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Convergence of generalized spherical harmonic expansions in the three nucleon bound state
Volume 40B, number 4
PHYSICS LETTERS
24 July 1972
CONVERGENCE OF GENERALIZED SPHERICAL HARMONIC EXPANSIONS IN THE THREE NUCLEON BOUND STATE T.R. SCHNEIDER* Department of Physics, Universityof Pennsylvania and National Centerfor Energy Management and Power** University of Pennsylvania, Philadelphia,Penn. 19104, USA Received 6 June 1972 Theorems are presented on the rate of convergence of approximate three nucleon binding energies obtained by expanding the bound state wavefunction in a truncated generalized spherical harmonic basis. The rate of convergence is related to the properties of the two-body potential. We clarify some convergence problems encountered in the three nucleon problem where the bound state wavefunction is expanded in a set of generalized spherical harmonics. These states, developed by Dragt [ 1] and others [2, 3], permit us to expand the three nucleon wavefunction in the form
qs(r, g~) = ~_j Rx(r)gx(~ )
(1)
where ~ represents the complete set of labels, r is a generalized radius vector, and ~2 represents the angular coordinates (five angular coordinates in the three nucleon problem). The introduction of the generalized radius vector and angular coordinates transforms the Schr6dinger equation to /2 [ d2+5 d
A 2-]
2mE&2
-~-J~+V(r,Sq)~=E$.
rdr
(2)
Here A 2 is a "Grand Angular Momentum" operator and can be related to a generalized impact parameter [4]. The corresponding classical operator is zero only when all three particles are at the same point in space at the same time, and it is a measure of the compactness of the physical system. The angular states used in eq. (1) are eigenvectors of A2with eigenvalues A2gx = X(X + 4)g x . * Supported in part by the National Science Foundation. ** Present address.
Using the wavefunction expansion in eq. (2) the Schr6dinger equation becomes a system of coupled radial equations, if the two-body potentials are local: /72 F d 2 +5 d
-UmL7 75
X(X_+4)IRx + ~_~XVx,Rx:ERx-
r2 J
(4)
For most potentials the solution of the above eigenvalue problem is a difficulat task. The number of coupled equations is infinite, so we must introduce a scheme to obtain approximate solutions. We do this by restricting the wavefunction expansion to only a finite number of terms. We then examine how the approximate binding energies vary as we increase the size of the basis. For most potentials we find that the convergence rate is proportional to an inverse power of ;~. This approach is not restricted to the threebody problem. Our result can be applied to any expansion of this type and can be applied to the fourbody problem and the N-body problem [5]. The theorems we present establish analytic bounds on the convergence rate of the approximate binding energy. Our approach is different from the work of De La Rippelle [6] since he was concerned only with numerical examples. Numerical examples are not sufficient and are not a substitute for analytic bounds. We generalize our results by using Dirac notation. We are looking for approximate solutions of the Schr6dinger equation
(3) Hlff)=Elff) ;
(5)
439
Volume 40B, number 4
PHYSICS LETTERS
H is a self-adjoint (Hermitian) Hamiltonian and E is the eigenvalue corresponding to the eigenvector I~b). The properties of H are such that it has a point spectrum and a smallest eigenvalue. We will suppress nonessential labels and assume that E is non-degenerate. Associated with H is a Hilbert space, I]. This space is spanned by a complete orthonormal basis, {[w 1, w 2 ..... Wr)}, which we will refer to as In). Corresponding to this basis is a set of operators, (W/} 7=1' In our notation we have,
Wiln) = wi(n)ln)
(6)
where wi(n ) is the eigenvalue of Wi in the nth basis vector. Any state If) in V can be represented by the expansion
If) = ~ a ( n ) l n )
24 July 1972
~E, but
<~NIHkoN> = @NIHNICN> = g N >~E and since each EN is an approximation to EN+ 1 we have
EN >~EN+ 1 ... >~E . To understand how EN converges to E we first want to know how INI$) converges to I~). This is the Fourier convergence problem where we have shown that [7]: Theorem 1 : If the quantities (Wi) oi[~) are bounded, then the expansion coefficients are bounded:
(7) la(n)l • II(Wi)°iltP)ll/lwi(n)ail.
where a(n) is the Fourier coefficient
a(n) = (nl~).
