doi:10.4236/am.2011.210165 Published Online October 2011 (http://www.SciRP.org/journal/am)
No Degeneracy of the Ground State for the Impact
Parameter Model
Héctor C. Merino1, Juan Héctor Arredondo2
1
Universidad Autónoma de Guerrero, Unidad Académica de Matemáticas,
Chilpancingo, México
2
Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa,
Distrito Federal, México
E-mail: [email protected], [email protected] Received May 25, 2011; revised July 6, 2011; accepted July 13, 2011
Abstract
A time dependent Hamiltonian associated to the impact parameter model for the scattering of a light particle and two heavy ones is considered. Existence and non degeneracy of the ground state is shown.
Keywords:Impact Parameter Model, Non Degeneracy of the Ground State
1. Introduction
In [1,2], the impact parameter model for the scattering of two heavy particles and a light one is studied, where it is assumed that the heavy particles are infinitely massive and that their motion along a classical trajectory is not affected by the light particle. Also, rigorous proof from first principles of the validity of Massey’s criterion is given [1,3].
The above mentioned results were proved for a simple Hamiltonian, by means of an adiabatic argumentation. Now we study a more complicated one than in [1], where a precise knowledge of the discrete spectrum of the cor-responding Hamiltonian was needed.
A physical ground state is a state of minimal energy, and therefore it has a relevant role in quantum theories. See for instance [4-17].
In this work we prove non degeneracy of the ground state for the Hamiltonian
1 1 1 2 2 1, 2 2,1
, 2
H t V V V V (1)
defined as an operator in the Hilbert space 2
nL
2 n of all complex valued Lebesgue measurable square
inte-grable functions on , with domain , the
Sobolev space of order two [18]. is the Laplace opera-tor [11].
n
H
2 2
2 2
1
, n
x x
with derivatives in the distribution sense, and, 1, 2, 1, 2
are positive constants. Also, for , we will
take the potentials of rank one:
1, 2
k
k
V
,
,k k
V g gk L2
n , (2) with g g1, 2 fixed elements in
2 n
L . Here ( , ) de-notes the scalar product in L2
n , antilinear on the factor on the left. Moreover,
, , ,
V g g , , g ,
x :g
x
t
, (3)
t n
being a continuous function on with values in
satisfying
0 0 n and
| |limt t .
We denote by the Fourier transform [19], as an uni-tary operator in 2
nL :
| |
ˆ lim eipx d ,
K x K
g g
p x
x 2
, n
gL
where the limit is taken in the L2-norm.
2. Main Theorem
From Weyl’s theorem [16], one knows that for each
t, H t
is a self-adjoint operator with discrete spectrum in
, 0
. The eigenvector corresponding to1192 H. C. MERINO ET AL.
ground state for H t
i
. The following theorem was proved in [20].
Theorem 2.1. For 1, 2, let
and 12 n i
g L
gˆnonnegative functions obeying More-over, we suppose the constants
2 1
ˆ
| p|g L
,
i i
. in Equation (1)
satisfy
1 1 2 1 2 2 0.
0E (2)
such that 0 E1 and 0 E2 E1. Here 2
1 0 and
1
, (
E E
2),E , E are the ground state ei-genvalues associated to
1 1
0
E
1 2 2
2 1 1 2
1 1
, ,
2 2
1 1
, ,
2 2
V V
V V
respectively. Then the following statements are valid: 1) The eigenvalue , corresponding to the ground state for the operator
1 2
1
1 2
21 2
0 ,
H V V
,
E
and the eigenvalue corresponding to the ground state for the operator
1 1 1 2
1
, 2
H V V
are strictly negative and the inequality E0 E holds.
2) The eigenvale E t
, corresponding to the ground state for H t
for all t lies in the interval
0
3) In the interval
,
E E
.
1
there are no eigenval-ues of,
E E
H t for every t.
We mention that for a given function 0 g L2
n ,0
one can find a sufficiently large positive constant such that the operator
1
,
2 g g
(4)
has a (unique) negative eigenvalue E for 0. In
fact, E is a negative eigenvalue iff [1]
2
1/ 2 2
ˆ 1
,
2
g
p E
(4)
where we denote 2 2
: .
p p Note also that for a given
g the right hand side of (5) is a monotone decreasing
function of E. Therefore, given functions gi in
2 n
L
one can find constants i, i
i1, 2
large enough forthe hypotheses of the theorem to hold.
