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Introduction Reflectionless potentials Main results

Inverse scattering for reflectionless Schrödingeroperators and generalized KdV solitons

Rostyslav Hryniv1 Yaroslav Mykytyuk2

1 Ukrainian Catholic University, Lviv, Ukraine

University of Rzeszów, Rzeszów, Poland

2 Lviv Franko National University, Lviv, Ukraine

Operator theory and Krein spaces

Vienna, 22 December 2019


Solitary waves and Korteweg–de Vries equation

In August 1834, JOHN SCOTT RUSSEL, a Scottish civil engineer, navalarchitect and shipbuilder observed an unusual solitary wave in achannel that kept its form and velocity for a long time

He experimentally established some intriguing properties of these“waves of translation” (nowadays known as

:::::::Russel

:::::::solitary

:::::::waves)

In 1870-ies, Lord RAYLEIGH suggested theoretical background for thisphenomenon and in 1877 BOUSSINESQ derived the equation for thewave profile φ:

∂tφ+ ∂3xφ− 6φ∂xφ = 0;

it was rediscovered in 1895 by KORTEWEG and DE VRIES and isnowadays known as the Korteweg–de Vries (KdV) equation

The KdV equation is a nonlinear dispersive PDE possessing infinitelymany first integrals and many interesting properties, n-soliton solutionsbeing one of them


Two-soliton solution


KdV via IST

The interest in KdV was essentially revived after GARDNER, GREENE,KRUSKAL and MIURA found in 1967 that KdV can be solved by theinverse scattering transform (IST) technique:

with a solution φ(x , t) to KdV, associatea family St of Schrödinger operators on the line

St := − d2

dx2 + φ(·, t)

then the scattering data of St

SD(t) := (r(·, t), {−κ2j (t)}nj=1, {αj(t)}nj=1)

evolve along a 1st order linear flow:

r(k , t) = r(k ,0)e−8ik3t , κj(t) ≡ κj(0), αj(t) = αj(0)e4κ3j t

the initial value φ(·,0) gives SD(0)then determine SD(t) from the flow and solvethe inverse scattering problem with SD(t) to find φ(·, t)


Solitons and IST

Solitons and Schrödinger operators:

Solitons for KdV correspond toSchrödinger operators with reflectionless potentials

Natural questions:how far the (classical) inverse scattering theory can be generalizedfor 1D Schrödinger operators with reflectionless potentials?what are the corresponding “soliton” solutions of the KdV?

Our results in a nutshell:Complete answers for integrable potentials ≡ integrable solitons


Potential scattering for Schrödinger operators:

In quantum mechanics, the Hamiltonian of the total energy for a “light”particle (electron) in the field of the “heavy” particle (nucleus) is theSchrödinger operator

Sq := − d2

dx2 + q(x) in the space L2(R)

When the potential q is real-valued and of compact support, then theequation −ψ′′ + qψ = k2ψ at the energy k2, k ∈ R, has the Jost solution

e+(x , k) =

{eikx , x � 1a(k)eikx + b(k)e−ikx , x � −1


The scattering solution

e+(x , k)

a(k)=

{t−(k)eikx , x � 1eikx + r−(k)e−ikx , x � −1

eikx

r−(k)e−ikx

t−(k)eikx

represents an incident wave eikx coming from −∞ which• partly reflects back to −∞ (term r−(k)e−ikx ) and• partly passes through to +∞ (term t−(k)eikx )


Scattering data:Here r−(k) := b(k)/a(k) is the left reflection coefficient and

t−(k) := 1/a(k) is the left transmission coefficientSimilarly define right reflection r+ and transmission t+ coefficients;then get the scattering matrix

S(k) :=

(t−(k) r+(k)r−(k) t+(k)

)Properties of the scattering matrix:

unitary for real kt−(k) = t+(k) =: t(k) admits meromorphic cont. in C+

r±(−k) = r±(k), t(−k) = t(k)

S(k) uniquely determined by r+ or r− and the poles of t

Discrete spectrum data:

Eigenvalues: −κ21 < −κ2

2 < · · · < −κ2n ⇐⇒ a(iκj) = 0

Norming constants: α1, α2, . . . , αn, αj := ‖e+(·, iκj)‖


Scattering on Faddeev–Marchenko potentials

Jost solutionsFor q ∈ L1(R, (1 + |x |)dx), the equation

−ψ′′ + qψ = k2ψ, k ∈ R,has Jost solutions e±(x , k) = e±ikx (1 + o(1)) as x → ±∞

Scattering coefficients

One then looks for a left scattering solution

ψ−(x , k) ∼{

eikx + r−(k) e−ikx if x → −∞,t−(k) eikx if x →∞

Discrete spectral dataSame as for q of compact support:

- finitely many eigenvalues −κ21 < −κ2

2 < · · · < −κ2n < 0

- corresponding norming constants α1, α2, . . . , αn


Direct and inverse scattering for Sq

Scattering problems

Direct scattering: q 7→(

r+, (−κ2j )n

j=1, (αj)nj=1

)scattering data

Inverse scattering (ISP):(

r+, (−κ2j )n

j=1, (αj)nj=1

)7→ q

The inverse scattering problem was completely solved for q in theFaddeev–Marchenko (FM) class

i.e. real-valued q in L1(R, (1 + |x |)dx)

by MARCHENKO, GELFAND, LEVITAN, and KREIN in 1950-ies:characterized reflection coefficients;suggested an algorithm for determining q from SD


Scattering for FM potentials

Jost solution: e+(x , k) = eikx(1 + o(1)), x →∞

transformation operator with kernel K+(x , t) s.t.

e+(x , t) = eikx +

∫ ∞x

K+(x , t)eikt dt

then K+ and

F+(s) :=1

2π

∫R

r+(k)eiks dk +∑

αje−κj s

are related via the Marchenko equation

K+(x , t) + F+(x + t) +

∫ ∞x

K+(x , s)F+(s + t) ds = 0, t > x


Solution to the ISP for FM potentials:

Algorithm:1 given SD, construct F+

2 solve the Marchenko equation for K+

3 then q(x) = −2 ddx K+(x , x)

Tasks:justify the algorithmestablish uniquenesscharacterize scattering data (SD) for q ∈ (FM)

Completed for potentials in (FM) by V. A. Marchenkoessential contributions byL. Faddeev, I. Gelfand, B. Levitan, M. Krein, P. Deift, E. Trubowitz a.o.


KdV and IST

finding solutions of the KdV equation

m

solving the inverse scattering problem for the Schrödinger oper.

A natural question:

How far can one generalize the IST beyond (FM)?

E.g., to include q = δ or other distributions;or to allow infinite discrete spectrum?


Reflectionless potentials

The inverse scattering problem is exactly soluble for a class of

:::::::::::::reflectionless potentials (r± ≡ 0)

Examples of reflectionless potentials producing just one negativeeigenvalue were constructed in

V. Bargmann, On the connection between phase shifts andscattering potential, Rev. Mod. Phys. 21 (1949), 30–45.

Later in 1956, I. Kay and H. E. Moses

I. Kay and H. E. Moses, Reflectionless transmission throughdielectrics and scattering potentials, J. Appl. Phys. 27 (1956),no. 12, 1503–1508.

obtained a formula for all classical reflectionless FM potentials


Reflectionless potentials ! soliton solutions of KdV

These reflectionless potentials have the form

q(x) = −2d2

dx2 log det(δjs + αjαs

e−(κj+κs)x

κj + κs

)1≤j,s≤n

, (1)

where (κj)nj=1 and (αj)

nj=1 are arbitrary sequences of positive numbers

the first of which is strictly decreasing

They generate the n-soliton solutions of KdV:

φ(x , t) = −2d2

dx2 log det(δjs + αjαs

e4(κ3j +κ

3s)t−(κj+κs)x

κj + κs

)1≤j,s≤n

(2)

Potentials in (1) are called classical reflectionless potentials anddenoted Qcl


Generalized reflectionless potentials?

A natural question arises,

Can one enlarge the class Qcl to getgeneralized soliton solutions of KdV?

F. Gesztesy, W. Karwowski and Z. Zhao, Limits of soliton solutions,Duke Math. J. 68 (1992), no. 1, 101–150.

gave a certain class Q∗ of potentials, for which analogues offormula (1) for the classical reflectionless potentialsformula (2) of soliton solutions

hold true


Gesztesy–Karwowsky–Zhao class

Namely, these potentials have the form

q(x) = −2d2

dx2 log det (I + C(x)), x ∈ R,

where C(x) is a trace class operator in `2 with matrix entries

cjs(x) := αjαse−(κj+κs)x

κj + κs, j , s ∈ N.