(8)
All indicatedsums may be infinite. If in the expansion of If) we include only a finite number of terms, the partial sum N [fN )= ~
a(n)ln)
will be an approximation to If). Our proceedure is to solve the eigenvalue problem (5) in a subspace that is of manageable size and which contains a large segment of the solution vector hb). We solve the eigenvalue problem
H N I~oN) = ENId:N)
(9)
where HN = IN H IN and IN = ~ N I n ) ( n l and I = IN+IN. /N is a simple projection operator, h0N) is an approximation to I~b)while EN is an approximation to E. Each approximate energy, EN, is a variational bound on E and EN+ 1 . This follows from the variational principal. For any Iq~) such that (~ Iq~)= 1 we have
It is important to realize that the oi's can all be very different and that the utility of this result depends on finding a Wi such that wi(n ) is an increasing function of n. We are not restricted to local differential operators or any particular representation. If we are treating local differential operators, this theorem connects the rate of decrease of a(n) with the differentiability of the function. This theorem gives us a bound on the component of Iff) lying outside of the subspace. If we use the projection operator
INI~) = ~
a(n)ln)
N+I we have IIINI~)II ~< [l(Wi)°ilqJ)ll ~a 1/Iwi(n)C'il . N+I That the Fourier convergence problem dominates the convergence o f E N t o e and I ~ ) t o Iff)is contained in the following [8] : Theorem 2: For the approximation to the solution of (5) obtained by solving (9) we have, for sufficiently large N, N > N O
(~b IHkb) = E~>E,
IEN- EI and hence for I~PN)we have and 440
~ clEl I[/NI~)[[ 2
Volume 40B, number 4
PHYSICS LETTERS
I kON>-PI¢N>I ~ll . Here P is the operator which projects out the solution of H. That is, for any I f)
( H - E ) P I ¢ ) = O. N must be large enough so that % v l ¢ , > >~ o .
If we combine the results of Theorems 1 and 2, we can relate the convergence rate o f E N to E and k0N> to Iff> with the operators, Wi, the eigenvalues wi(n ), and the number of states. Theorem 3: If the quantities II(Wi)ail@>tl are bounded, then the error terms are bounded:
IE EN[ <~clEI [I (Wi)oild2>ll2~V~+l 1/Iwi(n)°il) 2
sions about the properties of (A2)°~. I f f and g are square integrable then so is the product, fg, and the sum, f + g. If V is square integrable, then A 2 t~/r 2 must be square integrable. If (A2) a V is square integrable then A 2+2c~/r 2 is square integrable. Hence A2+2aff is square integrable. Since, with the usual definition of the norm, all functions which are square integrable are bounded, we have proved the desired result. Theorem 4: If A 2~ V is square integrable, then the quantity IIA2+2a~bll is bounded. It is worthwhile to consider several examples from the nuclear three-body problem. Commonly used potentials in nuclear physics are the Yukawa, square well, and Gaussian. These have the functional form: Yukawa
6aussian
These results tell us that the convergence rate will be fast if wi(n) °i increases rapidly with increasing n. In the extreme case where I~> is an eigenvector of Wi we get the exact answer by treating only a finite number of states. We have seen that the asymptotic convergence rate is dominated by the Fourier convergence problem. In any particular problem the precise rate will depend on the exact form of the Hamiltonian and the operators, Wi. These operators determine the basis. A different basis will yield a different convergence rate. In the three nucleon problem the generalized spherical harmonics are eigenvectors of the angular part of the field free Hamiltonian. The rate of convergence depends strongly on the commutator of the potential, V, with A 2. Consider the radial Schr6dinger equation written in the form
( L o - A 2 / r 2 + V)q/=Et~. The bound state solutions of this equation are square integrable. Recalling the elementary properties of square integrable functions [9], we can draw conclu-
V(r) = Voe-r/a/(r/a )
Square well V(r) = V 0 , r <. a =0, r>a
and 11Ig~v>-PI~N>II ~
24 July 1972
V(r) = V 0 e x p ( r 2 / a 2 ) .
