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On the relation between two formal definitions of resonances in non-relativistic scattering
Physica 52 (1971) 135-143 0 North-Holland Publishing Co.
ON THE
RELATION
OF RESONANCES
,BETWEEN
TWO FORMAL
IN NON-RELATIVISTIC
DEFINITIONS
SCATTERING
B. J. VERHAAR Afdeling VOOY Natuurkunde der Technische Hogeschool, Eindhoven, Nederland Received 1 September 1970
Synopsis A formulation of the resonance problem is given, which makes clear that an intimate relation exists between two formal theories of scattering of particles with in ternal degrees of freedom, one proposed by Humblet and Rosenfeld, the other by Fonda and Newton. In both theories resonances correspond to poles of the S matrix as a function of a variable 8, varying in a two-dimensional surface Z in the direct product space 81 @ 8s @ 8s . . ., where dr, &‘s, 8s . . . stand for the complex channel energies. The two theories differ in the choice of Z and 8.
1. Introduction. Qualitatively a resonance in a scattering process may be described as an anomaly in a cross section as a function of projectile energy, which’in its simplest possible form is illustrated in fig. 1. It is characterized by its resonance energy En and width rn. The index L is used here to distinguish different resonances. An alternative qualitative characterization of a resonance is possible in terms of the time-dependent description of scattering. Then it represents a situation in which the colliding particles “stick together” within the range of their mutual interaction during a time interval of the order of fi/Z’n.
Fig. 1. Breit-Wigner
resonance, characterized by resonance energy El and width Pa. 135
B. J. VERHAAR
136
Resonances ments,
have been observed
e.g., in neutron-nucleus
in various
types
l) and electron-atom
of scattering 2, scattering.
experiSince the
first of these observations several formal definitions of a scattering resonance have been proposed. One of these is due to Humblet and Rosenfelds), building on the “natural” definition of resonances by Siegert 4), which in turn was based on previous work by Gamow5) on the theory of nuclear CLdecay. A second theory, proposed by Fonda and Newtone), is intimately related to Feshbach’s work on this subject 7). In the following, after a concise description of the theories of H-R and F-N (abbreviations for the above-mentioned theories), we shall go into their relation. It will be shown that the F-N theory can be formulated in a form that reveals a close relationship to the H-R theory. To our knowledge this relationship has not been investigated in the literature?. Partly with the aim of avoiding intricacies of primarily mathematical nature, partly to exclude more essential complications such as the existence of three-particle channels, we shall concentrate on the most simple type of scattering problem for which each of the two scattering theories is meaningful. We consider the process of elastic and inelastic scattering of two particles with internal degrees of freedom. The hamiltonian of the total system is of the form
H = Hint(p) + -
P" 2m
+
VP, P)*
The first term, Hint, represents the sum of the internal hamiltonians of the two colliding particles with internal variables p, relative coordinate vector t and reduced mass m. The second and third terms represent the relative kinetic energy and mutual interaction, respectively. Only relative s waves are considered. Furthermore, I’ is supposed to vanish identically beyond a certain distance r = a and to be sufficiently regular at the origin. The spectrum of Hint is assumed to be discrete and to consist of a finite number n of points, the channel threshold energies Ed (CX= channel-index; El < Es <
E3 . ..).
2. Humblet-Rosenfeld theory. In this formalism resonances are defined as poles of the S matrix, analytically continued as a function of the total energy (to be specific we shall consider the S matrix whose open-channel sub-matrix is unitary and symmetric for real energies). An equivalent * t A comparative study has been devoted to the an international course in Triestes). Apparently, relation to be dealt with here, although each of the in detail. t The accidental situation of one or more bound is left out of consideration.
