Adiabatic Approximation and Necessary Conditions for the Existence of Bound States

Let

H (\vec{ρ}, \vec{r})

represent the (

M - 1

)-body Hamiltonian that results from fixing the center of mass

\vec{R}

of an

M

-body system, where

\vec{r}

is the relative coordinate of a particular pair of particles, and

\vec{ρ}

represents the

M - 2

remaining internal coordinates. With

E_{thr}

the lowest continuum threshold associated with

H

, the number of bound states of the system is the number of negative eigenvalues of

H - E_{thr}

. A simple lower bound on

H

was derived by Hahn and Spruch through the use of an adiabatic-like approximation in which the (

M - 1

)-body problem is attacked by considering first an (

M - 2

)-body problem and then a one-body problem. With

E_{a 0} (r)

the lowest energy of the system for

\vec{R}

and

\vec{r}

fixed, one finds

H (\vec{ρ}, \vec{r}) - E_{thr} >~ \hat{1} (\vec{ρ}) H^{(1)} (\vec{r}), where H^{(1)} \equiv T_{\vec{r}} + E_{a 0} (r) - E_{thr} \equiv T_{\vec{r}} + V^{(1)} (r) .

H^{(1)}

is a one-body Hamiltonian,

T_{\vec{r}}

is the kinetic energy operator for the relative motion of the particular pair, and

\hat{1} (\vec{ρ})

is the unit operator in the space of quadratically integrable functions of

\vec{ρ}

. The adiabatic potential

E_{a 0} (\dot{r})

has been tabulated for a number of systems, primarily atomic and molecular. A necessary condition for the existence of a bound state of

H

is that the lowest eigenvalue of

H^{(1)}

be negative.

This publication has 17 references indexed in Scilit: