A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications
Kira V. Khmelnytskaya Vladislav V. Kravchenko Sergii M. Torba




A representation of the transmutation kernels for the Schrödinger operator
in terms of eigenfunctions and applications
    Kira V. Khmelnytskaya^1, Vladislav V. Kravchenko^2,3, Sergii M.
Torba^3The authors acknowledge the support from CONACYT, Mexico
via the projects 284470 and 222478.
^1Faculty of Engineering, Autonomous University of
Queretaro, 
Cerro de las Campanas s/n, col. Las Campanas Querétaro,
Qro. C.P. 76010 México
^2Regional mathematical center of Southern Federal University, 
Bolshaya Sadovaya, 105/42, Rostov-on-Don, 344006, Russia,
^3Department of Mathematics, Cinvestav, Unidad
Querétaro
Libramiento Norponiente #2000, Fracc. Real de Juriquilla,
 Querétaro, Qro. C.P. 76230 México
khmel@uaq.edu.mx, vkravchenko@math.cinvestav.edu.mx,
storba@math.cinvestav.edu.mx
    December 2017 (Updated Feb 2018 and Dec 2018)
=======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================




The representations of the kernels of the transmutation operator and of its
inverse relating the one-dimensional Schrödinger operator with the second
derivative are obtained in terms of the eigenfunctions of a corresponding
Sturm-Liouville problem. Since both series converge slowly and in general only
in a certain distributional sense we find a way to improve these expansions
and make them convergent uniformly and absolutely by adding and subtracting
corresponding terms. A numerical illustration of the obtained results is
given.





§ INTRODUCTION


Since the first work by J. Delsarte <cit.>, <cit.> the
transmutation operator relating the one-dimensional Schrödinger operator
L:=d^2/dx^2-q(x) to a more elementary operator B:=d^2
/dx^2 has been subject of study  in hundreeds of publications devoted to
spectral theory and inverse problems (see, e.g., <cit.>,
<cit.>, <cit.>, <cit.>,
<cit.>, <cit.>). Recently in <cit.>,
<cit.>, <cit.> several representations of the integral kernel
of the transmutation operator in terms of series expansions in classical
orthogonal polynomials have been obtained equipped with convenient recurrent
formulas for the expansion coefficients. Every such representation leads to a
new functional series representation of the solutions of the Schrödinger
equation which enjoys a remarkable uniformness property. It admits a spectral
parameter independent estimate for the approximation of the solution by
partial sums of the series, which in practice allows one to compute huge
numbers of eigenvalues and eigenfunctions with a controllable accuracy
<cit.>, <cit.>, <cit.>. Similar results were obtained
for perturbed Bessel equations <cit.>, <cit.>.

Up to now no similar representation has been obtained for the integral kernel
of the inverse transmutation operator which is required in numerous
applications, especially when solving initial-boundary value problems for PDEs
with variable coefficients. Moreover, an apparently unanswered  question is to
find an eigenfunction expansion of both the direct and the inverse
transmutation kernels, resembling the well known expansion of the Green
function. Such eigenfunction series expansions additionally to their profound
theoretical value acquire also computational significance due to the
availability of the representations of solutions admitting the spectral
parameter independent estimates and allowing one to compute huge amounts of eigendata.

In the present work we obtain an eigenfunction expansion of the integral
transmutation kernels of both the direct and the inverse transmutation
operators. Quite naturally, since the transmutation operators are related to
pairs of differential operators, the corresponding eigenfunction expansions
contain the eigendata of both differential operators.

The series expansions of both the direct and the inverse transmutation
operators converge slowly and in general only in a certain distributional
sense. We find the way to improve these expansions and make them convergent
uniformly and absolutely by adding and subtracting corresponding terms. We
give a numerical illustration of the obtained results.



§ PRELIMINARIES


Let q be a real valued function belonging to L_2(0,π) and h, H be
two real numbers. Consider the Sturm-Liouville problem

    -y^''+q(x)y=λ y,
    
    y^'(0)-hy(0)=y^'(π)+Hy(π)=0.

It defines two sequences of real numbers, the eigenvalues {λ_n}  _n=0^∞ and the weight numbers (or normalizing
constants) {α_n}  _n=0^∞ such that
λ_n≠λ_m for n≠ m, α_n>0,

    ρ_n:=√(λ_n)=n+ω/π n+k_n/n
    ,    α_n=π/2+K_n/n,    {  k_n}
    _n=0^∞,{  K_n}  _n=0^∞∈ l_2
    .

The weight numbers are defined as follows α_n:=∫_0^π
c^2(ρ_n,x)dx where c(ρ,x) denotes the solution of the Cauchy
problem

    -c^''(ρ,x)+q(x)c(ρ,x)=ρ^2c(ρ,x),
    
    c(ρ,0)=1,     c^'(ρ,0)=h.

The value of the number ω in (<ref>) is
given by the formula

    ω=h+H+1/2∫_0^πq(t) dt.

Consider the Gel'fand-Levitan equation

    G(x,t)+F(x,t)+∫_0^xF(t,s)G(x,s) ds=0,    0≤
    t<x,

where F(x,t) has the form

    F(x,t)=∑_n=0^∞(  cosρ_nxcosρ_nt/α
    _n-cos nxcos nt/α_n^0)

with

    α_n^0=
    π/2,     n>0,
    π,     n=0

and G(x,t) is the kernel of a transmutation operator

    T[u](x):=u(x)+∫_0^xG(x,s)u(s) ds

relating the operator L:=d^2/dx^2-q(x) with the operator
B:=d^2/dx^2 as follows. Let u∈ C^2[0,π] and u^'(0)=0. Then LTu=TBu. Denote v:=Tu. Then v^'(0)-hv(0)=0. In
particular,

    T[cosρ x]=c(ρ,x).



Let H(x,t) denote the kernel of the inverse transmutation operator. That is

    T^-1[  u]  (  x)  =u(  x)
    +∫_0^xH(x,s)u(s) ds.

Recall that the kernel H(x,t) can be obtained from the equality <cit.>,

    H(x,t)-F(x,t)-∫_0^tF(x,s)G(t,s) ds=0,  0≤ t<x.



The integral kernels G and H are continuous functions in 0≤ t≤
x≤π and satisfy the equalities

    G(x,x)=-H(x,x)=h+1/2∫_0^xq(t) dt.





§ SERIES REPRESENTATIONS FOR THE TRANSMUTATION KERNELS



Equations (<ref>) and (<ref>) can be written in
the form

    G(x,t)=-T_x[ F] (x,t)

and

    H(x,t)=T_t[ F] (x,t),

respectively. That is, the kernels G(x,t) and H(x,t) are images of the
function ∓ F(x,t) under the action of the transmutation operator applied
with respect to the variable x and t, respectively.



The kernels G(x,t) and H(x,t) admit
the following representations

    G(x,t)=∑_n=0^∞(  c(  n,x)  cos nt/α_n^0-c(  ρ_n,x)  cosρ_nt/α_n
    )

and

    H(x,t)=∑_n=0^∞(  cosρ_nx c(  ρ
    _n,t)  /α_n-cos nx c(  n,t)  /α
    _n^0)  ,

where the series converge in the following distributional sense. Let the
integral kernels G(x,t) and H(x,t) be extended by zero for t>x. Then for
any f∈ AC[  0,π] the following limit (corresponding to
(<ref>) and (<ref>)) exists uniformly with respect to x,

    lim_N→∞∫_0^πf(t)I_N(x,t)dt      =∫_0^πf(t)(G(x,t)-H(t,x)) dt
         =-∫_0^πf(t)(  F(x,t)+∫_0^xG(x,s)F(s,t) ds)  dt,

where

    I_N(x,t)=∑_n=0^N(  c(  n,x)  cos nt/α_n^0-c(  ρ_n,x)  cosρ_nt/α_n
    )  .

Note that for x≠ t only one of the values G(x,t) or H(t,x) can be
different from zero.



 As can be concluded from results of Section
<ref> and Jordan's theorem <cit.>,
the series in (<ref>) and (<ref>) converge pointwise on [0,π
]×0,π], uniformly on any compact subset of [0,π
)×0,π) and their sums are continuous functions except a jump
discontinuity at x=t=π, the size of the jump being ω, where ω is the parameter from (<ref>).



Consider

    T_x[  cosρ_nxcosρ_nt/α_n-cos nxcos
    nt/α_n^0]  =c(  ρ_n,x)  cosρ_n
    t/α_n-c(  n,x)  cos nt/α_n^0.

Hence, formally,

    G(x,t)=-T_x∑_n=0^∞(  cosρ_nxcosρ_n
    t/α_n-cos nxcos nt/α_n^0)  =∑
    _n=0^∞(  c(  n,x)  cos nt/α_n^0
    -c(  ρ_n,x)  cosρ_nt/α_n)  ,

and similarly for H.

Due to (<ref>), (<ref>) and (<ref>) one has

    c(n,x)=cos nx+∫_0^xG(x,s)cos ns ds,    cosρ_nx=c(ρ
    _n,x)+∫_0^xH(x,s)c(ρ_n,s) ds,

hence

    I_N(x,t):      =∑_n=0^N(  c(n,x)cos nt/α_n^0
    -c(ρ_n,x)cosρ_nt/α_n)  
         =∑_n=0^N(  cos nxcos nt/α_n^0-cosρ_nxcosρ_nt/α_n)  
         +∑_n=0^Ncos nt/α_n^0∫_0^xG(x,s)cos
    ns ds-∑_n=0^Ncosρ_nt/α_n∫_0^x
    H(x,s)cosρ_ns ds
         =:Φ_N(x,t)+I_N,2(x,t)+I_N,4(x,t)

(here we follow notation from the proof of Theorem 1.3.1 <cit.>).

According to <cit.> for any f∈ AC[
0,π] the following limit exists

    lim_N→∞max_0≤ x≤π∫_0^πf(t)Φ
    _N(x,t) dt=0.

Moreover, uniformly with respect to x∈0,π],

    lim_N→∞∫_0^πf(t)I_N,2(x,t)dt      =∫_0
    ^xf(t)G(x,t)dt,
    lim_N→∞∫_0^πf(t)I_N,4(x,t)dt      =-∫_x^πf(t)H(t,x)dt.

Hence, extending G(x,t) and H(x,t) by zero for x<t we obtain that

    lim_N→∞I_N(x,t)=∫_0^π
    f(t)(G(x,t)-H(t,x))dt,

thus establishing (<ref>).



Our prime interest consists in application of the representation (<ref>) for
computing the preimages of functions under the action of the transmutation
operator. From Theorem <ref> we have that for any
f∈ AC[  0,π] the sequence

    f(x)+∑_n=0^N(  cosρ_nx /α_n∫_0
    ^xc(ρ_n,t)f(t) dt-cos nx /α_n^0∫_0
    ^xc(n,t)f(t) dt)  
    
    =f(x)+∫_0^x∑_n=0^N(  cosρ_nx c(ρ_n
    ,t)/α_n-cos nx c(n,t)/α_n^0)  f(t) dt

tends to T^-1[  f]  (  x) uniformly. This gives us
a practical way for computing the preimages of absolutely continuous functions
reducing such computation to a number of definite integrals from (<ref>).



The convergence rate of the series in the representations (<ref>), (<ref>)
and (<ref>) improves when the parameter ω in
(<ref>) equals zero, see Section
<ref> for details. Note that by appropriate choice of the
constant H in (<ref>) the parameter ω can always
be set to zero. Since the kernels G(x,t) and H(x,t) do not depend on H,
a right choice of the constant H can lead to a faster convergence of the
series in (<ref>), (<ref>) and (<ref>). In the next section we show how
the series (<ref>) and (<ref>) can be modified in order to improve the
convergence rate even when ω≠0.




