1. Introduction
By way of contextualizing, we begin with a summary of the cited article [1] , highlighting the results related to the amplitude
and the bounded potential on the half line. The last Theorem (10.2) stands out, where the estimates of the amplitude
appear with respect to the norm of the bounded potential on the half-line.
They start considering Schrödinger operators
(1.1)
in
for
or
and real-valued locally integrable q.
They are interested in cases for
, that is
Case 2: q is “essentially” bounded from below in the sense that
(1.4)
Case 3: (1.4) fails but (1.1) is limit point at
, that is, for each
,
(1.5)
has a unique solution, up to a multiplicative constant, which is
at
.
Case 4: (1.1) is limit circle at infinity; that is, every solution of (1.5) is
at infinity if
. We then pick a boundary condition by picking a nonzero solution
of (1.5) for
. Other functions u satisfying the associated boundary condition at infinity then are supposed to satisfy
(1.6)
1.1. The Function Weyl-Titchmarscht
, is defined for
as follows. Fix
. Let
be a nonzero solution of (1.5) which satisfies the boundary condition at b. That is, in Cases 2 and 3:
and
. (1.7)
In Case 4, it satisfies (1.6). And, more generally
. (1.8)
satisfies the Riccati equation (with
)
. (1.9)
is an analytic function of z for
, and
Case 2: For some
, m has an analytic continuation to
with m real on
.
Case 3: In general, m cannot be continued beyond
(there exist q's where m has a dense set of polar singularities on
).
Case 4: m is meromorphic in
with a discrete set of poles (and zeros) on
with limit points at both
and
.
Moreover, if
then
; so m satisfies a Herglotz representation theorem,
(1.10)
where
is a positive measure called the spectral measure, which satisfies
(1.11)
(1.12)
And it was found
1.2. Existence of Function Amplitude
Previous results
Theorem 1.2 ( [1] , Theorem 2.1). Let
. Then, there exists a function
on
so that
is continuous and satisfies (1.16)
Theorem 1. There exists a function
for all
so that
, for all
and
(1.15) (1)
as
with
.
if
and for all
,
. Moreover,
is continuous and
(1.16)
One of your purposes here is to prove this result if one only assumes (1.3) (i.e. in Cases 3 and 4).
Previous results
Theorem 2. ( [1] , Theorem 2.1) Let
. Then, there exists a function
on
so that
is continuous and satisfies (1.16) such that
,
. (1.17)
Theorem 1.1 in all cases follows from Theorem 1.2 and the following result which we will prove in Section 3.
Theorem 3. Let
be potentials defined on
with
for
. Suppose that
on
. Then, in the region
,
, we have that
, (1.18)
where
depends only on
, and
, where
is any number so that
,
.
1.3. The Connection between the Spectral Measure dρ and the A-Amplitude
Your basic formula says that
. (1.21)
gives nonzero weight to
, they interpret
Consistent with the fact that
defined on
extends to an entire function of
.
1.4. A Satisfies the Simple Differential Equation in the Distributional Sense
(1.26)
This is prove in [1] for
(and some other qs) and so holds in the generality of this paper since Theorem 1.3 implies
f or
is only a function of
for
.
Moreover, by (1.16), they have
, (1.27)
uniformly in x on compact subsets of the real line, so by the uniqueness theorem for solutions of (1.26), A on
determines q on
.
1.5. The Riccati Equation and the Atkinson Method and the Exponential Bounds for m
As explained in the introduction, the Riccati equation and a priori control on
allow one to obtain exponentially small estimates on
(Theorem 1.5).
Proposition 4. (Proposition 2.1) Let
be two absolutely continuous functions on
so that for some
,
,
then
.
As an immediate corollary, they have the following (this implies Theorem 1.3).
Theorem 5. (Theorem 2.2) Let
be functions defined for
and
some region of
. Suppose that for each
in K,
is absolutely continuous in x and satisfies (N.B.: q is the same for
and
),
.
