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Unstable bound states by an algebraic method
Volume 112A, number 6,7
PHYSICS LETTERS
4 November 1985
U N S T A B L E B O U N D S T A T E S BY AN ALGEBRAIC M E T H O D S. KAIS Department of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel Received 26 July 1985; accepted for publication 24 August 1985
A nonunitary discrete irreducible representation of the SU(1,1) Lie algebra is used to determine the eigenenergies of unstable bound states of the Coulomb potential.
Symmetry methods have been applied successfully in a wide variety of physical systems. Symmetry and dynamical algebras have proved useful tools in the analysis of both bound and scattering states, by the use of unitary representations of certain compact groups and analytic continuation to noncompact groups [1]. In a recent paper, Alhassid, Iachello and Levine [2] have shown that by using a nonunitary discrete irreducible representation of the appropriate Lie algebra it is also possible to determine positions and widths of resonances. Some years ago, Sannikov [3] showed that the Schr6dinger equation also admits of another class of solutions which may be considered as u n s t a b l e b o u n d states. Such states are characterized by complex energy E and complex angular momentum X. The purpose of this letter is to present a straightforward algebraic treatment of such states. In particular, we consider the case of the Coulomb potential with Schr6dinger equation (in suitable units) [d2/dr 2 - ;~(;k + 1)/r 2 + M / r + E ] ~ = 0 ,
Now the necessary condition for bound states is evidently - R e E + r -2 Re(X + 1/2) 2 < , M / r .
This condition is certainly satisfied if both Re E < 0 and Re(X + 1/2) 2 > 0 and the complexity of the an. gular momentum ~ leads naturally to complex energies and states showing a form of instability. We seek a relation E = E(X) where Re E(X) gives the position of an unstable state, while Im E(X) is connected with its lifetime. In the algebraic approach, we use the group SU(1,1) to determine the unstable states. For such states, the boundary conditions correspond to a nonunitary representation, and the energy eigenvalues may be complex. On the other hand, the spectrum is discrete, so that the relevant rionunitary representation will also be discrete. The Lie algebra associated with the noncompact group SU(1,1) is characterized by the commutator relationships: [T 1,/'2] = - i T 3 ,
0 < ~ r < °° ,
(3)
[T 2, Ta] = i T 1 ,
(1) IT 3, TI] = iT 2 .
(4)
where M is a real positive constant. Following Sannikov [3], we seek solutions with complex energy E which satisfy the usual boundary conditions:
The generators may be given the convenient realization [4]
g/ ~ C r X+ 1 ,
T 1 = 32/3y2 +a/y 2 +y2/16,
~ exp(-x/-Z-Er),
r -+ 0 ,
r~ ~
(Re E < 0).
(2)
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T2 =
--~ i ( y a / a y + ~ ) , 269
Volume 112A, number 6,7
PHYSICS LETTERS
T 3 = O2/Oy 2 + a/y 2 - y 2 / 1 6 ,
(5)
where a is an arbitrary parameter, while the Casimir invariant is given by C = T 2 - V2 - T 2 = - a l 4 - 3 1 1 6 .
(6)
r = y 2 ~ = (2y)l/2x
(7)
(d2/dy 2 +a/y 2 + b y 2 + c ) x = 0 ,
(8)
3/16,
a =-[4~,(~,+ 1)+3/4],
b =4E,
c =4M.
(9)
Eq. (8) involves the generators linearly, and we may diagonalize the Casimir and the generator T 3 simultaneously C l / m ) = / ( J "+ l ) l / m ) ,
T 3 1 / m ) = m l i m ).
(10)
The discrete representations D/.+ of SU(t, 1) are characterized by [4] 1,-1+ 2 .....
(11)
W h e n / i s real, these representations are unitary and describe bound states, whereas for c o m p l e x / ( = a + i/3) the representation D ; is nonunitary but remains discrete. Furthermore, the eigenenergies also become complex. Following Wybourne [4] the existence of a discrete eigenvalue spectrum associated with eq. (8) requires that b = -c2/16(x -/')2 = [-e2/16(3' 2 + 132)2] (3'2 -/32 + 2~3'i), x=0,1,2 .....
(13)
and write ~ = l - i8 where (14)
We obtain /3=8,
(12)
(15)
and eq. (12) becomes E = [-M214(n 2 + 82) 2] [(n 2
where
270
/ ( / + 1) = - a / 4 -
a=-0+l),
eq. (1) becomes
7=x-~,
In order to determine a and/3, we use eqs. (6) and (10) to obtain
0<6 ~ 1 .
Now, using the transformations
m=-/,-/+
4 November 1985
82) + 26ni] ,
(16)
where n=x+l+
1,
x = O , 1,2 . . . . .
I / O < 6 "~ 1 then eq. (14) gives Im E ~ 6/n 3 . We note that IRe Et increases with decreasing l when x is fixed and 8 satisfies (14), and similarly when l is fixed and x decreases as expected for the usual (real angular momentum) Coulomb spectrum. I would like to thank Professor R.D. Levine and Professor M. Cohen for some useful discussions. The Fritz Haber Research Center is supported by the Minerva Gesellschaft ftir die Forschung, mbH, Munich, FRG. References [ 1] Y. Alhassid, F. Gtirsey and F. lachello, Phys. Rev. Lett. 50 (1983) 873; Ann. Phys. 148 (1983) 346; F. lacheUo and R.D. Levine, J. Chem. Phys. 77 (1982) 3046. [2] Y. Alhassid, F. lachello and R.D. Levine, Phys. Rev. Lett. 54 (1985) 1746. [3] S.S. Sannikov, Phys. Lett. 19 (1965) 216. [4] B.G.Wybourne, Classical groups for physicists (Wiley, New York, 1974).
