Weyl-Titchmarsh type formula for periodic Schrödinger operator with Wigner-von Neumann potential

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1 Weyl-Titchmrsh type formul for periodic Schrödinger opertor with Wigner-von Neumnn potentil Pvel Kursov nd Sergey Simonov Abstrct Schrödinger opertor on the hlf-line with periodic bckground potentil perturbed by certin potentil of Wignervon Neumnn type is considered The symptotics of generlized eigenvectors for λ C + nd on the bsolutely continuous spectrum is estblished Weyl-Titchmrsh type formul for this opertor is proven Introduction L α := d c sinω + δ + q + + q d + γ, on the set of functions stisfying the boundry condition ψ cos α ψ sin α =, where c, ω, δ, R, α [, π, γ ; ] As it ws shown by the first uthor nd Nboko in [6], the bsolutely continuous spectrum of this Consider the one dimensionl Schrödinger opertor with the rel potentil which cn be represented s sum of three terms: certin periodic function, Wigner-von Neumnn potentil nd certin bsolutely integrble function More precisely, let q be rel periodic function with period such tht q L ; nd let q L R + Then the Schrödinger opertor L α is defined by the differentil epression 99 Mthemtics Subject Clssifiction 47E5,34B,34L4,34L,34E Key words nd phrses Asymptotics of generlized eigenvectors, Weyl- Titchmrsh theory, Schrödinger opertor, Wigner-von Neumnn potentil

2 PAVEL KURASOV AND SERGEY SIMONOV opertor hs multiplicity one nd coincides s set with the spectrum of the corresponding periodic opertor on R, 3 L per = d d + q Note tht the spectrum of L per hs multiplicity two Let ψ +, λ nd ψ, λ be Bloch solutions for L per nd ϕ α, λ be the solution of Cuchy problem ϕ α, λ + ϕ α, λ = sin α, ϕ α, λ = cos α q + c sinω+δ + q + γ ϕ α, λ = λϕ α, λ, The min result of the present pper is the following theorem tht reltes the spectrl density ρ α of the opertor L α nd the symptotics of the solution ϕ α We cll it the Weyl-Titchmrsh formul Theorem Let ω / Z nd q π L R +, then for lmost ll λ σl per there eists A α λ such tht 4 ϕ α, λ = A α λψ, λ + A α λψ +, λ + o s nd ρ αλ = π W ψ + λ, ψ λ A α λ Weyl-Titchmrsh formuls form n efficient tool to study the behvior of the spectrl density The bsolutely continuous spectrum of the opertor L α contins infinitely mny criticl resonnce points see 5 where the type of the symptotics of generlized eigenvectors chnges nd is not given by liner conbintion of ψ + nd ψ s in 4 Precisely t these points the embedded eigenvlues of L α my occur In the generic cse no eigenvlue occurs, but it is nturl to suspect tht the spectrl density of L α vnishes t these points Vnishing of the spectrl density divides the bsolutely continuous spectrum into independent prts nd hs cler physicl mening This phenomenon is clled pseudogp In the forthcoming pper we intend to study zeros of the spectrl density in more detil The study of Schrödinger opertors with Wigner-von Neumnn potentils begn from the clssicl pper [9] where it ws observed for c sinω + δ the first time tht the potentil my produce n eigenvlue inside the bsolutely continuous spectrum Lter on such oper- + tors ttrcted ttention of mny uthors [],[7],[8],[6],[],[3],[4],[4],[5],[],[3] The phenomenon of this nture, n embedded eigenvlue bound stte

3 WEYL-TITCHMARSH FORMULA 3 in the continuum, ws even observed eperimentlly in semiconductor geterostructures [7] Weyl-Titchmrsh formul for the spectrl density in the cse of zero periodic bckground potentil follows directly from the results of [7] This formul ws proved once gin in [] where the method of Hrris-Lutz trnsformtions [] ws used In the present pper we lso use modifiction of this method We would like to mention tht nother one pproch ws suggested in [5], but gin in the cse of zero periodic bckground potentil Preliminries The spectrum of L per consists of infinitely mny intervls [9, Theorem 3] σl per := [λ n ; µ n ] [µ n+ ; λ n+ ], where n= λ < µ µ < λ λ < µ µ 3 < λ λ 4 <, where λ j nd µ j re the eigenvlues of the Schrödinger differentil eqution on the intervl [, ] with periodic nd ntiperiodic boundry conditions Spectrl properties of L per re relted to the entire function Dλ discriminnt nd the function kλ qusi-momentum kλ := i ln trdλ + tr Dλ 4 We cn choose the brnch of kλ so tht this follows from the properties of Dλ, see [9, Theorem 3] kλ =, kµ = kµ = π, kλ = kλ = π,, kλ R, if λ σl per, kλ C +, if λ C + The eigenfunction eqution for L per, ψ + qψ = λψ, hs two solutions Bloch solutions ψ +, λ nd ψ, λ stisfying qusiperiodic conditions: ψ + +, λ e ikλ ψ +, λ, ψ +, λ e ikλ ψ, λ

4 4 PAVEL KURASOV AND SERGEY SIMONOV They re determined uniquely up to multipliction by coefficients depending on λ It is possible to choose these coefficients so tht Bloch solutions hve the following properties: ψ +, λ, ψ, λ for every nd their Wronskin W ψ + λ, ψ λ re nlytic functions of λ in C + nd continuous up to σl per \{λ n, µ n, n } For λ σl per \{λ n, µ n, n }, ψ +, λ ψ, λ 3 The Wronskin does not hve zeros nd for λ σl per \{λ n, µ n, n } W ψ + λ, ψ λ ir + Bloch solutions cn lso be written in the form ψ +, λ = e ikλ p +, λ, ψ, λ = e ikλ p, λ, where the functions p +, λ nd p, λ hve period in the vrible nd the sme properties s ψ +, λ nd ψ, λ with respect to the vrible λ As we mentioned erlier, the opertor L α ws studied in [6], where the symptotics of the generlized eigenvectors ws obtined The uthors showed tht in every zone of σl per [λ n ; µ n ] if n is even nd [µ n ; λ n ] if n is odd there eist two criticl points λ + n nd λ n determined by the equlities kλ 5 + n = π n + { } ω π, kλ n = π n + { } ω π They do not coincide with ech other nd with the ends of zones, if ω 6 / Z π 3 Reduction of the spectrl eqution to the discrete liner system of Levinson form In this section we trnsform the eigenufnction eqution for L α to liner system with the coefficient mtri being sum of the digonl nd summble mtrices Consider the eigenfunction eqution for L α : 7 ψ c sinω + δ + q + + q + γ ψ = λψ For every λ C + σl per \{λ n, µ n, n }

5 WEYL-TITCHMARSH FORMULA 5 let us mke the following substitution ψ ψ, λ ψ ψ = +, λ ψ, λ ψ +, λ 8 or ψ u := +, λ ψ +, λ W ψ + λ,ψ λ ψ, λ ψ, λ Writing 7 s ψ ψ = q + c sinω+δ + γ + q λ u, ψ ψ ψ ψ nd substituting 8 into it, we get: 9 c sinω+δ + q u + = γ ψ+, λψ, λ ψ+, λ W ψ + λ, ψ λ ψ, λ ψ +, λψ, λ Let us introduce nother one vector vlued function v e ikλ v := e ikλ u, or e ikλ u = e ikλ v nd the mtri R q p+, λp, λ :=, λ p +, λ W ψ + λ, ψ λ p, λ p +, λp, λ Then the system 9 is equivlent to u [ v ikλ c sinω + δ = ikλ + + γ W ψ + λ, ψ λ ] p+, λp, λ p +, λ p + R, λ p +, λp, λ, λ v Let us serch for differentible mtri-vlued function Q such tht Q, Q = O γ s nd such tht the substitution 3 v = e Q ṽ

6 6 PAVEL KURASOV AND SERGEY SIMONOV leds to system for the vector vlued function ṽ of the form [ ṽ ik c sinω + δ = ik + + γ W ψ +, ψ ] p+ p + R p + p ṽ, where the reminder R lso belongs to L, Using tht e ±Q = I ± Q + O, γ e ±Q = ±Q + O γ s we obtin [ 4 ṽ ik c sinω + δ = ik + + γ W ψ +, ψ p+ p p + p Q p + p [ ] ik Q, ik + R + O γ ] ṽ, where [ ] ik Q, ik is the commuttor of the two mtrices Our im is to cncel the ntidigonl entries of p+ p p + p p + p in 4 by properly choosing Q To this end Q hs to stisfy the following eqution: 5 [ ] Q ik c sinω + δ + Q, p ik = + + γ W ψ +, ψ p e ik The ltter is equivlent fter multipliction by e ik from the right nd by its inverse from the left to e ik e ik 6 e ik Q e ik c sinω + δ p = +e ik + γ W ψ +, ψ p e ik