Each of these is representative of a class of potentials. Each is less singular than r - 2 and falls off faster than r -3 . A potential made up of a sum of Yukawa potentials or square wells or Gaussian potentials has the same properties as the corresponding one term potential. For the Yukawa potential all the difficulties are introduced by the r -1 singularity. This potential is square integrable but A 2 V is not square integrable. We expect that the approximate binding energy will converge at least as fast as X-4. The square well is discontinuous at r = a, but it is bounded. Similarly A 2 ~ is discontinuous at r = a, but it is bounded. The quantity A 4 ~ involves derivatives of the step function and it is not square integrable. We expect the convergence rate to be at least as fast
as 3`-4. The Gaussian is a function of r 2 and is finite everywhere. It is infinitely differentiable, while it and all of its derivatives are square integrable. Hence A 2n is bounded for all values o f n . The convergence rate will be faster than any inverse power of 3, (exponentially fast). The Gaussian is a very special case and very different from the more singular Yukawa potential and square well. We can not extrapolate from numerical examples using Gaussian potentials, as sug44 1
Volume 40B, number 4
PHYSICS LETTERS
gested by De La Rippelle- [6], to gain insight into the convergence problems o f more singular potentials. The approach we have described is most useful when the physical problem and the choice o f basis is well matched. In this case the first few states would gives us a good approximation to the true wavefunction and the approximate binding energy would be close to the true binding energy. The theorems presented here do not tell us when the first states will be a good approximation, but relate general properties o f the two-body potential with the convergence rate. The sequence of approximate binding energies divides into two regions. It is in the region where ~2c~ is large relative to the quantity II(A2)~I 4>11 that these theorems will prove useful. These theorems are asymptotic convergence theorems for they are useful when k is large. To estimate how many states will make important contributions to the binding energy we must estimate IAI. A generalized kr argument can be constructed [4] to estimate IAI ~ lkl Irl. It is necessary t o carefully pick the estimates o f k and r so they exhibit dependence on both the two-body potential
442
24 July 1972
and the three-body bound state. In a later paper we will examine these arguments in more detail and present some numerical examples. The author would like to thank Professor Ralph D. A m a d o for his advice and encouragement. Without his guidance, this work would not have been realized.
References [1] A.J. Dragt, J. Math. Phys. 6 (1963) 533. [2] Yu.A. Simonov, Soviet J. Nucl. Phys. 3 (1966)461. [3] A.M. Badalyan and Yu. Simonov, Soviet J. Nucl. Phys. 3 (1966) 755. [4] F.T. Smith, Phys. Rev. 130 (1960) 1063. [5] H.W. Galbraith, J. Math. Phys. 12 (1971) 782. [6] M. Fabre De La Ripelle, Letter Nuo. Cim. 1 (1971) 584. [7] T.R. Schneider, J-. Math. Phys. 12 (1971) 1508. [8] G.M. Vainikko, U.S.S.R. Comp. Math. and Math. Phys. 4, No. 3 (1964) 9. [9] F. Reisz and B.Sz. Nagy, Functional analysis (F. Unger, New York, 1966).
PHYSICS LETTERS
24 July 1972
CONVERGENCE OF GENERALIZED SPHERICAL HARMONIC EXPANSIONS IN THE THREE NUCLEON BOUND STATE T.R. SCHNEIDER* Department of Physics, Universityof Pennsylvania and National Centerfor Energy Management and Power** University of Pennsylvania, Philadelphia,Penn. 19104, USA Received 6 June 1972 Theorems are presented on the rate of convergence of approximate three nucleon binding energies obtained by expanding the bound state wavefunction in a truncated generalized spherical harmonic basis. The rate of convergence is related to the properties of the two-body potential. We clarify some convergence problems encountered in the three nucleon problem where the bound state wavefunction is expanded in a set of generalized spherical harmonics. These states, developed by Dragt [ 1] and others [2, 3], permit us to expand the three nucleon wavefunction in the form
qs(r, g~) = ~_j Rx(r)gx(~ )
(1)
where ~ represents the complete set of labels, r is a generalized radius vector, and ~2 represents the angular coordinates (five angular coordinates in the three nucleon problem). The introduction of the generalized radius vector and angular coordinates transforms the Schr6dinger equation to /2 [ d2+5 d
A 2-]
2mE&2
-~-J~+V(r,Sq)~=E$.
rdr
(2)
Here A 2 is a "Grand Angular Momentum" operator and can be related to a generalized impact parameter [4]. The corresponding classical operator is zero only when all three particles are at the same point in space at the same time, and it is a measure of the compactness of the physical system. The angular states used in eq. (1) are eigenvectors of A2with eigenvalues A2gx = X(X + 4)g x . * Supported in part by the National Science Foundation. ** Present address.