PHYSICS LETTERS
4 November 1985
U N S T A B L E B O U N D S T A T E S BY AN ALGEBRAIC M E T H O D S. KAIS Department of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel Received 26 July 1985; accepted for publication 24 August 1985
A nonunitary discrete irreducible representation of the SU(1,1) Lie algebra is used to determine the eigenenergies of unstable bound states of the Coulomb potential.
Symmetry methods have been applied successfully in a wide variety of physical systems. Symmetry and dynamical algebras have proved useful tools in the analysis of both bound and scattering states, by the use of unitary representations of certain compact groups and analytic continuation to noncompact groups [1]. In a recent paper, Alhassid, Iachello and Levine [2] have shown that by using a nonunitary discrete irreducible representation of the appropriate Lie algebra it is also possible to determine positions and widths of resonances. Some years ago, Sannikov [3] showed that the Schr6dinger equation also admits of another class of solutions which may be considered as u n s t a b l e b o u n d states. Such states are characterized by complex energy E and complex angular momentum X. The purpose of this letter is to present a straightforward algebraic treatment of such states. In particular, we consider the case of the Coulomb potential with Schr6dinger equation (in suitable units) [d2/dr 2 - ;~(;k + 1)/r 2 + M / r + E ] ~ = 0 ,
Now the necessary condition for bound states is evidently - R e E + r -2 Re(X + 1/2) 2 < , M / r .
This condition is certainly satisfied if both Re E < 0 and Re(X + 1/2) 2 > 0 and the complexity of the an. gular momentum ~ leads naturally to complex energies and states showing a form of instability. We seek a relation E = E(X) where Re E(X) gives the position of an unstable state, while Im E(X) is connected with its lifetime. In the algebraic approach, we use the group SU(1,1) to determine the unstable states. For such states, the boundary conditions correspond to a nonunitary representation, and the energy eigenvalues may be complex. On the other hand, the spectrum is discrete, so that the relevant rionunitary representation will also be discrete. The Lie algebra associated with the noncompact group SU(1,1) is characterized by the commutator relationships: [T 1,/'2] = - i T 3 ,
0 < ~ r < °° ,
(3)
[T 2, Ta] = i T 1 ,
(1) IT 3, TI] = iT 2 .
(4)
where M is a real positive constant. Following Sannikov [3], we seek solutions with complex energy E which satisfy the usual boundary conditions:
The generators may be given the convenient realization [4]
g/ ~ C r X+ 1 ,
T 1 = 32/3y2 +a/y 2 +y2/16,
~ exp(-x/-Z-Er),
r -+ 0 ,
r~ ~
(Re E < 0).
(2)
0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
T2 =
--~ i ( y a / a y + ~ ) , 269
Volume 112A, number 6,7
PHYSICS LETTERS
T 3 = O2/Oy 2 + a/y 2 - y 2 / 1 6 ,
(5)
where a is an arbitrary parameter, while the Casimir invariant is given by C = T 2 - V2 - T 2 = - a l 4 - 3 1 1 6 .
(6)
r = y 2 ~ = (2y)l/2x
(7)
(d2/dy 2 +a/y 2 + b y 2 + c ) x = 0 ,
(8)
3/16,
a =-[4~,(~,+ 1)+3/4],
b =4E,
c =4M.
(9)
Eq. (8) involves the generators linearly, and we may diagonalize the Casimir and the generator T 3 simultaneously C l / m ) = / ( J "+ l ) l / m ) ,
T 3 1 / m ) = m l i m ).
(10)
The discrete representations D/.+ of SU(t, 1) are characterized by [4] 1,-1+ 2 .....
(11)
W h e n / i s real, these representations are unitary and describe bound states, whereas for c o m p l e x / ( = a + i/3) the representation D ; is nonunitary but remains discrete. Furthermore, the eigenenergies also become complex. Following Wybourne [4] the existence of a discrete eigenvalue spectrum associated with eq. (8) requires that b = -c2/16(x -/')2 = [-e2/16(3' 2 + 132)2] (3'2 -/32 + 2~3'i), x=0,1,2 .....
(13)
and write ~ = l - i8 where (14)
We obtain /3=8,
(12)
(15)
and eq. (12) becomes E = [-M214(n 2 + 82) 2] [(n 2
where
270
/ ( / + 1) = - a / 4 -
a=-0+l),
eq. (1) becomes
7=x-~,
In order to determine a and/3, we use eqs. (6) and (10) to obtain
0<6 ~ 1 .
Now, using the transformations
m=-/,-/+
4 November 1985
82) + 26ni] ,
(16)
where n=x+l+
1,
x = O , 1,2 . . . . .
I / O < 6 "~ 1 then eq. (14) gives Im E ~ 6/n 3 . We note that IRe Et increases with decreasing l when x is fixed and 8 satisfies (14), and similarly when l is fixed and x decreases as expected for the usual (real angular momentum) Coulomb spectrum. I would like to thank Professor R.D. Levine and Professor M. Cohen for some useful discussions. The Fritz Haber Research Center is supported by the Minerva Gesellschaft ftir die Forschung, mbH, Munich, FRG. References [ 1] Y. Alhassid, F. Gtirsey and F. lachello, Phys. Rev. Lett. 50 (1983) 873; Ann. Phys. 148 (1983) 346; F. lacheUo and R.D. Levine, J. Chem. Phys. 77 (1982) 3046. [2] Y. Alhassid, F. lachello and R.D. Levine, Phys. Rev. Lett. 54 (1985) 1746. [3] S.S. Sannikov, Phys. Lett. 19 (1965) 216. [4] B.G.Wybourne, Classical groups for physicists (Wiley, New York, 1974).