§ A FOURIER SERIES OF A DISCONTINUOUS FUNCTION AND IMPROVEMENT OF
CONVERGENCE


 Both series (<ref>) and (<ref>) converge
rather slowly, and, moreover, for x=t=π they converge to the exact values
G(π,π) and H(π,π)  if only the number ω from (<ref>)
equals zero.

A simple explanation (and an idea how to improve the convergence for an
arbitrary value of ω) can be seen in the proof of Lemma 1.3.4 from
<cit.>. We briefly repeat the formulas for the function F and
later present the proof for the functions G and H. Following
<cit.> let us introduce the following function

    a(x)=∑_n=0^∞(  cosρ_nx/α_n-cos
    nx/α_n^0)  .

The function F can be expressed in the terms of the function a as
follows:

    F(x,t)=1/2(a(x+t)+a(x-t)).

Note that the function F should be defined on the region (x,t)∈0,π]×0,π] to be able to consider equation
(<ref>), hence the function a should be defined for
x∈-π,2π].

The function a(x) can be represented as (see the proof of Lemma 1.3.4
<cit.> and notice that the factor 1/α_n^0=2/π is
missing in the proof)

    a(x)=-2ω x/π^2∑_n=1^∞sin nx/n
    +A_2(x),

where the function A_2(x) is continuous on [-π,2π]. As for the first
sum in (<ref>), we have

    -2ω x/π^2∑_n=1^∞sin nx/n=
    
    -ω|x|(π-|x|)/π^2,     |x|<2π,
    
    0,     x=2π.

One can easily see that whenever ω≠0, there is a jump discontinuity
at x=2π, corresponding to x=t=π for the function F(x,t). Moreover,
the series (<ref>) converges slowly which implies the slow
convergence of the series representing the function F. Same happens to the
functions G(x,t) and H(x,t).

The idea to improve the convergence is to consider the following expression
for the function a(x):

    a(x)=cosρ_0x/α_0-1/π+∑_n=1^∞(
    cosρ_nx/α_n-cos nx/α_n^0+2ω
    x/π^2sin nx/n)  -ω|x|(π-|x|)/π^2
    ,

that is, to subtract the slowly convergent series termwise and to add the
closed expression for the whole infinite sum. Note that as an additional
benefit of such reformulation, the series in (<ref>) is
uniformly convergent on the whole segment [-π,2π], and hence the
function a given by (<ref>) is continuous on [-π,2π].
Based on this idea the following result is obtained.


 The kernels G and H admit the following
representations

    G(x,t)     =∑_n=1^∞(  c( n,x) cos nt/α_n^0-
    c( ρ_n,x) cosρ_nt/α_n - 2ω/π^2
    n(xsin nx cos nt + tsin ntcos nx) ) 
         + c( 0,x)/π- c( ρ_0,x) cosρ_0t/α_0+
    ω/π^2(π x -x^2-t^2),    0≤ t≤ x≤π

and

    H(x,t)     =∑_n=1^∞( cosρ_nx c(ρ_n,t)/α_n- cos nx c(n,t)/α_n^0 + 2ω/π
    ^2n(xsin nx cos nt + tsin ntcos nx)) 
         +cosρ_0x c(ρ_0,t)/α_0- c(0,t)/π-ω/π^2(π x -x^2-t^2),     0≤ t≤
    x≤π

where the series converge uniformly and absolutely.



Let us show that the series for F(x,t) and H(x,t) given by (<ref>) and
(<ref>) differ from each other by an absolutely and uniformly convergent
series of continuous functions. Consider

    Δ_n     =(  cosρ_nx c(ρ_n,t)/α_n
    -cos nx c(n,t)/α_n^0)  -(  cosρ
    _nx cosρ_nt/α_n-cos nx cos nt/α_n^0
    )  
         =cosρ_nx/α_n(c(ρ_n,t)-cosρ
    _nt)-cos nx/α_n^0(c(n,t)-cos nt).