F-N and Feshbach theories during no attention has been paid to the F-N and H-R theories was treated states embedded in the continuum
DEFINITION
OF SCATTERING
RESONANCES
137
Ea+1
Fin. 2. Derivation of Humblet-Rosenfeld resonance + background expansion. Poles 8~ within G are included as resonance terms.
definition
is possible
in terms
of Gamow’s
“radioactive
states” : solutions
of the Schr6dinger equation of the system with outgoing waves in all channels. Clearly, such solutions are possible only for complex values of the total energy. The notion of “outgoing” waves is extended to complex values of the channel wave numbers in the following way. The radial wavefunction in each of the channels is required to be proportional to exp(ik,r)/r, where the channel wave number ka is given by k, = J(&
-
E&)2m/@,
(2)
in terms
of the total energy 8. To define the sign of the square root in this equation the branch points ca. o f the functions k&(6’) are of special importance (see fig. 2). We specify the signs of the square roots within an interval of the real energy axis between two thresholds by k, pos. real (open channels)
(3)
k, neg. imag. (closed channels) 1
and continue the square roots analytically into a region G of the complex plane. For the sake of simplicity we choose G such that E& < fie 8’ < e&+1. Within G the eigenvalue problem has thus been defined: the eigenvalues 8~ and eigenfunctions WA,labelled by 1, have to satisfy the equation Hyn = bays, together
with the above-mentioned
(4) boundary
condition.
B. J. VERHAAR
138
It is easily seen from the conservation
of particle
flux that all eigenvalues
are located in the lower half-plane part of G. (Note that for a more general choice of G, within which a curve such as r in fig. 2 is possible, eigenvalues may also be found in the upper half-plane.
For instance,
for each eigenvalue
&A in the lower half-plane, passing along r, the complex conjugated value 8; is also found to be an eigenvalue). Each of the eigenvalues is separated into its real and imaginary parts: ba = EA - +irn, with r~ > 0. The S matrix, analytically continued from the real energy interval into G, has poles located at the eigenvalues 8~. Assuming the existence of simple poles only, the following expansion is possible for any real energy E in G:
where the summation is over all poles in G, uLuja,l is the residue of S,,,(b) at br and S$L(E) is regular in G, having the form of an integral over its contour. Each term in the summation has the familiar Breit-Wigner form, En and ra functioning as resonance energy and width, respectively. The “background” term S::(E) is believed to have a relatively smooth form as a function of energy, if G has been chosen such that it includes a sufficient number of resonance poles. Note that different resonance + background expressions can be derived by a) expanding a different quantity*, e.g. (h&,)-i
(S -
l)n’a,
(6)
this specific type of expansion being advantageous for the discussion of threshold effects, b) choosing a different independent variable, e.g. the wave number k, of one of the channelss). In the following we shall see that a third “degree of freedom” exists, on the basis of which the Fonda-Newton theory
can be considered
3. Fonda-Newton ground expansion
as a special case.
theory. This theory leads to a similar resonance + back:
where, however, the significance of the symbols employed is different. The background term, which in the H-R theory is a regular remainder, is endowed with a specific physical significance in connection with direct reactions. The F-N theory is an attempt towards a unified reaction theory in which resonances and direct reactions are described in a combined frame* This type of the exponent
of ambiguity
includes
as a special
M in the Mittag-Leffler
case the arbitrarinesss)
expansion.
in the choice
DEFINITION
OF SCATTERING
RESONANCES
139
work. At first sight, this seems to go at the cost of the interpretation of resonances in terms of poles of the S matrix. It will be shown presently, however, that an interpretation in terms of a different type of S matrix poles remains possible. In the F-N theory the notion of “direct reaction” is interpreted as follows. One starts by specifying a real energy E for which the expression (7) shall be derived, and a corresponding division of the set of channels in open and closed ones. An arbitrary wavefunction y(p, r), not necessarily a stationary state, is expanded in terms of the internal eigenfunctions xa(p) :
(8) In the third member of this equation y,, and yC stand for the partial sums over open and closed channels, respectively. Eq. (8) thus corresponds to considering the total linear space of states y(p, r) to be a direct sum of “open and closed channel subspaces”. In view of this, y may be written in the form
yyyo,
0
(9)
YC
and correspondingly the hamiltonian posed into its components :
operator
of the system may be decom-
In principle, the direct part of the S matrix is defined as the S matrix obtained when the existence of the closed channels is left out of consideration, i.e. the S-matrix obtained when the total hamiltonian is replaced by Ho,. This definition is based on the assumption that direct processes are confined to the open-channel subspace without intervention of the closed channels, whereas resonance processes are indirect processes taking place through the intermediary of the closed-channel subspace. In Feshbach’s theory the resonance states are defined as eigenstates of subspace. This defiHCC> i.e. eigenstates of the isolated closed-channel nition looses much of its usefulness for stronger coupling between openand closed-channel subspaces. The F-N resonance definition includes this coupling: CPAin eq. (7) is defined by the eigenvalue problem (yC~ normalizable).