7 For every WEYL-TITCHMARSH FORMULA 7 µ σl per \{λ n, µ n, λ + n, λ n, n } nd for the vlues of λ from some neighbourhood of the point µ which we will specify lter let us tke the following solution of 6: e ik e ik c e ik Q, λ, µ e ik = W ψ + λ, ψ λ sinωt+δp + t,λeikλ t t+ γ sinωt+δp t,λeikλ t t+ γ sinωt+δp t,λeikµ t t+ γ this is our choice of constnts of integrtion tht depend on µ This leds to 7 Q, λ, µ := e ikλ In prticulr, for λ = µ, 8 Q, µ, µ = e ikµ c W ψ + λ, ψ λ e ikλ sinωt+δp t,λeikλ t t+ γ c W ψ + µ, ψ µ sinωt+δp + t,λeikλ t t+ γ t+ γ sinωt+δp t,λeikµ t e ikµ sinωt+δψ+ t,µ t+ γ sinωt+δψ t,µ t+ γ Formul 8 does not mke sense if µ C + due to the divergence of the integrl in the lower entry But it hs nlytic continution in its second rgument from the point µ Let us denote εµ := min n Z { kµ Consider some β > nd the set + ω + πn, kµ ω + πn Uβ, µ := {λ C + : ελ εµ, Im kλ/, } Re kλ kµ βim kλ}

8 8 PAVEL KURASOV AND SERGEY SIMONOV The set Uβ, µ is compct nd contins the point µ For every β < β it contins some neighbourhood of the verte of the sector Re λ µ β Im λ k µ Note tht k µ is positive for µ σl per \{λ n, µ n, n } Theorem Let β > nd µ σl per \{λ n, µ n, λ + n, λ n, n } Then there eists c β, µ, γ such tht for every nd λ Uβ, µ holds: Q, λ, µ, Q, λ, µ < c β, µ, γ + γ nd Proof Note first tht kλ R for λ σl per kλ C + for λ C + Let us denote the entries of Q, λ, µ s follows Q Q, λ, µ =, λ Q, λ, µ Let us estimte first the entry Q Let f be periodic function with period such tht f L ;, its Fourier coefficients will be denoted by f n fn := fe πin d Lemm If nd ξ C + is such tht then 9 eiξ e iξt ft t + γ {f n } + n= l Z ξ π / Z, + n= f n ξ πn + γ ie the epression on the left-hnd side eists nd is estimted by the right-hnd side

9 WEYL-TITCHMARSH FORMULA 9 Proof Consider > Since the Fourier series converges bsolutely, we hve e iξ e iξt + f t + γ n e πn t = n= + n= f n e iξ Integrting by prts nd estimting the bsolute vlue we get eiξ πn iξ+ t e t + γ + γ + + γ + γ Substituting into yields: eiξ e iξt ft t + γ ξ + πn By Cuchy s criterion the integrl t + γ+ + n= e iξt ft t + γ = ξ + πn f n ξ πn + γ πn iξ+ e t t + γ + γ eists nd the desired estimte 9 follows Formul 7 implies Q, λ = ce iω+iδ iw ψ + λ, ψ λ ce iωiδ iw ψ + λ, ψ λ kλ +ω ei kλ ω ei p +t, λe t + γ kλ i +ωt kλ i ωt p +t, λe t + γ Denote the Fourier coefficients of the function p +, λ by b n λ, b n λ := p +, λe πin d

10 PAVEL KURASOV AND SERGEY SIMONOV Then Lemm pplied to to gives c Q, λ W ψ + λ, ψ λ + γ + b n λ n= kλ + ω + πn + kλ ω + πn + c b n λ εµ W ψ + λ, ψ λ + γ n= Let us estimte now the entry Q Formul 7 implies ce ikλ Q, λ, µ = sinωt + δp t, λe ikλ t W ψ + λ, ψ λ t + γ sinωt + δp t, λe ikµ t t + γ ce ikλ sinωt + δp t, λ e ikλ t e ikµ t = W ψ + λ, ψ λ t + γ Denote nd Q I, λ, µ := Q II ce ikλ W ψ + λ, ψ λ ce ikλ W ψ + λ, ψ λ sinωt + δp t, λ, λ, µ := W ψ + λ, ψ λ so tht ce ikλ sinωt + δp t, λe ikµ t t + γ e ikλ t e ikµ t t + γ sinωt + δp t, λe ikµ t t + γ, Q, λ, µ = Q I, λ, µ + Q II, λ, µ

