Inverse scattering for reflectionless Schrdinger operators and - - PowerPoint PPT Presentation

Introduction Reflectionless potentials Main results

Inverse scattering for reflectionless Schrödinger

• perators 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

Introduction Reflectionless potentials Main results

Solitary waves and Korteweg–de Vries equation

In August 1834, JOHN SCOTT RUSSEL, a Scottish civil engineer, naval architect and shipbuilder observed an unusual solitary wave in a channel 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 this phenomenon and in 1877 BOUSSINESQ derived the equation for the wave profile φ: ∂tφ + ∂3

xφ − 6 φ ∂xφ = 0;

it was rediscovered in 1895 by KORTEWEG and DE VRIES and is nowadays known as the Korteweg–de Vries (KdV) equation The KdV equation is a nonlinear dispersive PDE possessing infinitely many first integrals and many interesting properties, n-soliton solutions being one of them

Introduction Reflectionless potentials Main results

Two-soliton solution

Introduction Reflectionless potentials Main results

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 the inverse scattering transform (IST) technique: with a solution φ(x, t) to KdV, associate a family St of Schrödinger operators on the line St := − d2 dx2 + φ(·, t) then the scattering data of St SD(t) := (r(·, t), {−κ2

j (t)}n j=1, {αj(t)}n j=1)

evolve along a 1st order linear flow: r(k, t) = r(k, 0)e−8ik3t, κj(t) ≡ κj(0), αj(t) = αj(0)e4κ3

j t

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

Introduction Reflectionless potentials Main results

Solitons and IST

Solitons and Schrödinger operators: Solitons for KdV correspond to Schrödinger operators with reflectionless potentials Natural questions: how far the (classical) inverse scattering theory can be generalized for 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

Introduction Reflectionless potentials Main results

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 the Schrödinger operator Sq := − d2 dx2 + q(x) in the space L2(R) When the potential q is real-valued and of compact support, then the equation −ψ′′ + qψ = k2ψ at the energy k2, k ∈ R, has the Jost solution e+(x, k) =

• eikx,

x ≫ 1 a(k)eikx + b(k)e−ikx, x ≪ −1

Introduction Reflectionless potentials Main results

The scattering solution e+(x, k) a(k) =

• t−(k)eikx,

x ≫ 1 eikx + 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)

Introduction Reflectionless potentials Main results

Scattering data:

Here r−(k) := b(k)/a(k) is the left reflection coefficient and t−(k) := 1/a(k) is the left transmission coefficient Similarly 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 k t−(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: −κ2

1 < −κ2 2 < · · · < −κ2 n ⇐

⇒ a(iκj) = 0 Norming constants: α1, α2, . . . , αn, αj := e+(·, iκj)

Introduction Reflectionless potentials Main results

Scattering on Faddeev–Marchenko potentials

Jost solutions For 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 data Same as for q of compact support:

• finitely many eigenvalues −κ2

1 < −κ2 2 < · · · < −κ2 n < 0

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

Introduction Reflectionless potentials Main results

Direct and inverse scattering for Sq

Scattering problems Direct scattering: q →

• r+, (−κ2

j )n j=1, (αj)n j=1

• scattering data

Inverse scattering (ISP):

• r+, (−κ2

j )n j=1, (αj)n j=1

• → q

The inverse scattering problem was completely solved for q in the Faddeev–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

Introduction Reflectionless potentials Main results

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−κjs

are related via the Marchenko equation K+(x, t) + F+(x + t) + ∞

x

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

Introduction Reflectionless potentials Main results

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 d

dx K+(x, x)

Tasks: justify the algorithm establish uniqueness characterize scattering data (SD) for q ∈ (FM) Completed for potentials in (FM) by V. A. Marchenko essential contributions by

• L. Faddeev, I. Gelfand, B. Levitan, M. Krein, P

. Deift, E. Trubowitz a.o.

Introduction Reflectionless potentials Main results

KdV and IST

finding solutions of the KdV equation

• 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;

• r to allow infinite discrete spectrum?