Using the wavefunction expansion in eq. (2) the Schr6dinger equation becomes a system of coupled radial equations, if the two-body potentials are local: /72 F d 2 +5 d
-UmL7 75
X(X_+4)IRx + ~_~XVx,Rx:ERx-
r2 J
(4)
For most potentials the solution of the above eigenvalue problem is a difficulat task. The number of coupled equations is infinite, so we must introduce a scheme to obtain approximate solutions. We do this by restricting the wavefunction expansion to only a finite number of terms. We then examine how the approximate binding energies vary as we increase the size of the basis. For most potentials we find that the convergence rate is proportional to an inverse power of ;~. This approach is not restricted to the threebody problem. Our result can be applied to any expansion of this type and can be applied to the fourbody problem and the N-body problem [5]. The theorems we present establish analytic bounds on the convergence rate of the approximate binding energy. Our approach is different from the work of De La Rippelle [6] since he was concerned only with numerical examples. Numerical examples are not sufficient and are not a substitute for analytic bounds. We generalize our results by using Dirac notation. We are looking for approximate solutions of the Schr6dinger equation
(3) Hlff)=Elff) ;
(5)
439
Volume 40B, number 4
PHYSICS LETTERS
H is a self-adjoint (Hermitian) Hamiltonian and E is the eigenvalue corresponding to the eigenvector I~b). The properties of H are such that it has a point spectrum and a smallest eigenvalue. We will suppress nonessential labels and assume that E is non-degenerate. Associated with H is a Hilbert space, I]. This space is spanned by a complete orthonormal basis, {[w 1, w 2 ..... Wr)}, which we will refer to as In). Corresponding to this basis is a set of operators, (W/} 7=1' In our notation we have,
Wiln) = wi(n)ln)
(6)
where wi(n ) is the eigenvalue of Wi in the nth basis vector. Any state If) in V can be represented by the expansion
If) = ~ a ( n ) l n )
24 July 1972
<~NIHkoN> = @NIHNICN> = g N >~E and since each EN is an approximation to EN+ 1 we have
EN >~EN+ 1 ... >~E . To understand how EN converges to E we first want to know how INI$) converges to I~). This is the Fourier convergence problem where we have shown that [7]: Theorem 1 : If the quantities (Wi) oi[~) are bounded, then the expansion coefficients are bounded:
(7) la(n)l • II(Wi)°iltP)ll/lwi(n)ail.
where a(n) is the Fourier coefficient
a(n) = (nl~).
(8)
All indicatedsums may be infinite. If in the expansion of If) we include only a finite number of terms, the partial sum N [fN )= ~
a(n)ln)
will be an approximation to If). Our proceedure is to solve the eigenvalue problem (5) in a subspace that is of manageable size and which contains a large segment of the solution vector hb). We solve the eigenvalue problem
H N I~oN) = ENId:N)
(9)
where HN = IN H IN and IN = ~ N I n ) ( n l and I = IN+IN. /N is a simple projection operator, h0N) is an approximation to I~b)while EN is an approximation to E. Each approximate energy, EN, is a variational bound on E and EN+ 1 . This follows from the variational principal. For any Iq~) such that (~ Iq~)= 1 we have
It is important to realize that the oi's can all be very different and that the utility of this result depends on finding a Wi such that wi(n ) is an increasing function of n. We are not restricted to local differential operators or any particular representation. If we are treating local differential operators, this theorem connects the rate of decrease of a(n) with the differentiability of the function. This theorem gives us a bound on the component of Iff) lying outside of the subspace. If we use the projection operator
INI~) = ~
a(n)ln)
N+I we have IIINI~)II ~< [l(Wi)°ilqJ)ll ~a 1/Iwi(n)C'il . N+I That the Fourier convergence problem dominates the convergence o f E N t o e and I ~ ) t o Iff)is contained in the following [8] : Theorem 2: For the approximation to the solution of (5) obtained by solving (9) we have, for sufficiently large N, N > N O
(~b IHkb) = E~>E,
IEN- EI and hence for I~PN)we have and 440
~ clEl I[/NI~)[[ 2
Volume 40B, number 4
PHYSICS LETTERS
I kON>-PI¢N>I ~
( H - E ) P I ¢ ) = O. N must be large enough so that % v l ¢ , > >~ o .