11 WEYL-TITCHMARSH FORMULA The second term cn be estimted in the sme mnner s Q, λ using Lemm Denote by ˆbn λ := the Fourier coefficients of p, λ Then p, λe πin d 3 Q II c, λ, µ W ψ + λ, ψ λ + γ + ˆb n λ n= kλ ω πn + kλ + ω πn + c ˆb n λ εµ W ψ + λ, ψ λ + γ n= using tht kµ R nd kλ C + To estimte Q I we shll need the following lemm: Lemm Let ε, β >, then there eists c ε, β, γ such tht for every ξ nd ξ such tht Im ξ, ξ ε, ξ R, ξ ε, Re ξ ξ βim ξ nd for every holds: eiξ e iξ t e iξ t t + γ < c ε, β, γ + γ Proof Integrting by prts we get e iξ e iξ t e iξ t = ieiξ ξ ξ + eiξξ t + γ iξ ξ iξ + γ + Consider the new constnt i ξ + + γeiξ ξ ξ γ iξ ξ γeiξ iξ c 3 γ := m γ e e iξt t + γ+ e iξ t e iξ t t + γ+

12 PAVEL KURASOV AND SERGEY SIMONOV For every nd ξ considered, Im ξ γ e Im ξ c 3γe Im ξ + γ ec 3γ + γ Using tht which is equivlent to Re ξ ξ βim ξ, ξ ξ β + Im ξ, nd the integrl cn be estimted s eiξ e iξ t e iξ t t + γ ec 3γ β + + ε + γ ε + γ + γ ε eiξ e iξ t e iξ t t + γ+ The lst term cn be split into three prts s follows: eiξ e iξ t e iξ t t + γ+ e Im ξ [ Im ξ + + Im ξ Let us estimte these three integrls seprtely e Im ξ Im ξ e iξ t e iξ t = e Im ξ Im ξ t + γ+ Introduce the constnt c 4 β := Since for the first intervl m β + e iξ ξ t ξ ξ Im ξ β +, ] e iξ t e iξ t t + γ+ e iξ ξ t t + γ+

13 we hve: e Im ξ Im ξ WEYL-TITCHMARSH FORMULA 3 e iξ t e iξ t e Im ξ Im ξ t + γ+ ec 3γc 4 β β + Im ξ γ + γ Im ξ c 4 β β + Im ξ t t + γ+ t + γ = ec 3γc 4 β β + + Im ξ γ Im ξ γ + γ γ For Im ξ e Im ξ with 3 Im ξ we hve e iξ t e iξ t e Im ξ For < Im ξ e Im ξ t + γ+ Im ξ γ ec 3 γc 4 β β + γ + γ e Im ξ t + e Im ξ e Im ξ t + γ+ the integrl is negtive e iξ t e iξ t t + γ+ t γ+ ec 3 γ γ+ γ + γ Im ξ γ+ < t + γ+ γ + γ Combining these estimtes we get eiξ e iξ t e iξ t t + γ+ < c ε, β, γ + γ c ε, β, γ := ec 3γ β + + ε ε + γ γ ec 3 γc 4 β β + + γ+ ec 3 γ + γ+ ε γ γ γ This completes the proof of the lemm

14 4 PAVEL KURASOV AND SERGEY SIMONOV Let us continue to estimte Q I Q I, λ, µ = + n= e + n= cˆb n λe iδ+iω+πin iw ψ + λ, ψ λ i kλ Applying Lemm we get ω πn t e t + γ cˆb n λe iδiω+πin iw ψ + λ, ψ λ kλ e i kλ ω ei πn kµ i +ω πn t e 4 Q I, λ, µ c c εµ, β, γ W ψ + λ, ψ λ ω πn t kλ +ω ei πn kµ i +ω πn t t + γ + n= ˆb n λ + γ Therefore combining the estimtes, 4 nd 3 the mtri Q cn be estimted s follows Q, λ, µ 4 + ε µ n= + γ c W ψ + λ, ψ λ b n λ + + εµ + c εµ, β, γ ˆb n λ n= Let us estimte now the Fourier coefficients For n we hve: b n λ = Thus p +, λe πin d = 4π n 5 b n λ 4π n p +, λ d p +, λ e πin d In terms of the corresponding Bloch solution the second derivtive of p + is p +, λ = e ikλ ψ +, λψ +, λ k λ ψ +, λ +ψ +, λ 4ikλ ψ +, λψ +, λ