~iX?ycn = EpdWCl
(1’)
Here, yen is a wavefunction with a closed-channel subspace component and ZCc is an effective closed-channel hamiltonian given by Zcc = Hcc +
Hco
1 E
+
iv
-
Ho0
Hoc
(7 +
f0).
only
(12)
B. J. VERHAAR
140
Its anti-hermitian part is negative. The operator XCC is non-hermitians). Hence, 8~ is in general complex and located in the lower half-plane : 8~ = Ez. - $il’A,
with
ra > 0.
(13)
Furthermore, XcC depends on E, so that 8’~ is a function of E. Finally, the quantities a a’a, Ain eq. (7), the specific form of which is of no importance for the following, also turn out to be energy dependents). 4. Relation. A point of similarity between the two theories is the definition of En and I’r, which in both cases are determined by an eigenvalue problem. One point of difference, however, bears upon the reason for the eigenvalue to be a complex number: in the H-R theory this is due to the type of boundary condition, whereas in the F-N theory the non-hermiticity of SIP,, itself is responsible. An additional difference is the fact that the F-N resonance states, in contrast to the H-R states yn, are confined to the closed-channel subspace. Both differences can be eliminated by transforming the F-N eigenvalue equation (11) into two coupled equations. We supplement yc~ to form a state ye in the total space by adding an open-channel component yOn, defined by 1 yen = E + irl _ Hoo H diJcn. The total wavefunction
ye then satisfies the equation
(15) which is identical to the H-R eigenvalue equation (4), except that in the latter the open- and closed-channel energies are kept equal. A strong similarity also appears with respect to the boundary condition: on the basis of eqs. (1 l), (13) and (14) the radial wavefunctions in all channels must have the form exp(ik,r)/r, where the wave numbers are defined as in section 2, except that k, is defined in terms of E for the open channels and in terms of 8, for the closed channels. Note that, apart from demonstrating a close resemblance to the H-R eigenvalue equation, the coupled differential equations (15) may possibly have the advantage of being more amenable to numerical computation than the integro-differential equation ( 11). A different possibility to express the intimate relation to the H-R theory is the statement that the F-N complex resonance energies 8~ are the poles of the S matrix, analytically continued as a function of the energy of the closed channels, for a fixed energy (= E) of the open channels. This follows from the previously mentioned boundary condition and the generalized
DEFINITION
OF SCATTERING RESONANCES
141
definitiong) of the S matrix, considered as a function of independent channel wave numbers or channel energies ba = lLI+ Pki/2m. This suggests a generalized definition of complex resonance energies and the introduction of a more general resonance + background expansion, of which the H-R and F-N theories may be considered as special cases. Each of the S-matrix elements can be written in the forms) * iV,,,(klk2...) D(klkz...)
