3.6 Newton and Berggren Completeness Relations Generated by Nonlocal. . .
111
by u(k, r) are still valid in the nonlocal case (see Sect. 2.1). Moreover, as Eq. (2.2)
is local for r > R, one can define the Jost function associated to u(k, r) using
r ≥ R as in Sect. 2.6.4. The definition of Jost functions in the general nonlocal
case is indeed not simple, as Wronskians are not independent of r therein. One can
also note that the demonstration followed in Sect. 2.6.8 to obtain the expressions for
a narrow width in Eqs. (2.197), (2.199), and (2.201) can be utilized as well in the
nonlocal case.
The existence and unicity of u(k, r) are not guaranteed in the nonlocal case.
Indeed, Eqs. (2.9) and (2.139) for nonlocal potentials become Fredholm equations
of the second kind, so that the mathematical properties of u(k, r) rely on Fredholm
theory [38]. In particular, a u(k, r) function verifying the boundary condition of
Eq. (2.6) does not exist if Eq. (2.9) possesses a nonvanishing solution whose function
and derivative vanish in r = r 0 [38]. Similarly, a u ± (k, r) function satisfying the
boundary condition of Eq. (2.7) in r = R cannot be constructed if Eq. (2.9) provides
a nonzero solution whose function and derivative vanish in r = R [38]. These
situations arise when the Fredholm operator acting on u(k, r) functions is noninvertible for certain energy values [38].
Moreover, the number of bound states generated by nonlocal potentials is not
necessarily equal to the number of zeros of u(k = 0, r) in the case of local potentials
(see Sect. 2.5). In fact, using convexity arguments on u(k, r) for r > R in the bound
energy region, one can show that the number of bound states is equal or smaller
than the number of zeros of u(k = 0, r) [10]. Nevertheless, the nonlocal integral
operators entering Eqs. (2.9) and (2.139) are diagonally dominant in practice, and
hence invertible for every k value in problems of physical interest. Therefore,
Eqs. (2.9) and 2.139) in practical applications always bear a unique solution, and the
relation between bound states and the number of zeros of u(k = 0, r) is fulfilled,
as in the local case. As the nonlocal integral operators in Eqs. (2.9) and (2.139)
are invertible in practice and analytic in k, hence u(k, r) bears the same analytic
properties as in the local case.
In fact, the main problem encountered with nonlocal Schrödinger equation is
related to the Newton completeness relation of Sect. 3.2.1. This arises because, in
general, the asymptotic solutions for |k| → +∞, devised in Sect. 2.6.2, no longer
hold in the nonlocal potential case. Real values of k pose no problem in this context.
One may show with integration by parts that the nonlocal term of Eq. (3.80) is
O(k −2 ) on the real k-axis using an analogous method as that presented in Ref. [10].
Nevertheless, the asymptotic expansion of u ± (k, r) and u(k, r) (see Sect. 2.6.2)
strongly diverges when (k) → +∞, by the way of additional terms diverging
exponentially with (k), induced by the nonlocal kernel (see Sect. 2.6.2).
Consequently, several integrals entering the demonstration of Newton completeness relation diverge in general when K → +∞, so that this procedure fails
in the non-local case (see Sect. 3.2.1 and Exercise IV). Therefore, the Newton
completeness relation involving non-local potentials must be demonstrated either
with an integral based on the Stieltjes-Lebesgue measure (see Sect. 3.2), or by using
a set of discrete eigenstates defined in a box, whose radius goes to infinity to recover
the continuum [10].
111
by u(k, r) are still valid in the nonlocal case (see Sect. 2.1). Moreover, as Eq. (2.2)
is local for r > R, one can define the Jost function associated to u(k, r) using
r ≥ R as in Sect. 2.6.4. The definition of Jost functions in the general nonlocal
case is indeed not simple, as Wronskians are not independent of r therein. One can
also note that the demonstration followed in Sect. 2.6.8 to obtain the expressions for
a narrow width in Eqs. (2.197), (2.199), and (2.201) can be utilized as well in the
nonlocal case.
The existence and unicity of u(k, r) are not guaranteed in the nonlocal case.
Indeed, Eqs. (2.9) and (2.139) for nonlocal potentials become Fredholm equations
of the second kind, so that the mathematical properties of u(k, r) rely on Fredholm
theory [38]. In particular, a u(k, r) function verifying the boundary condition of
Eq. (2.6) does not exist if Eq. (2.9) possesses a nonvanishing solution whose function
and derivative vanish in r = r 0 [38]. Similarly, a u ± (k, r) function satisfying the
boundary condition of Eq. (2.7) in r = R cannot be constructed if Eq. (2.9) provides
a nonzero solution whose function and derivative vanish in r = R [38]. These
situations arise when the Fredholm operator acting on u(k, r) functions is noninvertible for certain energy values [38].
Moreover, the number of bound states generated by nonlocal potentials is not
necessarily equal to the number of zeros of u(k = 0, r) in the case of local potentials
(see Sect. 2.5). In fact, using convexity arguments on u(k, r) for r > R in the bound
energy region, one can show that the number of bound states is equal or smaller
than the number of zeros of u(k = 0, r) [10]. Nevertheless, the nonlocal integral
operators entering Eqs. (2.9) and (2.139) are diagonally dominant in practice, and
hence invertible for every k value in problems of physical interest. Therefore,
Eqs. (2.9) and 2.139) in practical applications always bear a unique solution, and the
relation between bound states and the number of zeros of u(k = 0, r) is fulfilled,
as in the local case. As the nonlocal integral operators in Eqs. (2.9) and (2.139)
are invertible in practice and analytic in k, hence u(k, r) bears the same analytic
properties as in the local case.
In fact, the main problem encountered with nonlocal Schrödinger equation is
related to the Newton completeness relation of Sect. 3.2.1. This arises because, in
general, the asymptotic solutions for |k| → +∞, devised in Sect. 2.6.2, no longer
hold in the nonlocal potential case. Real values of k pose no problem in this context.
One may show with integration by parts that the nonlocal term of Eq. (3.80) is
O(k −2 ) on the real k-axis using an analogous method as that presented in Ref. [10].
Nevertheless, the asymptotic expansion of u ± (k, r) and u(k, r) (see Sect. 2.6.2)
strongly diverges when (k) → +∞, by the way of additional terms diverging
exponentially with (k), induced by the nonlocal kernel (see Sect. 2.6.2).
Consequently, several integrals entering the demonstration of Newton completeness relation diverge in general when K → +∞, so that this procedure fails
in the non-local case (see Sect. 3.2.1 and Exercise IV). Therefore, the Newton
completeness relation involving non-local potentials must be demonstrated either
with an integral based on the Stieltjes-Lebesgue measure (see Sect. 3.2), or by using
a set of discrete eigenstates defined in a box, whose radius goes to infinity to recover
the continuum [10].