15 WEYL-TITCHMARSH FORMULA 5 Let us estimte the norm in L ; of the function ψ +, λ From the eqution ψ +, λ = q λψ +, λ, we see tht ψ +, λ L ; q L ; + λ m [;] ψ +, λ Since the functions e ikλ, ψ +, λ, ψ +, λ re continuous in both vribles on the set [; ] Uβ, µ nd hence ttin their mimums, we hve: the integrl is bounded on Uβ, µ For n=, b λ = p +, λ d p +, λd = e ikλ ψ +, λd nd is lso bounded The sme rgument is vlid for ˆb n Finlly we see tht there eists c 5 β, µ such tht for every λ Uβ, µ b n λ, ˆb n λ c 5β, µ n + The Wronskin W ψ + λ, ψ λ does not hve zeros in Uβ, µ Also in the formul for the derivtive of Q, Q c sinω + δ p, λ, µ = +, λ + γ W ψ + λ, ψ λ p, λ [ ikλ ] Q, λ, µ,, ikλ the functions ± kλ, ±c sinω + δp ±, λ re bounded for ; λ [; + Uβ, µ Hence there eists c β, µ, γ such tht for every λ Uβ, µ nd the following estimtes hold Q, λ, µ, Q, λ, µ c β, µ, γ + γ This completes the proof of the theorem

16 6 PAVEL KURASOV AND SERGEY SIMONOV Let us study the properties of the reminder [ 6 R, λ, µ := e Q,λ,µ ikλ ikλ c sinω + δ p+, λp +, λ p +, λ + γ W ψ + λ, ψ λ p, λ p +, λp, λ +R, λ ] e Q,λ,µ ikλ ikλ c sinω + δ p+, λp, λ + γ W ψ + λ, ψ λ p +, λp, λ e Q,λ,µ e Q,λ,µ Lemm 3 The reminder R given by 6 possesses the following properties: R L ; nd the integrl 7 R, λ, µ d converges uniformly with respect to λ Uβ, µ R, µ, µ = R, µ, µ, R, µ, µ = R, µ, µ Proof The first ssertion follows directly from Theorem The property of mtrices from the second ssertion is preserved under summtion nd multipliction of such mtrices s well s tking the inverse We cn see from nd 8 tht R, µ nd Q, µ, µ possess this conjugtion property Therefore Q, µ, µ, e Q,µ,µ, e Q,µ,µ nd finlly R, µ, µ from 6 lso possess this property It should be tken into ccount tht W ψ + µ, ψ µ is pure imginry Let us denote ν, λ := ikλ c sinω + δp +, λp, λ + γ W ψ + λ, ψ λ Finlly we obtin system of the Levinson form: [ ] ν, λ 8 ṽ = + R ν, λ, λ, µ ṽ

17 WEYL-TITCHMARSH FORMULA 7 4 A Levinson-type theorem for systems In this section, we prove two sttements tht give uniform estimte nd symptotics of solutions to certin differentil systems The pproch is the sme s for the Levinson theorem [8], but the difference is tht we re interested in properties of solution with given initil condition Consider the system 9 u = [ λ λ ] + R u for, where u is two-dimensionl vector function nd R is mtri with comple entries 3 Lemm 4 Assume tht Rt < nd tht there eists constnt M such tht for every y it holds y Re λt M Then every solution u to 9 stisfies the estimte 3 u u e Re λt + e 4M e +e 4M Rt Proof First trnsform the system 9 by vrition of prmeters Denote λ Λ := λ nd tke 3 u = e Λt u, or u := e Λt u After substitution 9 becomes 33 u = e Λt Ru Integrting this from to nd returning bck to the function u on the left-hnd side we get 34 u = e Λt u + e t Λsds Rtu t Now multiply this epression by e Λsds nd denote 35 u 3 := e Λt u

18 8 PAVEL KURASOV AND SERGEY SIMONOV We get the following eqution for u 3 considered in L, ; C 36 u 3 = e λsds u + e t λsds Rtu 3 t The norm of the opertor V, V : u e t λsds Rtut is bounded by V + e 4M Rt nd the norm of the j-th power is bounded by Hence nd V j + e 4M Rt j j! u 3 = I V e λsds u u 3 L ;,C ep + e 4M Returning to u, we rrive t the estimte 3 Rt + e 4M u The second lemm sttes the symptotics of the solution Lemm 5 Let tht ll conditions of Lemm 4 be stisfied, then the following symptotics hold: If 37 Re λt < +, then every solution u of 9 hs the following symptotics: e u = λsds e λsds [u e + ] t λsds e t Rtu λsds t + o s 38 If Re λt = +,

