognizant military agency of their project or contract. All

other i)ersons and organizations should apply to the:

U.S. DEPAR1MENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

i.

Table of Contents

Page

Abstract 1

1. Introduction 2

2. Upper bound on cot(Ti - 0) using energy eigenfunctions 6

A - Potentials vhich vanish identically beyond r = R 6

B - Potentials which are solvable beyond r = R 10

5. Lower bound on t] 11

A - Use of the conditional inequality 12

B - Comparison with Risberg-Percival result 15

h. Upper bound on tj 18

A - Use of Wigner inequality 19

B - Use of energy eigenfunction expansion 19

C - Lower boxind on energy eigenvalue 21

5. Upper bound on cot(Tj - 0) using potential eigenfunctions 22

References 29

1.

Abstract

It has recently been shovn that rigorous upper bounds on scattering

lengths cBJi be obtained by adding to the Kohn variational expression cer-

tain Integrals involving approximate wave functions for each of the nega-

tive energy states. For potentials which vanish Identically beyond a cer-

tain point, it is possible to extend the method to positive energy scat-

tering; one obtains upper bounds on (- k cot T[) , where t| is the phase

shift. In addition to the negative energy states one must now take into

account a finite number of states with positive energies lying below the

8catteil.ng energy. The states in this associated energy eigenvalue prob-

lem are defined by the imposition of certain boundary conditions on the

vave functions. A second approach, involving an associated potential

strength eigenvalue problem, is also used. The second method includes

the first as a special case and, more significantly, can be extended to

scattering by compound systems. If some states are not accounted for, a

bound on cot tj is not obtained; nevertheless it is still possible to ob-

tain a rigorous lower bound on T). Upper bounds on tj may also be obtain-

ed, but in a way which is probably not too useful for many body scatter-

ing problems.

2.

1. IHTRODUCTION

A variational expression vMch provides a rigoroiia upper bovmd on

the scattering length has recently been obtained for the scattering of

one compoxmd system by another where only one channel is open and vhere

1-4

a finite number of composite bound states exist. The method consisted

of expressing the scattering length as a variational estimate plus an

error term which is of second order, and then bounding the error term.

This term is of the form / w(H - E )wdT, where H is the Hamlltonian,

E Is the sum of the unpertiirbed groxmd state energies of the two colli-

g

ding systems, and w is the difference between the exact and the trial

vave function, each appropriately normalized. It is first recognized

that if there are no composite bound states, the integral is non-negative,

1 2

I.e., one has a bound immediately. The extension to the case where

there are N composite bound states, represented by the functions with

eigenvalues E. , proceeds with the observation that the function

N

1=1

has no components with an energy less than E . It follows that the expec-

tation value of H - E with respect to the above function is non-negative.

Evaluating this expectation value, one then immediately finds that

|v(H - E^)wdx . g (E^ I E^) [/?^i(H - E^)x^,dx] (l.l)

vhere u Is the trial scattering function. Now of coiirse the are not

t i

3.

generally knovn. It proved to be possible, hovever, to show that the

bovmd Is rigorous even though the bound state functions employed In the

calculation are not exact solutions of the Schroedlnger equation; the

only reqvilrement, essentially. Is that they be sufficiently accurate to

give binding. If then one can find N orthonormal trial functions, ,

vhlch have the property that the N eigenvalues of the Hamiltonlan matrix

formed from them all lie belov E , the Inequality given by Eq. (l.l) re-

mains valid vhen the are replaced by the 0. . .