We will prove in this manuscript that under the hy-potheses of theorem 2.1, for t the ground state of
H t is not degenerate.
Let E t
be the ground state eigenvalue of the time dependent operator given by Equation (1). We define
: 2
2
p
p E
t and
2 2
1
ˆ ˆ
1 1
;
ii ii
i i
g g
a d
2
for i1, 2. (6)
Moreover,
1
12 1 1,
1
11 11 2
1
12 2 1,
1
21 22, 1
1
12 22 2,
ˆ, ˆ ,
ˆ , ˆ ,
ˆ , ˆ
ˆ , ˆ ,
ˆ , ˆ
a g g
b g g
b g g
b g g
d g g
, (7)Lemma 2.1. Let E t
be the ground state eigen-value of the time dependent operator H t
given by Equation (1). Then, the matrix equation4
11 12 11 21 12 22 12 11 11 12 11 12 21 11 12 22
0 ,
: ,
T
M
a a b b
a a b b
A B
M
b b d d
B D
b b d d
x
y
(8)
has a nontrivial solution. Furthermore
11 1212 22
det D det d d 0
d d
t
.Proof: Let
t the eigenvector for H t
with re-spective eigenvalue E t
, then the Fourier transform of
t is given by
1, 1
1 1 2 1,
2, 2
1 2 2 2,
ˆ ˆ
ˆ , ,
( )
ˆ ˆ
, , ,
g g
t g g
g g
g g
p p
p p
(9)
where
2
: . The Plancherel theorem im-
2
p E t
p
plies that
u v, u vˆ ˆ, u v, L2
n1
ˆ
. Taking inner roducts in (9) with
p g and gˆ1, for i1, 2, we get
1193
2
1 1 1
1
1 1 1 2 1, 1 1, 1 2 1 2 2 2, 1 2,
2 1,
1 1
1, 1 1 1, 1 2 1, 1 2 1, 2 2 2, 1, 2,
2 1 1 2
ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ , ˆ, ˆ , ˆ , ˆ ˆ , ˆ, ˆ ˆ , ˆ, ˆ ,
ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ , ˆ, ˆ , ˆ ˆ ˆ , ˆ , ˆ ˆ , ˆ , ˆ
ˆ ˆ
ˆ , ˆ , ˆ ,
g
g g g g g g g g g g g
g
g g g g g g g g g g g
g g g
1
,
2
1 1 2 1
1 2 1, 2 1, 1 2 2 2, 2 2,
2 2,
1 1 1
2, 1 1 2, 1 2 1, 2, 1, 1 2 2, 2 2 2,
ˆ
ˆ ˆ ˆ
ˆ ˆ , ˆ , ˆ ˆ , ˆ , ˆ , ˆ ,
ˆ
ˆ ˆ ˆ ˆ ˆ
ˆ , ˆ , ˆ , ˆ ˆ , ˆ , ˆ ˆ , ˆ , ˆ ˆ , ,
g
g g g g g g g g
g
g g g g g g g g g g g
.
(10)
This system of equations is represented in matrix form precisely by Equation (8), where
2 2
ˆ ˆ
1 g 1 g
2 2
1/ 2 1/ 2
2 2
1 2
2
2,
2 2
2 2
ˆ ˆ ,
2
E
R t
p p
E E
g g
p E
1 1 1
2 2 1,
1 2 1
2 2 2,
ˆ ˆ ,
: ;
ˆ
ˆ ,
ˆ ˆ , :
ˆ
ˆ ,
g x
x g
g y
y g
x
y
(11)
From Theorem 2.1 we deduce the existence of a non-trivial solution to Equation (8).
Now we fix For every t let us consider
the function, 0.
E
and observe that for E0(2)E,
2 4 2
2
2,
1 2 2 2 1 2 2
2
1/ 2 1/ 2 2 1/ 2
2 2 2
1 2 1 2 1 2 1 2
2
1 2 2
1/ 2 2
1 2 1 2
ˆ
ˆ ˆ ˆ
1 1
ˆ , 2
2 2 2
ˆ 1
0. 2
E
g
g g g
R t g
p
p p E p
E E
g p
E
E
(12)
The last inequality being true because of the remark following Equation (5). Also, we have used the Schwarz inequality and the Fourier transform property
1 2
1
1 2
1 0
2 2
H V V
is not degenerate.