19 [ u = e λsds WEYL-TITCHMARSH FORMULA 9 then every solution u of 9 hs the following symptotics: u + e t λsds Rtu t ] + o s Proof Asymptotics Consider the function u given by 3 nd integrte 33: 39 u = u + Since for every y we hve the estimte y Re λsds y Re λs ds e t Λsds Rtu t Re λs ds Re λsds + M, the eponent under the integrl in 39 is bounded The solution u t is lso bounded in this cse due to Lemm 4 Hence the integrl in 39 converges s nd there eists lim u = u + e t Λsds Rtu t Returning to u we obtin the nswer Asymptotics Consider the function u 3 given by 35 nd the corresponding eqution 36 It follows from Lemm 4 tht u 3 t is bounded, nd hence Lebesgue s dominted convergence theorem implies tht the following limit eists [ ] lim u 3 = u + Rtu 3 t This is equivlent to the nnounced symptotics for u 5 Asymptotics for the solution ϕ nd Weyl-Titchmrsh type formul In this section, we obtin the symptotics for the solution ϕ α, λ nd prove the Weyl-Titchmrsh type formul for the opertor L α Consider the set Uβ := Uβ, µ µ σl per\{λ n,µ n,λ + n,λ n,n }

20 PAVEL KURASOV AND SERGEY SIMONOV tht belongs to C + nd contins σl per \{λ n, µ n, λ + n, λ n, n } s prt of its boundry The number β is rbitrry here Theorem 3 Let ω π of the Cuchy problem ϕ α, λ + ϕ α, λ = sin α, ϕ α, λ = cos α / Z nd q L R +, then the solution ϕ α q + c sinω+δ + q + γ ϕ α, λ = λϕ α, λ, hs the following symptotics For every λ Uβ there eists A α λ such tht If λ C + Uβ, then ϕ α, λ = A α λψ, λ + o e Im kλ, ϕ α, λ = A α λψ, λ + o e Im kλ s If λ σl per \{λ n, µ n, λ + n, λ n, n }, then ϕ α, λ = A α λψ, λ + A α λψ +, λ + o, ϕ α, λ = A α λψ, λ + A α λψ +, λ + o s The function A α is nlytic in the interior of Uβ nd hs boundry vlues on σl per \{λ n, µ n, λ + n, λ n, n } Proof We re going to omit the inde α since the vlue of the boundry prmeter is fied throughout this proof According to 8 nd we write 4 ϕ ϕ = ψ, λ ψ +, λ ψ, λ ψ +, λ e ikλ e ikλ v ϕ, λ This is the definition of v ϕ, solution of corresponding to ϕ Let us fi the point µ σl per \{λ n, µ n, λ + n, λ n, n } nd consider λ Uβ, µ The function ṽ ϕ, λ, µ := e Q,λ,µ v ϕ, λ,

21 WEYL-TITCHMARSH FORMULA is solution to 8 corresponding to ϕ Let us see tht conditions of Lemm 4 re stisfied for the system 8 uniformly with respect to λ Uβ, µ First of ll we hve estimte 3 from Lemm 3 nd Re ν, λ = Im kλ Re c sinω + δp+, λp, λ + γ W ψ + λ, ψ λ Estimting the second term in the sme wy s in Theorem we hve: y Re c sinωt + δp +t, λp t, λ t + γ W ψ + λ, ψ λ c π W ψ + λ, ψ λ b n λ ω + n + ω n +, γ π π where n= bn λ := p +, λp, λe πin d re Fourier coefficients for p +, λp, λ Anlogously to 5 we hve: b n λ ψ 4π n +, λψ, λ d So there eists c 6 β, µ such tht for every λ Uβ, µ nd n while b n λ c 6β, µ n, b λ c 6 β, µ Eventully there eists c 7 β, µ such tht y Re c sinωt + δp +t, λp t, λ t + γ W ψ + λ, ψ λ c 7β, µ for every y nd λ Uβ, µ Thus we cn tke Mλ c 7 β, µ for these vlues of λ Lemm 4 gives the estimte 4 ṽ ϕ, λ, µ ṽ ϕ, λ, µ e Im kλ c8 β, µ, where c 8 β, µ := + e 4c 7β,µ ep + e 4c 7β,µ m λ Uβ,µ R t, λ