Suppose C is such that for each
and
,
, (2.2)
then
. (2.3)
They mention Theorem 2.2 places importance on a priori bounds of the form (2.2). Fortunately, by modifying ideas of Atkinson, we can obtain estimates of this form as long as
is bounded away from zero.
As a final result
Theorem 6. (Theorem 4.1) If
and
, then for all a:
. (4.1)
And they get
Corollary 7. (Corollary 4.8) Fix
,
, and
(let’s remember that:
(1.2),
, where
is shorthand for the Dirichlet boundary condition
) Fix
. Then, there exist positive constants C and
so that for all complex
with
,
1.6. The Bounds for the A Amplitude for the Potential q in the Half-Line
So far, it has been assumed that the potential
or
for all
and
.
Now, they assume examples with constant or bounded potentials defined on half-line
with
, and with
is shorthand for the Dirichlet boundary condition
.
See [1] : 10. Examples, I: Constant q.
Your claim
Theorem 8. (Theorem 10.1) If
and
,
, then if
,
(10.1)
where
is the Bessel function of order one (see, e.g. ( [1] , Chapter 9)); if
(10.2)
with
the corresponding modified Bessel function of order one (see, e.g. ( [1] , Chapter 9)). Since [1] , p. 375
(9.27)
This example is especially important because of a monotonicity property.
Theorem 9. (Theorem 10.2) Let
on
with
then
, on
. In particular, for any q satisfying
, they have that
(10.5)
where
(10.6)
In particular, (9.27) implies
(10.7)
and if q is bounded,
(10.8) (2)
The article is divided into the following sections.
In Section 2: Background. The results obtained in [2] and [3] are mentioned, which conclude with the existence of the Inverse Transformation Operator
, which transforms the solutions of an initial Sturm-Liouville equation into the solutions of a second Sturm-Liouville equation where the potential transformed by
is unique see [4] . Later, the Reduced Radial Schröndiger equation is considered where, the singular Bessel potential is the sum of the regular potential and a term with a singularity of quadratic order of the RRSE and the Inverse Transformation Operator
is applied to RRSE one obtains a Sturm-Liouville equation where, the potential
obtained is only the regular potential of the RRSE and the singular term of quadratic order does not appear, instead three additional terms. This is the bounded potential proposed on half-line as our mentioned example.
In Section 3: The bounded potential
on the half-line. This section is the main one of the article, because it is proved that the three additional terms to the potential obtained
are bounded on half-line: Theorem 25. The Proof is done through several consecutive steps: Lemmas 17, 18, 19, 21, 23, 24; in which the bounding of each summand of the proposed potential is proven. Highlighting, the bounding achieved by the Jost function through the magnitudes of the Eigen values and the regular potential [5] .
Finally, in Section 4: Estimates for the
amplitude for a bounded potential on half-line. We quote the corresponding Theorem 10.3 of [3] and the proposed potential is exhibited.
Other articles, where the A amplitude is mentioned as a function of the phase
and estimates are established are: [6] [7] [8] .
2. Background
Following what we name the formulation of Marchenko [2] in [3] , we obtained the following results.
Consider two problems with symmetrical boundary value problems and defined by for
through:
(3)
(4)
where
,
are different real numbers,
(5)
represents the same family of eigenvalues for both problems,
are continuous real valued functions. Their uniqueness is determined through their respective spectral distribution function
. The aim of the paper is to relate both previous problems in the following way. We will assume the uniqueness of the first problem and determine the uniqueness of the second problem by linking: both spectral distribution functions
, both boundary conditions
(6)
and both potential
. The stated theorem was: [2] , Section 4, pp. 481-483.