(16)
’
the functions D and N,,, being integral in each of the complex channel wave numbers, whereas D is independent of the channels 01 and 01’ considered t . Taken as functions of the channel energies, D and N,,, are regular, except for branch points at the thresholds. Representing each combination of channel energies as a point in the 2n-dimensional space bi 18 8s @ 83 . . . spanned by n perpendicular complex 8, planes, the following sets of points are of special importance (see fig. 3) : a) The line 81 = 8s = . . . = real, containing physically realizable combinations of channel energies. For a point Qi = 8s = . . . = E on this line, not coincident with a threshold, a resonance + background expansion of the S matrix is to be derived. hypersurface deb) The 292 - 2 dimensional, in general non-connected termined by D = 0. c) In D = 0 a set of points is selected to give rise to resonances. The selection is achieved by requiring this set to consist of the points of intersection between D = 0 and
Fig. 3. Location of resonance poles contributing to the S matrix. Each 8~ is a point of intersection between D = 0 and 27. For the sake of clarity the latter has been omitted. * Our notation is related to that of ref. 9 by N,,
= ga and
Nasa(a’ f a) = h-f,*,.
B. J. VERHAAR
142
d) A 2-dimensional surface Z:, which contains the point 6i= 8s = . . . = E. In order to arrive at a resonance + background expression of the form (5) or (7) we choose Z, like D = 0, to be an analytic manifoldlo), i.e. the set of (common) variables.
zeros of one or more analytic
functions
of a number
of complex
Finally, it is of importance to choose a complex variable d to parametrize the points in 2. It is chosen such that the channel energies varying in Z are regular functions of 8. Obviously, the functions D and N,,, are then regular in an environment G of the point 6’1 = 8s = . . . = E, provided G is chosen such that it does not contain possible branch points due to thresholds. The points of intersection mentioned under c) are the zeros 8~ of the function D in L’, and hence, discarding accidental coincidence of zeros of D and NLYfLY, the poles of the S matrix considered as a function of 8. Excluding also the accidental situation that .Z is tangent to the hypersurface D = 0, all poles are simple and a resonance + background expansion as in eq. (5) is possible. Note that a necessary condition for the possibility of this expansion is the absence of points of accumulation in the set 6~ within G. This is guaranteed by the choice of Z, since a point of accumulation would imply D = 0 in G, and therefore the existence of a bound state embedded in the continuum at the energy E, a situation which has been excluded above. From the point of view outlined here, the H-R and F-N theories differ only in the choice of Z and in the parametrization. In the H-R case Z is is defined by 61 = 8s = . . . (= complex, in general). The parametrization introduced by d = F, (01 arbitrary). On the other hand, to obtain the F-N theory we define Z by the requirement that all open-channel energies shall be equal to E, whereas all closed-channel energies be identical. The variable d is chosen to be any of the closed-channel energies. Note that in this case 2 depends on 6, explaining the energy-dependence of the quantities 6’~ and aala,A in eq. (7), whereas in the H-R theory Z contains the entire line mentioned under a) and is independent of E. We have thus described a formulation of the resonance problem, which amounts essentially to adding a third “degree of freedom” to the HumbletRosenfeld theory. This extended formulation includes the Fonda-Newton theory as a special case. Acknowledgement. several enlightening
The author discussions.
is indebted
to Vc’. C. Hermans
for
DEFINITION
OF SCATTERING
RESONANCES
143
REFERENCES 1) Blatt, J. M. and Weisskopf, V. F., Theoretical Nuclear Physics, p. 468, John Wiley (New York, 1952). 2) Smith, K., Rep. Progr. Phys. 29 (1966) 373. 3) Humblet, J. and Rosenfeld, L., Nuclear Phys. 26 (1961) 529. 4) Siegert, A. F. J., Phys. Rev. 56 (1939) 750. 5) Gamow, G., Z. Phys. 51 (1928) 204. 6) Fonda, L. and Newton, R. G., Ann. Physics 10 (1960) 490. 7) Feshbach, H., Ann. Physics 5 (1958) 357 and 19 (1962) 287. 8) Fundamentals in Nuclear Theory, editors De-Shalit, A. and Villi, C., Ch. 7 and 13, International Atomic Energy Agency (Vienna, 1967). 9) Newton, R. G., J. math. Phys. 2 (1961) 188. 10) Bochner, S. and Martin, W. T., Several Complex Variables, p. 205, Princeton University Press (Princeton, 1948).