2, 2
ˆ i t ˆ .
g p e pg p When E E t
is the ei- Proof: Lemma (2.1) assures that exists.Equa-tion (8) implies,
1
D
genvalue for H t
,
E t
then the determinant of matrix D in
Equation (8) satisfies, Theorem 2.1
states that
det D R E t .
0,
E E
and E E1E0
2 .
D 0. Then, (12) gives det
1
,
D B
y x
AB D BT 1
x0. (13)We take 1 so that,
: T ,
C A B D B
The main result will be proved by showing that the dimension of the eigenspace associated to the ground sate remains constant over time.
11 12 12 22
,
c c C
c c
where
1194 H. C. MERINO ET AL.
2 2
11 22 11 21 12 21 11 11 11
12
2
11 12 22 11 21 11 11 12 12 21 12 12
2 2
11 11 11 12 12 21 22 22 22 2 , det , det 2 . det
b d b b d b d c a
D c
b b d b b d b d b b d a
D b d b b d b d c a D (14)
From Theorem 2.1, we know that there exists a
non-trivial solution to system (8). Thus
Accord-ingly,
detC0.
11 12 11 12 , c c C kc kc
(15)
for some constant kk t
. Moreover, for t0 thematrix is not null. In fact, for this value of t,
the following terms simplify
CC t
2 1/ 2 2
12 0 1
2 1/ 2 2
12 0 2
1 2
21 12 11 1 0 2
ˆ , 2
ˆ , 2
ˆ , ˆ .
2
p
a E g
p
d E g
p
b b b g E g
It follows that,
2
2
11 11 22 12 11 1
12 12 2 1/ 2
0
2 2
2 1 2 2
1 2 1/ 2
2 1 2 0 0 2 2
1/ 2 1
2 2
0 2 1 0 2
2 ˆ
det
2
ˆ ˆ
ˆ , 2
2 2
det
ˆ ˆ ˆ
2 ,
2 2
det 0,
b d d d b g c a D p E g g g p p E E D p p
E g g E g
D (16)
where we use equation (5), the hypothesis E0
2 E1 and statement (3) of theorem 2.1. Therefore,2 1/ 2 2 1 2 0 2 1 2 ˆ . 2 2 p
E g
Equations (11)-(16) imply,
11 12 22 22 12 21 12 11 22
1 2 1
12
11 22 11 12 12 11 12 21 11
2 2, 1
12 ˆ ˆ , det det ˆ ˆ , det det
c b d b d
b d b d
g x
D c D
c b d b d
b d b d
g x
D c D
(17)
Substitution of these equalities in Equation (9) gives,
1 1,1 2 1 2
0 0
2, 2
2 2 3 2
0 0 ˆ ˆ ˆ 0 2 2 ˆ ˆ . 2 2 g g x k p p E E g g k k p p E E (18) Here,
11 1 1212 21 12 11 12 22 12 11 22 11 22 12
2
12
12 11 12 11 22 11 12 21 11 11 12 12
3 12 det . det c k c
c b d c b d c b d c b d k
c D
c b d c b d c b d c b d k c D (19)
This determines the vector up to a
multiplica-tive constant, and from the Plancherel theorem, also the eigenspace associated to the ground state for
ˆ 0
0 ,H
proving the statement of the lemma.
Theorem 2.2.Let H t
be defined by Equation (1)and suppose the hypotheses of theorem 2.1 hold true. Moreover, we take the curve :n so that
t a vt, M,
, n
t
for some positive constant M and fixed vectors Then the dimension of the spectral projection onto the interval
.
a v
E0,E
,asso-ciated with the selfadjoint operator H t , is equal to one for each t.
