2D Schrodinger Operator, (2+1) Systems and New Reductions. The 2D Burgers Hierarchy and Inverse Problem Data

P .Grinevic h, A.Mirono v, S.No vik o v 1 2D Sc hro dinger Op erator, (2+1) Systems and New Reductions. The 2D Burgers Hierarch y and In v erse Problem Data 2 Abstract . The Theory of (2+1 ) Sy stems based on 2D Sc hro dinger Op- erator w as star t ed b y S.Manak o v, B.Dubro vin, I.Kric hev er and S.Nov ik o v in 1976 (see[1,2]). The Analog of Lax P airs in tro duced in [1], has a fo rm L t = [ L, H ] − f L (”The L, H , f -triples”) where L = ∂ x ∂ y + G∂ y + S and H , f -some linear PDEs . Their Algebro-Geometric Solutions and therefore the full higher order hierarc hies w ere constructed in [2]. The Theory of 2D In v erse Sp ectral Pro blems for the Elliptic Op erator L with x, y replaced by z , ¯ z , w as started in [2]: The Inv ers e Sp ectral Problem Dat a are tak en from the comple x ”F ermi-Curv e” cons isting of all Bloch-Flo quet Eigenfunctions Lψ = const . Many in teresting systems w ere found later [3]. Ho w ev er, sp e- cific prop erties of the v ery first system offered in [1 ] f or the v erification of new method only , w ere not studied more than 10 y ears un til B.Konop elc henk o found in 1 9 88 (see [6]) analogs of Bac kund T ransformations for it. He p oin ted out on the “Burgers-T yp e Reduction” 3 . Indee d, t he presen t authors quite re- cen tly found v ery interesting extensions, reductions and applications of that system b oth in the theory o f nonlinear ev olution systems (The Self- Adjoin t and 2 D Burgers Hierar hies were in v en ted, and cor r esp onding r eductions of In v erse Problem Dat a found) and in the Sp ectral Theory of Imp ortant Ph ys- ical Op erato rs (”The Purely Magnetic 2D Pauli Op erators”). W e call this system G KMMN by the names of authors who s tudied it. Let us consider the 2nd order op erato r s L, H and scalar function f , re- 1 All Novik ov’s works quoted here, can be taken directly fro m his ho mepage ht tp://www.mi.r as.ru/ ˜ snovik ov, click Publi cations . Our e-mails ar e pg g@landau.ac.ru (Grinevich), mir ono v@nsc .math.ru (Mir onov), novik ov@ipst.umd.edu (Noviko v) 2 This work was accepted for publication in the Journa l Russia n Math Surveys. It will app ear in 2010 , v 6 5 n 3 3 Unfortunately , in the w or k [6 ] this system is presented as a new one. At the same time, the work [1] wher e it was orig inally found, is included in the list of r eferences o f [6]. The work [2] where Algebro-Geometric Solutions of such systems w ere found in 1976 , is not quoted at a ll in [6 ]. It leads to the wrong impre ssion that ther e was no developmen t till 1 980, reversing the chronological order of participating a uthors to the opp osite 1 duced to the follo wing form by the gauge transformations L = ∂ x ∂ y + G∂ y + S, H = ∂ 2 x + F ∂ y + A Using L, H , f -triple (see Abstract) we define corresp onding ( 2+1)nonlinear ev olution system. W e call it ”The G K MMN System”. Making calculation, w e obtain following Prop osition. The GKMMN System has a fo rm(I) G t = G xx − G y y + ( F 2 ) x − ( G 2 ) x − A x + 2 S y , S t = − S xx + S y y + 2 ( GS ) x − 2( F S ) y F x = 2 G y , A y = 2 S x , f = 2 G x − F y Let us formulate some useful corollaries of that sys tem. Corollary 1. The system