1 Xw

It vould be extremely useful to generalize the above result to the

case for which the initial relative kinetic energy of the systems is

greater than zero. More precisely, ve seek bounds on tan t\, vhere t^ is

one of the phase shifts. (Actually, it will prove convenient as well as

\iseful to seek bounds on cot(r) - 0), where ^ 9 < jt.) For purposes of

simplicity, the present paper is confined to consideration of the zero

angular momentum scattering of a splnless particle by a short range cen-

tral potential. (The case of a short range central potential plus a

Coulomb potential will also be treated. See Sec. 2B.) As in the zero

energy case, many of the results are applicable to a wider class of prob-

lems, and in fact throughout the present paper stress is placed upon

those methods which do Indeed allow extensions. The wider class of

problems, which are always restricted to those for which only one channel

is open. Include the scattering of one system by another, with the effects

of the Paull principle taken fully into account . The results can also be

extended to the case of an arbitrary anguleir momentum. On the other hand,

in contrast to the zero energy case, the inclusion of tensor forces wovild

require a major modification . The origin of this difference is that the

k.

mixing parameter, the parameter vhlch characterizes the relative admix-

tures of states of different orbital angular momentum, vanishes In general

only at zero energy.

Henceforth, then, we restrict our attention to the one body problem,

for zero angular momentum. Since the Initial kinetic energy, to be de-

noted by E, Is greater than 0, there are nov an Infinite number of solu-

tions of the Hamlltonlan with energies less the energy of the system under

consideration, the N bound states and the continuum below E. If, however,

ve restrict ourselves to potentials which vanish Identically beyond r = R,

it will prove possible by the Imposition of appropriate boundary conditions,

to require the subtraction of only a finite number of states with energies

less than E. The problem Is then almost Identical In form with that at

zero energy.

The technique Involved utilizes a connection between the scattering

pi^blem at energy E for a potential, V(r), which cuts off beyond r = R and

the bound state problem for the potential V(r) for ^ r < R followed by

an Infinitely rep\il8lve potential for r ^ R. The spectrum of the bound

state problem Is of course discrete, and a boirnd on / w(H - E)wdr may be

obtained by peirformlng a subtraction that Involves the N negative energy

7

states and the finite niomber of positive energy elgenstates of the bound

state problem which lie below the energy E. This Is shown In Sees. 2 and 5-

The reduction of the positive energy problem to a form similar to

that of the zero energy problem previously studied has of course been

achieved only at a price, namely, the restriction on the potential. One

would expect, however, that In practice, for low energy scattering, R can

be chosen to be sufficiently large so that the neglect of the potential

tall vhlch exists beyond r = R gives rise to a negligible error in q; it

may be possible. In fact, to make reasonably reliable estimates of the

error Inciirred. Of course at higher scattering energies the labor involved

in the calculation is Increased, due to the larger number of states to be

8\ibtracted off, for R fixed. At higher scattering energies, therefore, one

vould have to strike a balance in the choice of R; it would have to be suf-

ficiently large so as to give reasonable accuracy, but not so large as to

unduly increase the necessary labor.

The results of Sees. 2 and 3 lead to a lover bound on t]. Two methods

of obtaining an upper boiind on t] are discussed in Sec. k.

The method of Sees. 2 and 3 involves the introduction of an associated

bound state energy eigenvalue problem. It is shown in Sec. 5 that a more

general result (which includes the above as a special case) can be obtain-

ed by the introduction of an associated eigenvalue problem in which the

eigenvalues are certain appropriate strengths of an auxiliary arbitrarily

chosen positive definite potential, p(r); this will be referred to as the

associated potential strength eigenvalue problem. The significant feature

of this alternate approach is not simply that its greater generality leads

to an Inrproved bovmd on cot(yi - 0) and on t^. Rather, it is that it makes

possible the extension of the determination of a bound on tj to the scat-

tering of one system by another, with the Pauli prtnciple fully accoimted

for ^ .