22 PAVEL KURASOV AND SERGEY SIMONOV Conditions of Lemm 5 re lso stisfied: 37 holds for λ R Uβ, µ nd 38 holds for λ C + Uβ, µ So Lemm 5 gives the following symptotics: + for λ C + Uβ, µ, ṽ ϕ, λ, µ = e ikλ c sinωt+δp + t,λp t,λ t+ γ W ψ + λ,ψ λ [ e ikλ t + t c sinωs+δp + s,λp s,λds s+ γ W ψ + λ,ψ λ R t, λ, µṽ ϕ t, λ, µ for λ = µ, 4 ṽ ϕ, µ, µ = ṽ ϕ, µ, µ + eikµ c sinωt+δp + t,µp t,µ t+ γ W ψ + µ,ψ µ c sinωt+δp + t,µp t,µ t+ γ W ψ + µ,ψ µ e ikµ + t eikµ + t c sinωs+δp + s,µp s,µds s+ γ W ψ + µ,ψ µ ṽ ϕ, λ, µ ] + o, e ikµ t t c sinωs+δp + s,µp s,µds s+ γ W ψ + µ,ψ µ R t, µ, µṽ ϕ t, µ, µ + o ] Since Q, λ, µ = O, we cn denote for λ Uβ, µ: + γ Aλ, µ :=, e c sinωt+δp + t,λp t,λ [ t+ γ W ψ + λ,ψ λ e Q,λ,µ v ϕ, λ ] + e ikλ t + t c sinωs+δp + s,λp s,λds s+ γ W ψ + λ,ψ λ R t, λ, µe Qt,λ,µ v ϕ t, λ where <, > stnds for the sclr product in C nd hve: 43 lim v ϕ, λe ikλ Aλ, µ = From this we see tht the coefficient Aλ, µ does not depend on µ, so we will denote it by Aλ Reltion 4 cn be written s 44 v ϕ, λ = ψ +, λϕ, λ ψ +, λϕ, λ W ψ + λ, ψ λ ϕ, λψ, λ ϕ, λψ, λ so v ϕ, is nlytic in C + nd continuous up to σl per \{λ n, µ n, n } From the estimte 4 nd properties of Q, λ, µ nd R, λ, µ given by Theorem nd Lemm 3 it follows tht Aλ is continuous in,

23 WEYL-TITCHMARSH FORMULA 3 Uβ, µ nd nlytic in its interior Thus A is nlytic in the interior of Uβ hving non-tngentil boundry limits on σl per \{λ n, µ n, λ + n, λ n, n } tht coincide with its vlues on this set The solution ϕ, λ nd its derivtive re rel if λ is rel Thus 44 shows tht the upper nd the lower components of the vector v ϕ, λ re comple conjugte for λ σl per \{λ n, µ n, n } This property is preserved if we multiply the vector by mtri X such tht X = X, X = X, like 7 It follows from Lemm 3 tht the upper nd the lower components of the vectors in the equlity 4 re comple conjugte to ech other Hence for λ = µ we hve Aµe ikµ 45 v ϕ, µ = + o s Aµe ikµ The symptotics of the solution ϕ nd its derivtive follows from 4, 43 nd 45 Using the obtined symptotics both on the spectrum nd in C + we now prove the Weyl-Titchmrsh type formul Theorem 4 Let ω / Z nd q π L R +, then for lmost ll λ σl per the spectrl density of the opertor L α, defined by, is given by ρ αλ = π W ψ + λ, ψ λ A α λ, where A α is the sme s in Theorem 3 Proof In ddition to ϕ α consider nother one solution of 7, to be denoted by θ α := ϕ α+ π, stisfying the initil conditions θ α, λ = cos α, θ α, λ = sin α The Wronskin of ϕ α nd θ α is equl to one Theorem 3 yields for λ Uβ C +, θ α, λ = A α+ π λψ, λ + o e Im kλ s Since the opertor L α is in the limit point cse, the combintion θ α + m α ϕ α belongs to L, where m α is the Weyl function for L α It hs the symptotics θ α, λ+m α λϕ α, λ = A α+ π λ+m αa α λψ, λ+oe Im kλ