Here,(κj)j∈N is an arbitrary bounded sequence of pairwise distinctpositive numbers;(αj)j∈N is an arbitrary sequence of positive numbersthe trace-class condition

∑∞j=1 α

2j /κj <∞ assumed to hold


Marchenko class

Earlier in 1985 and 1991, D. S. Lundina and V. A. Marchenko

D. S. Lundina, Teoriya Funkts., Funkts. Analiz i ikh Prilozh. 44 (1985),57–66.

V. A. Marchenko, in What is integrability?, Springer Ser. NonlinearDynam., Springer, Berlin, 1991, pp. 273–318.

studied the properties of the closure B(κ) of

B(κ) := {q ∈ Qcl | σ(Sq) ⊂ [−κ2,∞)}, κ > 0,

in the topology of uniform convergence on compact subsets of RThe elements of the set

Q :=⋃κ>0

B(κ)

are called generalized reflectionless potentials; observe that Q∗ ⊂ Q.


Classes Qp

Set Qp := Q∩ Lp(R), 1 ≤ p ≤ ∞; then it can be shown thatQp is closed in Lp(R)

for q ∈ Q∞ \ Qcl, the negative spectrum of Sq coincides with{−κ2

j }j∈N with κj → 0

for an eigenvalue −κ2j , a norming constant αj can be defined

WLOG, take κj strictly decreasing, introduce

Q∞ \ Qcl 3 q 7→ κ(q) := (κj)j∈N,

Q∞ \ Qcl 3 q 7→ α(q) := (αj)j∈N,

and define the scattering map Υp via

Qp \ Qcl 3 q 7→ Υp(q) :=(κ(q),α(q)

)Denote by `+p the set of all positive strictly decreasing sequences in `p


Interesting questionsIn view of the Lieb–Thirring inequality, for q ∈ Lr (R) with r ∈ [1,∞) onehas κ(q) ∈ `p with p := 2r − 1 and

‖κ(q)‖`p ≤ Cr‖q−‖Lr ,

with q− := max{−q,0} and an absolute constant Cr .For p = 1 and negative q ∈ L1(R) this can be sharpened (T. WEIDL’96,D. HUNDERTMARK A.O.’98) to

‖κ(q)‖`1 ≤12‖q‖L1

The following questions seem of importance:1 Is it true that κ(Qr ) = `+p ? If so, is there cr > 0 s.t.

cr‖q−‖Lr ≤ ‖κ(q)‖`p?2 Can one describe the isospectral set

{α(q) | q ∈ Qr , κ(q) = κ(q0)}3 Is Υp injective? If so, can one reconstruct q ∈ Qr \ Qcl from(

κ(q),α(q))?


Main results

For p = 1, Que. 1 was answered in the paper by Gesztesy a.o.

Our aim is to give complete answers to Que. 2 and 3 for p = 1.

Theorem (Scattering map is one-to-one)

The mapping Υ1 is injective and onto `+1 × RN+.

Proofessentially uses the first FZ trace formula and relies on the followingobjects:

1 K is a positive diagonal operator in `2 with simple spectrumκ = (κj)j∈N ∈ `+1 , (i.e., K is of trace class);

2 There is a nonzero ρ = (ρj) ∈ `2 and a positive operator G : `2 → `2s.t.

KG + GK = ( · |ρ)ρ;

then ρ ∈ dom G−1/2 and ‖G−1/2ρ‖ = 2 tr K


Generalized reflectionless potentials

Theorem (Generalized reflectionless potentials)

Under the above assumptions, let A = diag{aj} be any positive diagonaloperator in `2. Then

1 the nonnegative decreasing function

ϕ(x) := ‖AexK (A2e2xK + G)−1/2ρ‖2, x ∈ R, (3)

can be analytically continued into the strip

Π := {z = x + iy | x , y ∈ R, |y | < π2‖K‖}; (4)

2 all components ρj of ρ are nonzero;3 the function q := 2φ′ is a reflectionless potential in Q1 satisfying

κ(q) = κ, while the sequence of norming constants α(q) = (αj)satisfy αj = aj |ρj |


Generalized soliton solutions to KdV equation

Using the above construction of generalized reflectionless potentials,one can construct generalized soliton solutions to the KdV equation, viz.

Theorem (Generalized soliton solutions)

In the above constructions, take A(t) := eK 3tA in place of A and define

ϕ(x , t) := ‖A(t)exK (A(t)2e2xK + G)−1/2ρ‖2, x ∈ R, (5)

and u(x , t) := 2∂xϕ(x , t). Then u(x , t) is a generalized soliton solutionto KdV equation s.t. u(·, t) ∈ Q1 for all t ≥ 0.


Thank you for your attention!

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