Proof: The resolvent i of a self-adjoint opera-tor A at
R A
i is defined by with I denoting
the identity operator on
1A iI
2 n
L .We take H2 H t
2 ,
,1 1
H H t for two distinct values and and
calculate the difference
1
t t2
2
1i R Hi
.R H
1 2 1 22 1 2 2 1
2 2 1, 1, 1
2 2 2, 2, 1
i i i i
i i
i i
R H R H R H H H R H R H V V R H R H V V R H
1195 [6] K. Chadan and P. C. Sabatier. “Inverse Problems in
Quantum Scattering Theory,” Springer-Verlag, New York, 1989.
Here
1
1,
V is given as in Equation (3) with
,
i i
g x g x t replaced with
1
1, 1 1
.g x g x t Also
2 1
1, , 2, ,
V V
2
2,
V
and [7] T. Cubel, B. K. Teo, V. S. Malinoysky, J. R. Guest, A. Reinhard, B. Knuffman, P. R. Berman and G. Raithel, “Coherent Population Transfer of Ground-State Atoms into Rydberg States,” Physical Review A, Vol. 72, 2005, pp. 023405 (1-4).
being defined similarly. It follows from Equation (1) and standard arguments that
2
1 2 ,i i
R H R H t t1
where is a constant uniform in t t1, 2 [M M, ]
depending on pg and g , This implies
that is uniformly continuous on with
respect to the norm topology. Let denote the
spectral projection of a self-adjoint operator B
corre-sponding to the Borel set By functional
calcu-lus, we get
1, 2.
SP B
t
1 i
R H
.
S
[8] H. L. Cycon, R. G. Froese, W. Kirsh and B. Simon. “Schrödinger Operators with Application to Quantum Mechanics and Global Geometry,” Springer-Verlag, New York, 1987.
[9] E. Farhi, J. Goldstone, S. Gutmann and M. Sipser, “Quantum Computation by Adiabatic Evolution,” 2000. arXiv:quant-ph/0001106v1
[10] P. D. Hislop and I. M. Sigal, “Introduction to Spectral Theory with Applications to Schrödinger Operators,” Springer-Verlag, New York, 1996.
E0, E
2
E0, E
1
P H t P H t as t2t1,
[11] T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin and New York, 1984.
in the operator norm. Therefore, by standard arguments
0,
2
0,
dimPE E H t dimPE E H t , [12] J. O. Lee and J. Yin, “A Lower Bound on the Ground State Energy of Dilute Bose Gas,” Journal of Mathe-matical Physics, Vol. 51, No. 5, 2010, pp. 053302 (1-31).
For 2 close enough to . It follows from lemma
2.2 that
t t1
[13] E. L. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, “The Ground State of the Bose Gas,” 2003.
arXiv:math-ph/0204027v2
0,
dimP E E H t 1
t
.[14] H. E. Puthoff, “Ground State of Hydrogen as a Zero- Point-Fluctuation-Determined State,” Physical Review D, Vol. 35, No. 10, 1987, pp. 3266-3269.
doi:10.1103/PhysRevD.35.3266
Remark: We mention that the hypothesis for the curve
t can be relaxed to the condition that
t is as-ymptotic to a straight line.[15] M. Reed and B. Simon, “Methods of Modern Mathe-matical Physics, Vol. IV,” Academic Press, New York, 1975.
3. References
[1] J. H. Arredondo, “Asymptotic Transition Probabilities,” Journal of Mathematical Physics, Vol. 30, No. 10, 1989, pp. 2291-2296. doi:10.1063/1.528558
[16] W. Thirring, “A Course in Mathematical Physics. Vol. III,” Springer-Verlag, New York, 1981.
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[2] J. H. Arredondo, “Adiabatic Approximation on the Im-pact-Parameter Model,” Few-Body Systems, Vol. 10, No. 2, 1991, pp. 59-72. doi:10.1007/BF01352402
[3] H. S. W. Massey, “Collision between Atoms and Mole-cules at Ordinary Temperatures,” Reports on Progress in Physics, Vol. 12, No. 1, 1949, p. 248.
doi:10.1088/0034-4885/12/1/311
[18] P. D. Lax, “Functional Analysis,” John Wiley & Sons Inc., New York, 2002.
[19] M. Reed and B. Simon, “Methods of Modern Mathe-matical Physics, Vol. II,” Academic Press, New York, 1975.
[4] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd and O. Regev, “Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation,” 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS'04), Rome, 17-19 October 2004.
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