No Degeneracy of the Ground State for the Impact Parameter Model

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 corresponding 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


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 corresponding Hamiltonian was needed.
In this work we prove non degeneracy of the ground state for the Hamiltonian   defined as an operator in the Hilbert space  of all complex valued Lebesgue measurable square integrable functions on , with domain , the Sobolev space of order two [18]. is the Laplace operator [11]. are positive constants.Also, for , we will take the potentials of rank one: with 1 2 , g g fixed elements in

 
2 n L  .Here ( , )   denotes the scalar product in , antilinear on the factor on the left.Moreover, being a continuous function on with values in satisfying We denote by  the Fourier transform [19], as an unitary operator in where the limit is taken in the -norm.

Main Theorem
From Weyl's theorem [16], one knows that for each   in Equation (1) Here are the ground state eigenvalues associated to respectively.Then the following statements are valid: 1) The eigenvalue , corresponding to the ground state for the operator and the eigenvalue corresponding to the ground state for the operator where we denote .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 i g in large enough for the hypotheses of the theorem to hold.We will prove in this manuscript that under the hypotheses of theorem 2.1, for the ground state of be the ground state eigenvalue of the time dependent operator given by Equation (1).We define be the ground state eigenvalue of the time dependent operator   H t given by Equation (1).Then, the matrix equation , where     2 : . The Plancherel theorem im- Taking inner roducts in (9) with p g and 1, ĝ  for we get This system of equations is represented in matrix form precisely by Equation (8), where , From Theorem 2.1 we deduce the existence of a nontrivial solution to Equation (8).Now we fix For every t let us consider the function, 0. E    and observe that for 0 (2)  , The last inequality being true because of the remark following Equation (5).Also, we have used the Schwarz inequality and the Fourier transform property We take so that, 1 : , The main result will be proved by showing that the dimension of the eigenspace associated to the ground sate remains constant over time.
From Theorem 2.1, we know that there exists a nontrivial solution to system (8).Thus Accordingly, for some constant Moreover, for  .
is not null.In fact, for this value of t, the following terms simplify where we use equation ( 5), the hypothesis Equations ( 11)-( 16 Substitution of these equalities in Equation (9) gives, Here,

 
H t be defined by Equation (1) and suppose the hypotheses of theorem 2.1 hold true.Moreover, we take the curve so that : for some positive constant M and fixed vectors Then the dimension of the spectral projection onto the interval Copyright © 2011 SciRes.AM V  is given as in Equation ( 3) with

2 L
vector up to a multiplicative constant, and from the Plancherel theorem, also the eigenspace associated to the ground state for of the lemma. Theorem 2.2.Let