ON THE
RELATION
OF RESONANCES
,BETWEEN
TWO FORMAL
IN NON-RELATIVISTIC
DEFINITIONS
SCATTERING
B. J. VERHAAR Afdeling VOOY Natuurkunde der Technische Hogeschool, Eindhoven, Nederland Received 1 September 1970
Synopsis A formulation of the resonance problem is given, which makes clear that an intimate relation exists between two formal theories of scattering of particles with in ternal degrees of freedom, one proposed by Humblet and Rosenfeld, the other by Fonda and Newton. In both theories resonances correspond to poles of the S matrix as a function of a variable 8, varying in a two-dimensional surface Z in the direct product space 81 @ 8s @ 8s . . ., where dr, &‘s, 8s . . . stand for the complex channel energies. The two theories differ in the choice of Z and 8.
1. Introduction. Qualitatively a resonance in a scattering process may be described as an anomaly in a cross section as a function of projectile energy, which’in its simplest possible form is illustrated in fig. 1. It is characterized by its resonance energy En and width rn. The index L is used here to distinguish different resonances. An alternative qualitative characterization of a resonance is possible in terms of the time-dependent description of scattering. Then it represents a situation in which the colliding particles “stick together” within the range of their mutual interaction during a time interval of the order of fi/Z’n.
Fig. 1. Breit-Wigner
resonance, characterized by resonance energy El and width Pa. 135
B. J. VERHAAR
136
Resonances ments,
have been observed
e.g., in neutron-nucleus
in various
types
l) and electron-atom
of scattering 2, scattering.
experiSince the
first of these observations several formal definitions of a scattering resonance have been proposed. One of these is due to Humblet and Rosenfelds), building on the “natural” definition of resonances by Siegert 4), which in turn was based on previous work by Gamow5) on the theory of nuclear CLdecay. A second theory, proposed by Fonda and Newtone), is intimately related to Feshbach’s work on this subject 7). In the following, after a concise description of the theories of H-R and F-N (abbreviations for the above-mentioned theories), we shall go into their relation. It will be shown that the F-N theory can be formulated in a form that reveals a close relationship to the H-R theory. To our knowledge this relationship has not been investigated in the literature?. Partly with the aim of avoiding intricacies of primarily mathematical nature, partly to exclude more essential complications such as the existence of three-particle channels, we shall concentrate on the most simple type of scattering problem for which each of the two scattering theories is meaningful. We consider the process of elastic and inelastic scattering of two particles with internal degrees of freedom. The hamiltonian of the total system is of the form
H = Hint(p) + -
P" 2m
+
VP, P)*
The first term, Hint, represents the sum of the internal hamiltonians of the two colliding particles with internal variables p, relative coordinate vector t and reduced mass m. The second and third terms represent the relative kinetic energy and mutual interaction, respectively. Only relative s waves are considered. Furthermore, I’ is supposed to vanish identically beyond a certain distance r = a and to be sufficiently regular at the origin. The spectrum of Hint is assumed to be discrete and to consist of a finite number n of points, the channel threshold energies Ed (CX= channel-index; El < Es <
E3 . ..).
2. Humblet-Rosenfeld theory. In this formalism resonances are defined as poles of the S matrix, analytically continued as a function of the total energy (to be specific we shall consider the S matrix whose open-channel sub-matrix is unitary and symmetric for real energies). An equivalent * t A comparative study has been devoted to the an international course in Triestes). Apparently, relation to be dealt with here, although each of the in detail. t The accidental situation of one or more bound is left out of consideration.