If we combine the results of Theorems 1 and 2, we can relate the convergence rate o f E N to E and k0N> to Iff> with the operators, Wi, the eigenvalues wi(n ), and the number of states. Theorem 3: If the quantities II(Wi)ail@>tl are bounded, then the error terms are bounded:
IE EN[ <~clEI [I (Wi)oild2>ll2~V~+l 1/Iwi(n)°il) 2
sions about the properties of (A2)°~. I f f and g are square integrable then so is the product, fg, and the sum, f + g. If V is square integrable, then A 2 t~/r 2 must be square integrable. If (A2) a V is square integrable then A 2+2c~/r 2 is square integrable. Hence A2+2aff is square integrable. Since, with the usual definition of the norm, all functions which are square integrable are bounded, we have proved the desired result. Theorem 4: If A 2~ V is square integrable, then the quantity IIA2+2a~bll is bounded. It is worthwhile to consider several examples from the nuclear three-body problem. Commonly used potentials in nuclear physics are the Yukawa, square well, and Gaussian. These have the functional form: Yukawa
6aussian
These results tell us that the convergence rate will be fast if wi(n) °i increases rapidly with increasing n. In the extreme case where I~> is an eigenvector of Wi we get the exact answer by treating only a finite number of states. We have seen that the asymptotic convergence rate is dominated by the Fourier convergence problem. In any particular problem the precise rate will depend on the exact form of the Hamiltonian and the operators, Wi. These operators determine the basis. A different basis will yield a different convergence rate. In the three nucleon problem the generalized spherical harmonics are eigenvectors of the angular part of the field free Hamiltonian. The rate of convergence depends strongly on the commutator of the potential, V, with A 2. Consider the radial Schr6dinger equation written in the form
( L o - A 2 / r 2 + V)q/=Et~. The bound state solutions of this equation are square integrable. Recalling the elementary properties of square integrable functions [9], we can draw conclu-
V(r) = Voe-r/a/(r/a )
Square well V(r) = V 0 , r <. a =0, r>a
and 11Ig~v>-PI~N>II ~
24 July 1972
V(r) = V 0 e x p ( r 2 / a 2 ) .
Each of these is representative of a class of potentials. Each is less singular than r - 2 and falls off faster than r -3 . A potential made up of a sum of Yukawa potentials or square wells or Gaussian potentials has the same properties as the corresponding one term potential. For the Yukawa potential all the difficulties are introduced by the r -1 singularity. This potential is square integrable but A 2 V is not square integrable. We expect that the approximate binding energy will converge at least as fast as X-4. The square well is discontinuous at r = a, but it is bounded. Similarly A 2 ~ is discontinuous at r = a, but it is bounded. The quantity A 4 ~ involves derivatives of the step function and it is not square integrable. We expect the convergence rate to be at least as fast
as 3`-4. The Gaussian is a function of r 2 and is finite everywhere. It is infinitely differentiable, while it and all of its derivatives are square integrable. Hence A 2n is bounded for all values o f n . The convergence rate will be faster than any inverse power of 3, (exponentially fast). The Gaussian is a very special case and very different from the more singular Yukawa potential and square well. We can not extrapolate from numerical examples using Gaussian potentials, as sug44 1
Volume 40B, number 4
PHYSICS LETTERS
gested by De La Rippelle- [6], to gain insight into the convergence problems o f more singular potentials. The approach we have described is most useful when the physical problem and the choice o f basis is well matched. In this case the first few states would gives us a good approximation to the true wavefunction and the approximate binding energy would be close to the true binding energy. The theorems presented here do not tell us when the first states will be a good approximation, but relate general properties o f the two-body potential with the convergence rate. The sequence of approximate binding energies divides into two regions. It is in the region where ~2c~ is large relative to the quantity II(A2)~I 4>11 that these theorems will prove useful. These theorems are asymptotic convergence theorems for they are useful when k is large. To estimate how many states will make important contributions to the binding energy we must estimate IAI. A generalized kr argument can be constructed [4] to estimate IAI ~ lkl Irl. It is necessary t o carefully pick the estimates o f k and r so they exhibit dependence on both the two-body potential
442
24 July 1972
and the three-body bound state. In a later paper we will examine these arguments in more detail and present some numerical examples. The author would like to thank Professor Ralph D. A m a d o for his advice and encouragement. Without his guidance, this work would not have been realized.
References [1] A.J. Dragt, J. Math. Phys. 6 (1963) 533. [2] Yu.A. Simonov, Soviet J. Nucl. Phys. 3 (1966)461. [3] A.M. Badalyan and Yu. Simonov, Soviet J. Nucl. Phys. 3 (1966) 755. [4] F.T. Smith, Phys. Rev. 130 (1960) 1063. [5] H.W. Galbraith, J. Math. Phys. 12 (1971) 782. [6] M. Fabre De La Ripelle, Letter Nuo. Cim. 1 (1971) 584. [7] T.R. Schneider, J-. Math. Phys. 12 (1971) 1508. [8] G.M. Vainikko, U.S.S.R. Comp. Math. and Math. Phys. 4, No. 3 (1964) 9. [9] F. Reisz and B.Sz. Nagy, Functional analysis (F. Unger, New York, 1966).