The function c(ρ,t) satisfies the following asymptotic relation
<cit.>

    c(ρ,t)=cosρ t+q_1(t)sinρ t/ρ+∫_0^t
    q(s)sinρ(t-2s)/2ρds+O(  1/ρ^2)
    ,    ρ→∞,

where q_1(t)=h+1/2∫_0^tq(s) ds. Hence

    Δ_n     =q_1(t)(  cosρ_nx sinρ_nt/α
    _nρ_n-cos nx sin nt/nα_n^0)  
         +∫_0^tq(t)(  cosρ_nx sinρ_n
    (t-2s)/2α_nρ_n-cos nx sin n(t-2s)/2nα_n^0
    )  ds+O(  1/n^2)  ,     n→∞.

We obtain for the first term that

    Δ_n^1     :=cosρ_nx sinρ_nt/α_nρ_n
    -cos nx sin nt/nα_n^0
         =cosρ_nx sinρ_nt-cos nx sin nt/α_nρ_n
    +(  1/α_nρ_n-1/nα_n^0)  cos
    nx sin nt
         =sin(ρ_n-n)(x+t)/2cos(ρ_n+n)(x+t)/2
    -sin(ρ_n-n)(x-t)/2cos(ρ_n+n)(x-t)/2/α
    _nρ_n-α_nρ_n-nα_n^0/nα_nα
    _n^0ρ_ncos nx sin nt.

Taking into account (<ref>) one can see that
ρ_n-n=O(1/n) and α_nρ_n-nα_n^0=O(1), hence
Δ_n^1=O(1/n^2), n→∞. Similarly for the second
term in (<ref>).

Now, combining (<ref>) with (<ref>) we obtain that

    F(x,t)       = ∑_n=1^∞( cosρ_n x  cosρ_n
    t/α_n - cos nx cos nt/α_n^0+ω
    (x+t)/π^2sin n(x+t)/n +ω(x-t)/π^2sin
    n(x-t)/n) 
         +cosρ_0 x  cosρ_0 t/α_0 - 1/π-
    ω|x+t|(π-|x+t|)/2π^2-ω|x-t|(π-|x-t|)/2π
    ^2
         = ∑_n=1^∞( cosρ_n x  cosρ_nt/α_n - cos nx cos nt/α_n^0+2ω/π^2
    n( xsin nx cos nt + tsin nt cos nx)) 
         +cosρ_0 x  cosρ_0 t/α_0 - 1/π -
    ω/π^2(πmax{x,t} -x^2-t^2),

where the series converges absolutely and uniformly and the equality holds
whenever x<π or t<π, i.e., min{x,t}<π.

Since

    H(x,t) = F(x,t) - ∑_n=0^∞Δ_n,

where the series ∑_n=0^∞Δ_n converges absolutely and
uniformly, we immediately obtain (<ref>) from (<ref>)
and (<ref>) for all x,t which satisfy min{x,t}<π. Note
also that the kernel H is a continuous function on 0≤ t≤ x≤π, and
the right hand side of (<ref>) is also a continuous function on the
same region. Hence the equality holds for x=t=π as well.

The proof of (<ref>) is completely similar.




§ AN EXPLICIT EXAMPLE


Let us consider an exactly solvable example which reveals some important
features of the representations (<ref>) and (<ref>). Let q≡1 and
h=0. Then

    c(ρ,x) =T[ cosρ x] =cosμ x

where μ:=√(ρ^2-1). In particular,

    c( 0,x) =T[ 1] =cosh x.



The integral kernel G for this example is given by, see <cit.> and <cit.>

    G(x,t) =
    x I_1(√(x^2-t^2))/√(x^2-t^2),     t<x,
    x/2,     t=x,

where I_1 is the modified Bessel function of the first kind.

Consider first the corresponding Sturm-Liouville problem with H=0. Then
c(  ρ_n,x)  =cos√(ρ_n^2-1)x, ρ_n
=√(n^2+1), α_n=α_n^0. Hence (<ref>) gives

    G_1(x,t)     =∑_n=0^∞1/α_n^0(
    cos√(n^2-1)x cos nt-cos nx cos√(n^2+1)t) 
        = cosh x - cos t/π+ ∑_n=1^∞2/π(
    cos√(n^2-1)x cos nt-cos nx cos√(n^2+1)t) .