F-N and Feshbach theories during no attention has been paid to the F-N and H-R theories was treated states embedded in the continuum
DEFINITION
OF SCATTERING
RESONANCES
137
Ea+1
Fin. 2. Derivation of Humblet-Rosenfeld resonance + background expansion. Poles 8~ within G are included as resonance terms.
definition
is possible
in terms
of Gamow’s
“radioactive
states” : solutions
of the Schr6dinger equation of the system with outgoing waves in all channels. Clearly, such solutions are possible only for complex values of the total energy. The notion of “outgoing” waves is extended to complex values of the channel wave numbers in the following way. The radial wavefunction in each of the channels is required to be proportional to exp(ik,r)/r, where the channel wave number ka is given by k, = J(&
-
E&)2m/@,
(2)
in terms
of the total energy 8. To define the sign of the square root in this equation the branch points ca. o f the functions k&(6’) are of special importance (see fig. 2). We specify the signs of the square roots within an interval of the real energy axis between two thresholds by k, pos. real (open channels)
(3)
k, neg. imag. (closed channels) 1
and continue the square roots analytically into a region G of the complex plane. For the sake of simplicity we choose G such that E& < fie 8’ < e&+1. Within G the eigenvalue problem has thus been defined: the eigenvalues 8~ and eigenfunctions WA,labelled by 1, have to satisfy the equation Hyn = bays, together
with the above-mentioned
(4) boundary
condition.
B. J. VERHAAR
138
It is easily seen from the conservation
of particle
flux that all eigenvalues
are located in the lower half-plane part of G. (Note that for a more general choice of G, within which a curve such as r in fig. 2 is possible, eigenvalues may also be found in the upper half-plane.
For instance,
for each eigenvalue
&A in the lower half-plane, passing along r, the complex conjugated value 8; is also found to be an eigenvalue). Each of the eigenvalues is separated into its real and imaginary parts: ba = EA - +irn, with r~ > 0. The S matrix, analytically continued from the real energy interval into G, has poles located at the eigenvalues 8~. Assuming the existence of simple poles only, the following expansion is possible for any real energy E in G:
where the summation is over all poles in G, uLuja,l is the residue of S,,,(b) at br and S$L(E) is regular in G, having the form of an integral over its contour. Each term in the summation has the familiar Breit-Wigner form, En and ra functioning as resonance energy and width, respectively. The “background” term S::(E) is believed to have a relatively smooth form as a function of energy, if G has been chosen such that it includes a sufficient number of resonance poles. Note that different resonance + background expressions can be derived by a) expanding a different quantity*, e.g. (h&,)-i
(S -
l)n’a,
(6)
this specific type of expansion being advantageous for the discussion of threshold effects, b) choosing a different independent variable, e.g. the wave number k, of one of the channelss). In the following we shall see that a third “degree of freedom” exists, on the basis of which the Fonda-Newton theory
can be considered
3. Fonda-Newton ground expansion
as a special case.
theory. This theory leads to a similar resonance + back:
where, however, the significance of the symbols employed is different. The background term, which in the H-R theory is a regular remainder, is endowed with a specific physical significance in connection with direct reactions. The F-N theory is an attempt towards a unified reaction theory in which resonances and direct reactions are described in a combined frame* This type of the exponent
of ambiguity
includes
as a special
M in the Mittag-Leffler
case the arbitrarinesss)
expansion.
in the choice
DEFINITION
OF SCATTERING
RESONANCES
139
work. At first sight, this seems to go at the cost of the interpretation of resonances in terms of poles of the S matrix. It will be shown presently, however, that an interpretation in terms of a different type of S matrix poles remains possible. In the F-N theory the notion of “direct reaction” is interpreted as follows. One starts by specifying a real energy E for which the expression (7) shall be derived, and a corresponding division of the set of channels in open and closed ones. An arbitrary wavefunction y(p, r), not necessarily a stationary state, is expanded in terms of the internal eigenfunctions xa(p) :
(8) In the third member of this equation y,, and yC stand for the partial sums over open and closed channels, respectively. Eq. (8) thus corresponds to considering the total linear space of states y(p, r) to be a direct sum of “open and closed channel subspaces”. In view of this, y may be written in the form
yyyo,
0
(9)
YC
and correspondingly the hamiltonian posed into its components :
operator
of the system may be decom-
In principle, the direct part of the S matrix is defined as the S matrix obtained when the existence of the closed channels is left out of consideration, i.e. the S-matrix obtained when the total hamiltonian is replaced by Ho,. This definition is based on the assumption that direct processes are confined to the open-channel subspace without intervention of the closed channels, whereas resonance processes are indirect processes taking place through the intermediary of the closed-channel subspace. In Feshbach’s theory the resonance states are defined as eigenstates of subspace. This defiHCC> i.e. eigenstates of the isolated closed-channel nition looses much of its usefulness for stronger coupling between openand closed-channel subspaces. The F-N resonance definition includes this coupling: CPAin eq. (7) is defined by the eigenvalue problem (yC~ normalizable).