Theorem 10. (Theorem 1) Let
be a spectral distribution function of the boundary value problem 2 with
and for
(7)
the corresponding solution of the equation:
(8)
respectively. We will assume the following hypothesis:
(9)
and
(10)
where kernel
was defined in Lemma 1. Then
is a spectral distribution function of the boundary value problem
(11)
On the other hand, following a series of problems proposed by V. Marchenko 4, that we will name Marchenko’s formulation, and relating it to a generalized version of Theorem 1 given in 3. The main theorem was proved: [2] .
Theorem 11. (Theorem 1) Let’s consider two Sturm-Liouville equations
(12)
continuous only in the interior points of
. Consider in particular, the following pair of boundary value problems of Sturm-Liouville on the Half Line
(13)
and
(14)
Let
are continuous on
. If
is a fixed solution of the first equation
for
and let
an arbitrary solution of
for
, then
and (15)
(16)
Suppose
(17)
If
(18)
is solution of
for
where the wronskian
(19)
. Then,
satisfies the equation
(20)
where
(21)
According to (16), we define the corresponding Inverse Transformation Operator as
(22)
In [3] , Section 3: The Two Examples: Reducced Radial Schrödinger Equation and Schrödinger Equation on the Half-Line, pp. 492-501, we apply Inverse Transformation Operator
previous to the Reduced Radial Schrödinger Equation (RRSE)
(23)
where
the partial wave of angular momentum l and wave number k. (whose main characteristic is the addition a singular term of quadratic order (named Bessel Singular Potential) to a regular potential
if
(24)
If
(25)
finally one gets
, (119) (26)
we have obtained the uniqueness of the potential which is regular when
and, bounded with exponential decrease fast enough when
. See [3] , Section 3, Formula 119, p. 501.
We will use the following estimation to
, that is:
(170) - (171) (27)
(172) (28)
See Formulas 170 - 172, p. 522, Appendix of 3.
3. The Bounded Potential q(x) on the Half-Line
3.1. Preliminaries
We start with the following preliminary results
Theorem 12. Of 26
(29)
is continuous on
,
, l even
Proof. According to Theorem 1 [3] , Section 2, pp. 485-493: (11)
where
and
are continuous on
. And (15) and (16)
and
If we define to
as (18)
then one got 20
Now, according to the proof of Theorem 1 after laborious calculus we get the successive equations: (49), (50) and (51), see pp. 491-492 of [3] .
(49)
that is
(50)
then if (21)
(51)
must be continuous by hypothesis: see Theorem 1 [3] , Section 2, pp. 485-493, we got (20)
(30)
In the case of Reduced Radial Schrödinger Equation (RRSE) agree to (25)
then the uniqueness and continuity of
is obtained (26)
. (31)
The Potential q Is “Essentially” Bounded
Definition 13.
(32)
If
named q integrated.
Definition 14
(33)
If
named q essentially bounded.
Definition 15. The potential q is “essentially” bounded if
(1.4) (34)
See [1] , Introduction, Case 2, Formula 1.4 and we get the
Lemma 16.
(35)
Proof. Let
, using (34) let
, since
and
then
, and
,
then
and therefore
. ■
3.2. Existence of a Bounded Potential Defined on Half-Line
The existence and uniqueness of the potential
(31) was established through the Inverse Transformation Operator
. See [3] , Section 3, Formula 119, p. 501.
The main result of the article: the bounded potential
(31) obtained as the image of the Inverse Transformation Operator
which is composed of three fundamental addends
1)
, 2)
and 3)
.
The fundamental result consists of the uniform bounded of three addends separately
1)
, (36)
2)
, (37)
3)
, (38)
and conclude that the potential obtained
given in (31) is such that
.
Let’s observe that initial potential
of named Problem 1 of Theorem 1: ( [3] , Section 3, Formula 54, p. 492) is
for the Reduced Radial Schrödinger Equation (RRSE) see (25) where
satisfy (24):
implies
(39)
then
(40)
We start the first addend of (38).
