Introduction Reflectionless potentials Main results Inverse scattering for reflectionless Schrödinger operators and generalized KdV solitons Rostyslav Hryniv 1 Yaroslav Mykytyuk 2 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, J OHN S COTT R USSEL , 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 R AYLEIGH suggested theoretical background for this phenomenon and in 1877 B OUSSINESQ derived the equation for the wave profile φ : ∂ t φ + ∂ 3 x φ − 6 φ ∂ x φ = 0 ; it was rediscovered in 1895 by K ORTEWEG and DE V RIES 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 G ARDNER , G REENE , K RUSKAL and M IURA 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 S t of Schrödinger operators on the line S t := − d 2 dx 2 + φ ( · , t ) then the scattering data of S t SD ( t ) := ( r ( · , t ) , {− κ 2 j ( t ) } n j = 1 , { α j ( t ) } n j = 1 ) evolve along a 1 st order linear flow: α j ( t ) = α j ( 0 ) e 4 κ 3 r ( k , t ) = r ( k , 0 ) e − 8 ik 3 t , j t κ j ( t ) ≡ κ j ( 0 ) , 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 S q := − d 2 L 2 ( R ) dx 2 + q ( x ) in the space When the potential q is real-valued and of compact support, then the equation − ψ ′′ + q ψ = k 2 ψ at the energy k 2 , k ∈ R , has the Jost solution � e ikx , x ≫ 1 e + ( x , k ) = a ( k ) e ikx + b ( k ) e − ikx , x ≪ − 1
Introduction Reflectionless potentials Main results The scattering solution � t − ( k ) e ikx , e + ( x , k ) x ≫ 1 = e ikx + r − ( k ) e − ikx , a ( k ) x ≪ − 1 r − ( k ) e − ikx t − ( k ) e ikx e ikx represents an incident wave e ikx coming from −∞ which • partly reflects back to −∞ (term r − ( k ) e − ikx ) and • partly passes through to + ∞ (term t − ( k ) e ikx )
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 � t − ( k ) � r + ( k ) S ( 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 : − κ 2 1 < − κ 2 2 < · · · < − κ 2 n ⇐ ⇒ a ( i κ j ) = 0 Norming constants : α 1 , α 2 , . . . , α n , α j := � e + ( · , i κ j ) �
Introduction Reflectionless potentials Main results Scattering on Faddeev–Marchenko potentials Jost solutions For q ∈ L 1 ( R , ( 1 + | x | ) dx ) , the equation − ψ ′′ + q ψ = k 2 ψ, 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 � e ikx + r − ( k ) e − ikx if x → −∞ , ψ − ( x , k ) ∼ t − ( k ) e ikx if x → ∞ Discrete spectral data Same as for q of compact support: - finitely many eigenvalues − κ 2 1 < − κ 2 2 < · · · < − κ 2 n < 0 - corresponding norming constants α 1 , α 2 , . . . , α n
Introduction Reflectionless potentials Main results Direct and inverse scattering for S q Scattering problems � � r + , ( − κ 2 j ) n j = 1 , ( α j ) n Direct scattering: q �→ j = 1 scattering data � � r + , ( − κ 2 j ) n j = 1 , ( α j ) n �→ q Inverse scattering (ISP): j = 1 The inverse scattering problem was completely solved for q in the Faddeev–Marchenko (FM) class i.e. real-valued q in L 1 ( R , ( 1 + | x | ) dx ) by M ARCHENKO , G ELFAND , L EVITAN , and K REIN 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 ) = e ikx � � 1 + o ( 1 ) , x → ∞ transformation operator with kernel K + ( x , t ) s.t. � ∞ e + ( x , t ) = e ikx + K + ( x , t ) e ikt dt x then K + and F + ( s ) := 1 � r + ( k ) e iks dk + � α j e − κ j s 2 π R are related via the Marchenko equation � ∞ K + ( x , t ) + F + ( x + t ) + K + ( x , s ) F + ( s + t ) ds = 0 , t > x x
Introduction Reflectionless potentials Main results Solution to the ISP for FM potentials: Algorithm: given SD , construct F + 1 solve the Marchenko equation for K + 2 then q ( x ) = − 2 d dx K + ( x , x ) 3 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; or 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. obtained 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 d 2 e − ( κ j + κ s ) x � � d x 2 log det δ js + α j α s 1 ≤ j , s ≤ n , (1) κ j + κ s where ( κ j ) n j = 1 and ( α j ) n j = 1 are arbitrary sequences of positive numbers the first of which is strictly decreasing They generate the n -soliton solutions of KdV: e 4 ( κ 3 j + κ 3 s ) t − ( κ j + κ s ) x φ ( x , t ) = − 2 d 2 � � d x 2 log det δ js + α j α s (2) κ j + κ s 1 ≤ j , s ≤ n Potentials in (1) are called classical reflectionless potentials and denoted Q cl
Introduction Reflectionless potentials Main results Generalized reflectionless potentials? A natural question arises, Can one enlarge the class Q cl 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 d 2 d x 2 log det ( I + C ( x )) , x ∈ R , where C ( x ) is a trace class operator in ℓ 2 with matrix entries e − ( κ j + κ s ) x c js ( x ) := α j α s , j , s ∈ N . κ j + κ s Here, ( κ j ) j ∈ N is an arbitrary bounded sequence of pairwise distinct positive numbers; ( α j ) j ∈ N is an arbitrary sequence of positive numbers the trace-class condition � ∞ j = 1 α 2 j /κ j < ∞ assumed to hold
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