GKMMN is compatible with the purely real reduction whe re all co efficien ts are real. Corollary 2. The system GKMMN admits a Reduction S = 0. W e c all it ” The 2D Burgers System” and denote b y B 2 . The whole Hie rarch y can be naturally cons tructed after Theorem 2 b elo w. Corollary 3. F or the GKMMN system and its statio nary pro blem the elliptic op erator H can be self-a djoin t only in the trivial cases reducible to the functions of one v ariable. Here H = ∆ + F ∂ y + A is suc h that the magnetic field B = F x /i, i 2 = − 1 , a nd electric p otential U = A − F 2 / 4 − F y / 2 are real and smo oth. Pro of of this corollary requires calculations. Under these restrictions the system GKMMN b ecame strongly o v er-determined leading to the complete degeneration. Conjecture. F or the smo oth p erio dic second or der self-adjoint elliptic 2D op erators the complete complex manifold of the Blo c h-Flo quet Eigen- functions W (except some trivial cases reducible t o one v ariable), cannot con tain Zar iski O p en P art of the Complex Algebraic Curv e Γ ⊂ W except of the lev els ǫ = const found in 1976 in [2]. Corollary 4. The substitution G = − (log c ) x , F = − 2(log c ) y , A = − 2 u x , S = − u y transforms our system in to the f ollo wing sy stem (I I): [( c t − c xx + c y y ) c − 1 ] x = 2( u y y − u xx ) , u t = u y y − u xx + 2( u y c x /c ) x − 2( u y c y /c ) y 2 The B 2 Reduction S = 0 reduce s system to the linear form (II I): c t − c xx + c y y = ( U ( x ) + V ( y )) c exactly in the same w a y as the ordinary 1D Burgers System (i.e. our system with U = V = 0 not dep ending on the v ariable y ). The sp ectral meaning of this v ariables and substitution will b e clarified b elo w for the Algebro-G eometric (AG) Solutions immediately leading to the full Hierarch y of suc h systems. The Algebro-Geometric (AG) Inv erse Sp ectral Problem Data. T ak e Riemann Surface Γ with 2 ”infinite” p oin ts ∞ 1 , ∞ 2 and lo cal pa- rameters 1 /k 1 , 1 /k 2 near them, 1 /k j ( ∞ j ) = 0. Select the ”D ivisor of p oles” D = P 1 + ... + P g in Γ. Construct ”The 2 -p oin t Ba ker-Akhiez er F unction” ψ ( P , x, y , t ) in v en ted in [2]. It should be meromorphic in the v ariable P ∈ Γ outside of infinities, with divisor of poles D whic h is x, y , t -indep enden t. Its asymptotic b ehavior near infinities is fo llo wing: ψ = ce k 1 x + k 2 1 t (1 + v /k 1 + O (1 /k 2 1 )) , ψ = e k 2 y + k 2 2 t (1 + u/k 2 + O (1 /k 2 2 )) This function satisfies to the equation Lψ = 0 and to the (2+1)-systems (I,I I,I II) with parameters ( c, u ) en tering it. The Real AG Reduction of (I) is follo wing: There is a n antiholomor- phic in volution σ : Γ → Γ , σ 2 = 1, s uc h that σ ( ∞ j ) = ∞ j , σ ∗ ( k j ) = − ¯ k j , σ ( D ) = D Easy to formulate conditions suc h that real solutions ( written through the Θ-functions) are smoo t h nonsingular. F or the dense family of data they a re p erio dic. In general they are quasiperio dic as usual. The Stationary A G Solutions are suc h that [ L, H ] = f L and H ψ = λ ( P ) ψ where H is an elliptic op erator as ab o v e. They corresp ond to the algebraic curv es Γ with algebraic function λ ha ving exactly 2 p o les on Γ of the second order in b oth infinite p oints ∞ 1 , ∞ 2 . Ho w ev er, t hey a re no n- self- adjoin t in