~iX?ycn = EpdWCl
(1’)
Here, yen is a wavefunction with a closed-channel subspace component and ZCc is an effective closed-channel hamiltonian given by Zcc = Hcc +
Hco
1 E
+
iv
-
Ho0
Hoc
(7 +
f0).
only
(12)
B. J. VERHAAR
140
Its anti-hermitian part is negative. The operator XCC is non-hermitians). Hence, 8~ is in general complex and located in the lower half-plane : 8~ = Ez. - $il’A,
with
ra > 0.
(13)
Furthermore, XcC depends on E, so that 8’~ is a function of E. Finally, the quantities a a’a, Ain eq. (7), the specific form of which is of no importance for the following, also turn out to be energy dependents). 4. Relation. A point of similarity between the two theories is the definition of En and I’r, which in both cases are determined by an eigenvalue problem. One point of difference, however, bears upon the reason for the eigenvalue to be a complex number: in the H-R theory this is due to the type of boundary condition, whereas in the F-N theory the non-hermiticity of SIP,, itself is responsible. An additional difference is the fact that the F-N resonance states, in contrast to the H-R states yn, are confined to the closed-channel subspace. Both differences can be eliminated by transforming the F-N eigenvalue equation (11) into two coupled equations. We supplement yc~ to form a state ye in the total space by adding an open-channel component yOn, defined by 1 yen = E + irl _ Hoo H diJcn. The total wavefunction
ye then satisfies the equation
(15) which is identical to the H-R eigenvalue equation (4), except that in the latter the open- and closed-channel energies are kept equal. A strong similarity also appears with respect to the boundary condition: on the basis of eqs. (1 l), (13) and (14) the radial wavefunctions in all channels must have the form exp(ik,r)/r, where the wave numbers are defined as in section 2, except that k, is defined in terms of E for the open channels and in terms of 8, for the closed channels. Note that, apart from demonstrating a close resemblance to the H-R eigenvalue equation, the coupled differential equations (15) may possibly have the advantage of being more amenable to numerical computation than the integro-differential equation ( 11). A different possibility to express the intimate relation to the H-R theory is the statement that the F-N complex resonance energies 8~ are the poles of the S matrix, analytically continued as a function of the energy of the closed channels, for a fixed energy (= E) of the open channels. This follows from the previously mentioned boundary condition and the generalized
DEFINITION
OF SCATTERING RESONANCES
141
definitiong) of the S matrix, considered as a function of independent channel wave numbers or channel energies ba = lLI+ Pki/2m. This suggests a generalized definition of complex resonance energies and the introduction of a more general resonance + background expansion, of which the H-R and F-N theories may be considered as special cases. Each of the S-matrix elements can be written in the forms) * iV,,,(klk2...) D(klkz...)