On the other hand choosing the constant H in (<ref>)
equal to -π/2 we obtain that ω=0. Let us construct the
corresponding series representation for G(x,t). Thus, q≡1, h=0 and
H=-π/2. Then c( ρ_n,x) =cosμ_nx, where μ_n=√(ρ
_n^2-1) are solutions of the characteristic equation

    μsinμπ+π/2cosμπ=0

and α_n=π/2+sin2μ_nπ/4μ_n. Notice that
the problem possesses one negative eigenvalue ρ_0^2≈-1.468. The
corresponding series representation for G(x,t) takes the form

    G_2(x,t)       =cosh x/π-cosμ_0x cosρ_0t/α_0
         +∑_n=1^∞(  2cos√(n^2-1)x cos nt/π-cosμ_nx cosρ_nt/α_n)  .



Additionally, for the case h=H=0 we consider the representation given by
(<ref>),

    G_3(x,t)      =∑_n=1^∞2/π(  cos√(n^2
    -1)x cos nt-cos nx cos√(n^2+1)t-xsin nx cos nt+tsin
    nt cos nx/2n)  
         +cosh x-cos t/π+ω/π^2(π
    x-x^2-t^2).



We stress that the representations (<ref>), (<ref>) and (<ref>)
correspond to the same kernel G(x,t).

We computed approximate integral kernels by truncating the series in
(<ref>), (<ref>) and (<ref>) and compared with the exact integral
kernel at x=π. On Figure <ref> we present the absolute
value of the differences, 10 and 100 terms of the series were used. In
accordance with Remark <ref> the difference between
partial sums of the series (<ref>) and the exact value at t=π remains
close to π/2, while both series (<ref>) and (<ref>) converge
uniformly and faster. All computations were realized in Matlab 2017a. Notice
that opposite to (<ref>) and (<ref>), the approximation obtained from
(<ref>) requires μ_n to be computed numerically from
(<ref>). This was done by converting the function μsinμπ+π/2cosμπ into a spline and finding its zeros with the aid
of the Matlab routine .



Now let us compare the convergence rate of the series applying all three
representations for computing T[1]. Of course, in our example,
T[1]=c(  0,x)  =cosh x. Thus, using (<ref>), (<ref>) and
(<ref>) we construct three approximations of the function cosh x,

    cosh x     ≈1+xcosh x-sin x/π+∑_n=1^N2/π(  cos√(n^2-1)x sin nx/n-cos nx sin√(n^2+1)x/√(n^2+1)) ,
    cosh x     ≈1+xcosh x/π-cosμ_0x sinρ_0
    x/α_0ρ_0 +∑_n=1^N(  2cos√(n^2-1)x sin
    nx/π n-cosμ_nx sinρ_nx/α_nρ_n)
    ,

and

    cosh x     ≈1+xcosh x-sin x/π +ω x^2/π -
    4ω x^3/3π^2
         +∑_n=1^N2/π(  cos√(n^2-1)x sin
    nx/n-cos nx sin√(n^2+1)x/√(n^2+1) + xcos
    2nx/2n^2 - sin2nx/4n^3) ,

respectively.

For N=10 the absolute error of the first approximation was 9.5·10^-2, of the second 5.5·10^-4 and of the third 3.3·10^-4. For
N=100 the absolute error of the first approximation was 9.9·10^-3,
of the second 4.0·10^-6 and that of the third 3.8·10^-7.
Finally, for N=1000 the absolute error of the first approximation was
1.0·10^-3, of the second 9.8·10^-9 and that of the third
3.9·10^-10. All three series converge slowly. However the convergence
rate greatly improves either considering the second representation
(<ref>) corresponding to ω=0 or the third representation
(<ref>). On Figure <ref> we present the absolute errors of
representations (<ref>), (<ref>) and (<ref>) as
functions of N.



99                                                                                               
=-1pt


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