the non trivial cases. The Burgers Reduction B 2 is esp ecially in teresting. Here S = u y = 0 . Theorem 1. T ak e r educible Riemann Surface Γ = Γ ′ S Γ ′′ suc h that Γ ′ \ Γ ′′ = Q = Q 0 [ ... [ Q k , ∞ 1 ∈ Γ ′ ∞ 2 ∈ Γ ′′ 3 and divisor D = D ′ + D ′′ , D ′ ⊂ Γ ′ , D ′′ ⊂ Γ ′′ , | D ′ | = g ′ + k , | D ′′ | = g ′′ where g ′ =gen us of Γ ′ , D ′′ =gen us of Γ ′′ , all p oin ts ∞ j , Q, D are distinct. Construct ψ as a standard one-p oint Bak er Akhiezer function ψ ′′ on Γ ′′ with divisor D ′′ and asymptotic ψ ′′ = e k 2 x + k 2 2 t (1 + O (1 /k 2 )) On the part Γ ′ our function ψ should coincide with ψ ′ . It has the divisor of p oles D ′ , asymptotic ψ ′ = ce k 1 y + k 2 1 t (1 + O (1 /k 1 ) and conditions (*) ψ ′ ( Q s ) = ψ ′′ ( Q σ ( s ) ) where σ is some permu tation of the set Q . Then w e ha v e L ( ψ ) = 0 and ( L t − [ L, H ]) ψ = 0 with S = u y = 0. Remark. W e can drop the surface Γ ′′ and divisor D ′′ . T ak e any set of solutions ψ ′′ s ( x, t ) t o the equation ψ ′′ s,t = ψ ′′ s,xx , s = 0 , 1 , ..., k . Define ψ ′ ( x, y , t, P ) using conditio ns ψ ′ ( x, y , t, Q s ) = ψ ′′ s ( x, t ) instead of conditions (*). Our function ψ = ψ ′ satisfies to the equations Lψ = 0 , L t = [ L, H ] − f L for all p oin ts P ∈ Γ ′ , and S = u y = 0. Corresp onding hierarh y with higher times w e call ” The 2D Burgers Hier- arh y” B 2 . There are 2 cases in our theory: 1.( x, y ) ∈ R . This is the system GK M M N − I and reduction B 2 − I 2. x → z , y → ¯ z and ∂ x → ∂ = ∂ x − i∂ y , ∂ y → ¯ ∂ = ∂ x + i∂ y , ∂ ¯ ∂ = ∆. This is the system GK M M N − I I and reduction B 2 − I I . Theorem 2 . F or the system GK M M N − I I in t he v ariables z , ¯ z the reduction to the class of self-adjoint op erators L with real magnetic field − 2 B = 2 G ¯ z = F z and p oten tial S ∈ R is compatible with time dy namics in the v ariable it, t ∈ R (V): [( c t − 4 c xy ) c − 1 ] z = 8 u xy , ( u + 4 u xy ) ¯ z = 2 /i [( u ¯ z c z /c ) − ( u ¯ z c ¯ z /c ) ¯ z ] Here w e hav e S = u ¯ z ∈ R, c = e 2Φ ∈ R , and system can b e written in the form (VI): c t − 4 c xy = 8 a y c = − 4 I m ( u z ) c, S t + 4 S xy = 8[ S Φ xy − S x Φ y − S y Φ x ] with u = a + ib, S = a x − b y , a y + b x = 0. The condition S = 0 leads to the linear system B 2 − I I (form ula (VI I)): c t − 4 c xy = T ( x, y , t ) c, ∆ T = 0 , T = 8 a y ∈ R 4 Here G = 1 / 2(log c ) z , B = − 1 / 2∆(log c ) ∈ R . F or the self-adjoint factorizable op erator L (i.e. S = 0 ), the Reducible Riemann Surface Γ admits an an ti-in v olution σ : Γ ′ → Γ ′′ and bac k, σ 2 = 1. The sp ectrum of this op erator determines the sp ectrum of P urely Magnetic Nonrelativistic 2D Pa uli Op erator for the particles with spin 1/2. The t heory of ground states for the Algebro-Geom etric P auli Op erators is dev elop ed in [4] . Results o f the presen t w ork (without theorem 2) can b e found in our art icle in arXiv (see [5]). References. 1.S.Manak o v, Usp ekhi Math Nauk, 1976 v 31 n 5 pp 245-246 2.B.Dubrov in, I.Krichev er, S.No vik o v, Doklady AN SSSR, 1976, v 229 n 1 pp 15-1 8 3.A.V ese lov , S.Novik ov, Doklady AN SSSR, 1984, v 279 v 4 pp 784-788 4.P .Grinevic h, A.Mironov, S.No vik ov, arXiv: 1004.1157 5.P .Grinevic h, A.Mironov, S.No vik ov, arXiv: 10014300 6.B.Konop elc henk o, In v erse Problems, 1988, v.4, pp 151-163. 5