Introduction Reflectionless potentials Main results

Reflectionless potentials

The inverse scattering problem is exactly soluble for a class of

✿✿✿✿✿✿✿✿✿✿✿✿✿

reflectionless potentials (r± ≡ 0) Examples of reflectionless potentials producing just one negative eigenvalue were constructed in

• V. Bargmann, On the connection between phase shifts and

scattering 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 through

dielectrics and scattering potentials, J. Appl. Phys. 27 (1956),

• no. 12, 1503–1508.
• btained a formula for all classical reflectionless FM potentials

Introduction Reflectionless potentials Main results

Reflectionless potentials soliton solutions of KdV

These reflectionless potentials have the form q(x) = −2 d2 dx2 log det

• δjs + αjαs

e−(κj+κs)x κj + κs

• 1≤j,s≤n,

(1) where (κj)n

j=1 and (αj)n j=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) = −2 d2 dx2 log det

• δjs + αjαs

e4(κ3

j +κ3 s)t−(κj+κs)x

κj + κs

• 1≤j,s≤n

(2) Potentials in (1) are called classical reflectionless potentials and denoted Qcl

Introduction Reflectionless potentials Main results

Generalized reflectionless potentials?

A natural question arises, Can one enlarge the class Qcl to get generalized 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 of formula (1) for the classical reflectionless potentials formula (2) of soliton solutions hold true

Introduction Reflectionless potentials Main results

Gesztesy–Karwowsky–Zhao class

Namely, these potentials have the form q(x) = −2 d2 dx2 log det (I + C(x)), x ∈ R, where C(x) is a trace class operator in ℓ2 with matrix entries cjs(x) := αjαs e−(κj+κs)x κj + κs , j, s ∈ N. Here, (κj)j∈N is an arbitrary bounded sequence of pairwise distinct positive numbers; (αj)j∈N is an arbitrary sequence of positive numbers the trace-class condition ∞

j=1 α2 j /κj < ∞ assumed to hold

Introduction Reflectionless potentials Main results

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. Nonlinear

Dynam., 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 R The elements of the set Q :=

• κ>0

B(κ) are called generalized reflectionless potentials; observe that Q∗ ⊂ Q.

Introduction Reflectionless potentials Main results

Classes Qp

Set Qp := Q ∩ Lp(R), 1 ≤ p ≤ ∞; then it can be shown that Qp 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 −κ2

j , a norming constant αj can be defined

WLOG, take κj strictly decreasing, introduce Q∞ \ Qcl ∋ q → κ(q) := (κj)j∈N, Q∞ \ Qcl ∋ q → α(q) := (αj)j∈N, and define the scattering map Υp via Qp \ Qcl ∋ q → Υp(q) :=

• κ(q), α(q)
• Denote by ℓ+

p the set of all positive strictly decreasing sequences in ℓp

Introduction Reflectionless potentials Main results

Interesting questions

In view of the Lieb–Thirring inequality, for q ∈ Lr(R) with r ∈ [1, ∞) one has κ(q) ∈ ℓp with p := 2r − 1 and κ(q)ℓp ≤ Crq−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)ℓ1 ≤ 1

2qL1

The following questions seem of importance:

1

Is it true that κ(Qr) = ℓ+

p ? If so, is there cr > 0 s.t.

crq−Lr ≤ κ(q)ℓp?

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)
• ?

Introduction Reflectionless potentials Main results

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 ℓ+

1 × RN +.

Proof essentially uses the first FZ trace formula and relies on the following

• bjects:

1

K is a positive diagonal operator in ℓ2 with simple spectrum κ = (κj)j∈N ∈ ℓ+

1 , (i.e., K is of trace class);

2

There is a nonzero ρ = (ρj) ∈ ℓ2 and a positive operator G : ℓ2 → ℓ2 s.t. KG + GK = ( · |ρ)ρ; then ρ ∈ dom G−1/2 and G−1/2ρ = 2 tr K

Introduction Reflectionless potentials Main results

Generalized reflectionless potentials

Theorem (Generalized reflectionless potentials) Under the above assumptions, let A = diag{aj} be any positive diagonal

• perator in ℓ2. Then

1

the nonnegative decreasing function ϕ(x) := AexK(A2e2xK + G)−1/2ρ2, x ∈ R, (3) can be analytically continued into the strip Π := {z = x + iy | x, y ∈ R, |y| <

π 2K};

(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|

Introduction Reflectionless potentials Main results

Generalized soliton solutions to KdV equation

Using the above construction of generalized reflectionless potentials,

• ne 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)−1/2ρ2, x ∈ R, (5) and u(x, t) := 2∂xϕ(x, t). Then u(x, t) is a generalized soliton solution to KdV equation s.t. u(·, t) ∈ Q1 for all t ≥ 0.

Introduction Reflectionless potentials Main results

Thank you for your attention!