The technique of the associated potential strength eigenvalue problem

8,9

was Introduced into scattering theory some time ago by Kato, who obtain-

ed some rather striking results for zero angular momentum scattering by a

static central potential. He was able to get both bounds on cot(Ti - 0)>

6.

and he did not need to cut off the potentleil. The differences between the

present approach and that of the original work of Kato, for the one bound

that ve obtain, are however essential, both as matters of practice and of

principle, if results are to be obtained for compound systems. The matter

of practice is that we need only calculate matrix elements of H while the

2

Kato method requires the calculation of matrix elements of H ; the former

is difficult enough -vrtille the latter is all but impossible for even the

slurplest compoxind systems. The matter of principle is that the result de-

duced by Kato will be less accurate, for the same choice of trial scatter-

ing function, than the result obtained in Sec. 5Â«

2. UPPER BOUND ON COT(ti - O) USING ENERGY EIGENFUNCTIONS

A. Potentials which vanish identically beyond r = R

The scattering problem is defined by the differential equation

(H - E)u(r) = (- 1^ -^ + V(r) - ^ju(r) = 0,

with boundary conditions

u(0) =

u(r) = co8(kr + 9) + cot(Ti - 0)sin(kr +9), r * R.

The last result follows from the restriction, in line with the previous

dlsciisslon, to potentials V(r) which vanish identically for r > R.

We Introduce the trial function u (r) vMch satisfies

u^(0) - 0,

(2.1)

u^(r) Â» co8(kr + 9) + cot(Ti^- 0)sln(kr + e), r S R .

We then have the Identity

R

k cot(Ti - 9) x= k cot(Ti^- 9) + (t^/2n) I u^(H - E)u^dr

o

R

- (t^/2n) I v(H - E)vdr, (2.2)

o

vhere the difference function

v(r) = u^(r) - u(r),

has the properties

v(0) Â« 0,

v(r) = [cot(i]^- 9) - cot(T] - 9)]sln(kr + 9)^ r * R (2.5)

(H - E)u (r) Â» (H - E)w(r), for all r,

and where ve have made use of the fact that

(H - E)u^(r) = (H - E)v(r) =0, r > R.

The Identity Is due to Kato, but Is here specialized to the case of a cut-

off potential. The last term in Eq.. (2.2) is of second order; if dropped

ve remain with a one parameter family of variational principles. It is

8.

our present purpose, however, to boiind this term.

One procedxire for doing so is to adjust and R such that

kR + = (P + l)rt,

where P is an arbitrary non-negative integer . It then follows from

Eqs. (2.5) that w(r) vanishes at r = R as well as at r = 0. These bound-

ary conditions are precisely those that would be placed on an energy

eigenfunction for a particle in a spherical box, with a rigid wall at a

radius R, within which a potential may exist. The desired bound on

R

w(H - E)wdr

o

may then be obtained by considering the associated energy eigenvalue prob-

lem in which the potential Is V(r) for r < R and is + Â» for r > R. Sup-

pose this potential supports M states with energies below E. (These M

states Include N negative energy states.) Denote the normalized eigen-

functlons by 0. (r) with corresponding eigenvalues E . We now assume that

we have been able to find M trial functions 0. . (r) with the properties

0it(O) = ?f,^(R) = 0,

(^It'^st) = ^is' (^-^^

(^it'^^st^ = Vis' ^it N), and write

Ej - â€¢fe^j^/2n . (5.4)

The sequence then Is E,,...E ,...E.,... . The bound states are charac-

1 n' J'

terlzed by the requirement that the function vanish at r = R. Now consider

that function which Is Identical with the J'th bound state solution for

r S R and which Is a solution of the free wave equation for the energy E

for r> R, with contlnuoiis value and slope at r = R. From \inlqueness, this

amst be Identical with the scattering solution, which Is a multiple of

BlnTk r + Tj(k.)3 In the external region. Since this must vanish at r = R,

It follows that k R + Ti(k ) must be a multiple of n.

J J

To prove the equivalence of the two different definitions of k., Eqs.

(5.2) and (5.4), it must still be shown that the multiple of Â« is in fact

J itself. To see this, we note firstly that by Levlnson's theorem.

13a

Figure 1

{N> Z)ir

N-fl

N+2

A schematic plot of ^(k) a kR + T](k) versus k; ^(k) need be

defined only for non-negative energies. Since there are N negative

energy states, Le-vlnson's theorem gives ^(O) = Nrt. By Wigner's causal

Inequality, d5(k)/dk > for any interval of k vhlch satisfies

Jx ^ C(l^) - (J + 2)jt> where J is an integer, which, from the above dis-

cussion, must be greater than N. By continuity the slope will be posi-

tive for ^(k) less than but sufficiently close to (j + l)jt. The eigen-

values k are defined by C(k.) = Jrt- Once ^(k) has passed through the

u J

value Jjt, which occurs at k = k., it can never retiim to that value.