24 4 PAVEL KURASOV AND SERGEY SIMONOV Therefore for λ Uβ C + nd m α λ = A α+ π λ A α λ m α λ + i = A α+ π λ A α λ for λ σl per \{λ n, µ n, λ + n, λ n, n } It follows from the subordincy theory [] tht the spectrum of L α on this set is purely bsolutely continuous nd 46 ρ αλ = π Im m αλ + i = A αλa α+ π λ A αλa α+ π λ πi A α λ Theorem 3 yields for these vlues of λ: θ α, λ = A α+ π λψ, λ + A α+ π λψ +, λ + o, θ α, λ = A α+ π λψ, λ + A α+ π λψ +, λ + o, s Substituting these symptotics nd the symptotics of ϕ α nd ϕ α into the epression for the Wronskin we get: = A α λa α+ π λ A αλa α+ π λw ψ +λ, ψ λ the term o cncels, since both sides re independent of Combining with 46 we hve ρ αλ = πiw ψ + λ, ψ λ A α λ which completes the proof = π W ψ + λ, ψ λ A α λ, 6 Acknowledgements The second uthor epresses his deep grtitude to Professor SN Nboko for his constnt ttention to this work nd for mny fruitful discussions on the subject nd lso to the Mthemtics Deprtment of Lund Institute of Technology for finncil support nd hospitlity The work ws supported by grnts RFBR , INTAS nd Swedish Reserch Council 8554

25 WEYL-TITCHMARSH FORMULA 5 References [] S Albeverio On bound sttes in the continuum of N-body systems nd the viril theorem Ann Physics, 7:67 76, 97 [] H Behncke Absolute continuity of hmiltonins with von Neumnn Wigner Potentils I Proceedings of the Americn Mthemticl Society, : , 99 [3] H Behncke Absolute continuity of hmiltonins with von Neumnn Wigner potentils II Mnuscript Mthemtic, 7:63 8, 99 [4] H Behncke The m-function for Hmiltonins with Wigner-von Neumnn potentils Journl of Mthemticl Physics, 354:445 46, 994 [5] BM Brown, MSP Esthm, nd DKR McCormck Absolute continuity nd spectrl concentrtion for slowly decying potentils Journl of Computtionl nd Applied Mthemtics, 94:8 97, 998 rxiv:mth/9855v [6] VS Buslev nd VB Mtveev Wve opertors for the Schrödinger eqution with slowly decresing potentil Theoreticl nd Mthemticl Physics, 3:66 74, 97 [7] F Cpsso, C Sirtori, J Fist, DL Sivco, SNG Chu, nd AY Cho Observtion of n electronic bound stte bove potentil well Nture, 358: , 99 [8] EA Coddington nd N Levinson Theory of ordinry differentil equtions McGrw-Hill, New York, 955 [9] MSP Esthm The spectrl theory of periodic differentil equtions Edinburgh, 973 [] DJ Gilbert nd DB Person On subordincy nd nlysis of the spectrum of one-dimensionl Schrödinger opertors Journl of mthemticl nlysis nd pplictions, 8:3 56, 987 [] WA Hrris nd DA Lutz Asymptotic integrtion of dibtic oscilltors J Mth Anl Appl, 5:76 93, 975 [] DB Hinton, M Klus, nd JK Shw Embedded hlf-bound sttes for potentils of Wigner-von Neumnn type Proceedings of the London Mthemticl Society, 33:67 646, 99 [3] M Klus Asymptotic behvior of Jost functions ner resonnce points for Wigner-von Neumnn type potentils Journl of Mthemticl Physics, 3:63 74, 99 [4] P Kursov Zero-rnge potentils with internl structures nd the inverse scttering problem Letters in Mthemticl Physics, 54:87 97, 99 [5] P Kursov Scttering mtrices with finite phse shift nd the inverse scttering problem Inverse Problems, 3:95 37, 996 [6] P Kursov nd S Nboko Wigner-von Neumnn perturbtions of periodic potentil: spectrl singulrities in bnds Mthemticl Proceedings of the Cmbridge Philosophicl Society, 4:6 83, 7 [7] VB Mtveev Wve opertors nd positive eigenvlues for Schrödinger eqution with oscillting potentil Theoreticl nd Mthemticl Physics, 53: , 973 [8] VB Mtveev nd MM Skrignov Scttering problem for rdil Schrödinger eqution with slowly decresing potentil Theoreticl nd Mthemticl Physics, :56 64, 97

26 6 PAVEL KURASOV AND SERGEY SIMONOV [9] J von Neumnn nd EP Wigner Über merkwürdige diskrete Eigenwerte Z Phys, 3: , 99 Deprtment of Mthemtics, Lund Institute of Technology, Lund, Sweden, E-mil ddress: kursov@mthslthse Deprtment of Mthemticl Physics, Institute of Physics, St Petersburg University, Ulinovski, St Petergoff, St Petersburg, Russi, 9894 E-mil ddress: sergey simonov@milru