3.3. The First Addend 1)
Lemma 17. Let
, for each fixed
, let
(39), l even and (31)
(41)
then
1)
. (42)
Proof. For
fixed and (26):
(43)
For the uniqueness of
(44)
and in this case it is fulfilled:
(15) then
, by (31) implies
(45)
that is
(46)
We have demonstrated the boundedness of each of the above addends
(47)
of each of the previous summands of the hypothesis
.
The summands:
and
are actually integrable, namely
since
by (39). And
(48)
Then
(49)
Therefore, from (34)
(50)
Now, let’s estimate term:
, since
: (15), (25) then
(51)
then
(52)
and (50) then
(53)
Then by (51)
(54)
Therefore, from (45), (46), (50) and (54), we conclude
1)
. ■
Lemma 18. Let x such that
, l even, fixed
, of (24)
and of (44)
1)
,
then
1)
. (55)
Proof. According to (26), the term:
, for
, and by (44) implies
(56)
then
. Since
is continuous on
, (15), let
, in particular
is continuous on
, since
, for
, there exists
such that
. In particular,
is continuous, since
is compact then
is bounded on
, that is, there exists
such that
for all
. Then
(57)
So, then it is possible that the function
is discontinuous only on the interval
, but its Lebesgue measure:
, We conclude that
is bounded except, in the interval
that has zero Lebesgue measure. Therefore
(58)
Now, let’s estimate the term:
and consider the multiplication:
, for
, since (24)
then
(59)
And let
the term
is continuous on
and therefore boun-ded on
and, similarly to
then
and
is bounded except in the interval
which has zero Lebesgue measure. Then
(60)
From (58), (59) and (60), we conclude that
Therefore, of (42) and (55), we conclude
(61)
■
3.4. Second Addend 2)
Lemma 19. For each fixed
f, the term
(62)
Proof.
for all
.
Now, one obtains the following estimates: the term
, be independent of the x-coordinate.
3.5. Third Addend 3)
Remark 20. Two cases: First case:
and Second case:
.
Lemma 21. First case: let
,
and
(63)
then
(64)
Proof. According to the Estimate (27)
for l even and
. Which is given in terms of the Jost functions
,
and the pair of eigenvalues
. Since [5] , Formula 1.4.5, p. 12.
(65)
then
(66)
(67)
(68)
Now, for
, we have for
,
,
(69)
entonces
, that is
(70)
then (68) and (70) implies
(71)
Now, using the hypothesis (63)
(72)
Since (67):
That is
(73)
Now, coming back to 27
using (72) and (73)
moreover, since for
,
, then
That is
So,
is bounded uniformely respect to the x-coordinate on the interval
.
Then
And we get (64)
■
Remark 22. Since:
Therefore
(74)
Lemma 23. Second case:
, l even, fixed
then
(75)
Proof. For l even, fixed
, according to (28)
then
. ■
And we conclude with the following
Lemma 24. Let
,
,
,
,
then
(76)
Finally, summarizing the conclusions of the previous lemmas: 17, 18, 19, 21, 23 and 24, we obtain the main theorem.
Theorem 25. Let
,
, l even,
and
Then
(77)
where the potential
is given by (31).
We finish by displaying the bounds for the amplitude A, in terms of the potential
established in [1] .
4. Estimates for Amplitude
of a Bounded Potential on Half-Line
We begin by citing the alluded theorem of [1] , see Introduction: the bounds for the A amplitude for the potential q on half-line.
Theorem 26. (Theorem 10.3) Let
and
). Suppose
. Then
with a converget integral and no error term.
The example on the half-line displayed is above (31)
And
according to (61), (62) and (64).
Acknowledgements
Thank the referee for the comments and suggestions that helped the final presentation of the article. To Nancy Ho, thank you very much for your patience and professional work. I am also grateful to Ibrahim Serroukh for his comments and talks on the importance of phase measurability in the Inverse Problems in Imaging.
Dedication
The dedication of the article to my deceased sisters: Alicia Blancarte and Rosa Blancarte.