J

The k. as defined above determine the positive energy eigenstates, E.,

J J

of the particle constrained to the interval to R in the presence of

the potential, V(r), by the relationship E. = "fe ic /2|a.

Ik,

5(0) = Tj(0) Â« Nrt. Further, it follows from Eq.8. (5.2) and (5.5) that once

5(k) has passed through a given multiple of Â«, it can never return to that

15

value -'. Thus ^(k) starts, at k = 0, at Njt; at the first positive energy

boimd state, vhlch is the state labelled by N + 1, it has the value (N + l)n.

In general, then, we do in fact have that

k R + Ti(k ) = JÂ« , J = N+l,N+2, ...

and that E. = "h k. /3^ represents the J'th bound state. It follows that

J J

if Jjt < 5(k) < (j + l)rt, precisely J bound states of the associated energy

eigenvalue problem exist with energies less than "fe. k. /2\i,

The fact that the k. defined by Eq. (3.2) are identical with the k.

J

defined by Eq. (5.^) can also be proved , with the aid of continuity argu-

ments (the case E = must therefore be excluded in this parti ciilar proof),

by studying the phase shift as the potential strength is built up from zero

to its true strength.

We are now in a position to obtain a rigorous lower bouad on t). Thus,

suppose that M' bound stetes have been found with energies below the energy

E. If M' = M, where M is the true number of such states, then the inequa-

lity satisfied by k cot(Ti - 9), Eq. (2.5), is valid. If, however, M' < M,

the above mentioned inequality, in which M' trial bo'und state functions,

, are employed need not be valid. We may, however, simply assume that

M' = M, which Implies that

M'jt ^ kR + Ti(k) ^ (M' + l)rt.

If the aaeitmption is correct, Eq. (2.5) will provide a lower bound on t],

namely t) , where tj is a known number which is less than (M + l)jt - kR.

15.

It, on the other hand, the assiomption is Incorrect we have r\ > (M+l)jt-kR.

Therefore the Inequality t] > tu Is correct. Independent of the validity

of the original assumption, and we then have a rigorous lower bound on t].

In summary then, we have, if we have proved the existence of at least

M' eigenstates of our bound state problem with energies less than the scat-

tering energy E, that

n > Tjj^ / (3.5a)

where r\j Is defined by the relations

k cot(TV^ - e) = right hand side of Eq. (2.5), (5.5h)

and

M'k - kR S Ti^ s (M' + l)rt - kR. (5.5c)

The technique of treating Eq. (2.5) as a conditional Inequality which

leads nevertheless to a rigorous lower bound on the phase shift is closely

17

analogoiis to a procedure used earlier in a study of the Kato method.

It should be remeirked that while the lower bound on t\ is rigorous even

vhen one has not accounted for all of the bound states, a situation quite

different from that for the upper boxind on cot(Ti - 0), the lower bovmd on tj

deduced when M j^ M' is a very poor one, being too small by roughly (M - M')jt.

B. Comparison with Risberg Fercival Res\ilt

During the course of the last derivation, we arrived at the result

that 5(k) > Jjt if k >k . This inequality itself clearly provides a lower

J

bound on the phase shift for a scattering energy which lies above the appro-

18

prlate level. With the method of Hylleraas and Undheim upper bounds, E ,

16.

may be found, yielding a discrete set of scattering energies for vhlch

a lower bound on the phase shift Is determined. This method for getting

a boxond on Tj(k) is essentially that given earlier by Risberg and by

Of)

Perclval . The simileirlty betveen their result and that deduced in the

previous subsection, vhlch grew out of a quite different approach, is

rather striking and it may be of some interest to compare the two. We

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