(16)
’
the functions D and N,,, being integral in each of the complex channel wave numbers, whereas D is independent of the channels 01 and 01’ considered t . Taken as functions of the channel energies, D and N,,, are regular, except for branch points at the thresholds. Representing each combination of channel energies as a point in the 2n-dimensional space bi 18 8s @ 83 . . . spanned by n perpendicular complex 8, planes, the following sets of points are of special importance (see fig. 3) : a) The line 81 = 8s = . . . = real, containing physically realizable combinations of channel energies. For a point Qi = 8s = . . . = E on this line, not coincident with a threshold, a resonance + background expansion of the S matrix is to be derived. hypersurface deb) The 292 - 2 dimensional, in general non-connected termined by D = 0. c) In D = 0 a set of points is selected to give rise to resonances. The selection is achieved by requiring this set to consist of the points of intersection between D = 0 and
Fig. 3. Location of resonance poles contributing to the S matrix. Each 8~ is a point of intersection between D = 0 and 27. For the sake of clarity the latter has been omitted. * Our notation is related to that of ref. 9 by N,,
= ga and
Nasa(a’ f a) = h-f,*,.
B. J. VERHAAR
142
d) A 2-dimensional surface Z:, which contains the point 6i= 8s = . . . = E. In order to arrive at a resonance + background expression of the form (5) or (7) we choose Z, like D = 0, to be an analytic manifoldlo), i.e. the set of (common) variables.
zeros of one or more analytic
functions
of a number
of complex
Finally, it is of importance to choose a complex variable d to parametrize the points in 2. It is chosen such that the channel energies varying in Z are regular functions of 8. Obviously, the functions D and N,,, are then regular in an environment G of the point 6’1 = 8s = . . . = E, provided G is chosen such that it does not contain possible branch points due to thresholds. The points of intersection mentioned under c) are the zeros 8~ of the function D in L’, and hence, discarding accidental coincidence of zeros of D and NLYfLY, the poles of the S matrix considered as a function of 8. Excluding also the accidental situation that .Z is tangent to the hypersurface D = 0, all poles are simple and a resonance + background expansion as in eq. (5) is possible. Note that a necessary condition for the possibility of this expansion is the absence of points of accumulation in the set 6~ within G. This is guaranteed by the choice of Z, since a point of accumulation would imply D = 0 in G, and therefore the existence of a bound state embedded in the continuum at the energy E, a situation which has been excluded above. From the point of view outlined here, the H-R and F-N theories differ only in the choice of Z and in the parametrization. In the H-R case Z is is defined by 61 = 8s = . . . (= complex, in general). The parametrization introduced by d = F, (01 arbitrary). On the other hand, to obtain the F-N theory we define Z by the requirement that all open-channel energies shall be equal to E, whereas all closed-channel energies be identical. The variable d is chosen to be any of the closed-channel energies. Note that in this case 2 depends on 6, explaining the energy-dependence of the quantities 6’~ and aala,A in eq. (7), whereas in the H-R theory Z contains the entire line mentioned under a) and is independent of E. We have thus described a formulation of the resonance problem, which amounts essentially to adding a third “degree of freedom” to the HumbletRosenfeld theory. This extended formulation includes the Fonda-Newton theory as a special case. Acknowledgement. several enlightening
The author discussions.
is indebted
to Vc’. C. Hermans
for
DEFINITION
OF SCATTERING
RESONANCES
143
REFERENCES 1) Blatt, J. M. and Weisskopf, V. F., Theoretical Nuclear Physics, p. 468, John Wiley (New York, 1952). 2) Smith, K., Rep. Progr. Phys. 29 (1966) 373. 3) Humblet, J. and Rosenfeld, L., Nuclear Phys. 26 (1961) 529. 4) Siegert, A. F. J., Phys. Rev. 56 (1939) 750. 5) Gamow, G., Z. Phys. 51 (1928) 204. 6) Fonda, L. and Newton, R. G., Ann. Physics 10 (1960) 490. 7) Feshbach, H., Ann. Physics 5 (1958) 357 and 19 (1962) 287. 8) Fundamentals in Nuclear Theory, editors De-Shalit, A. and Villi, C., Ch. 7 and 13, International Atomic Energy Agency (Vienna, 1967). 9) Newton, R. G., J. math. Phys. 2 (1961) 188. 10) Bochner, S. and Martin, W. T., Several Complex Variables, p. 205, Princeton University Press (Princeton, 1948).