A commutator method for the diagonalization of Hankel operators
aa r X i v : . [ m a t h . SP ] S e p A COMMUTATOR METHOD FOR THE DIAGONALIZATIONOF HANKEL OPERATORS
D. R. YAFAEV
To the memory of Mikhail Shl¨emovich Birman
Abstract.
We present a method for the explicit diagonalization of some Han-kel operators. This method allows us to recover classical results on the diag-onalization of Hankel operators with the absolutely continuous spectrum. Itleads also to new results. Our approach relies on the commutation of a Hankeloperator with some differential operator of second order. Introduction
Hankel operators can be defined (see, e.g., book [8]) as integral operatorsin the space L ( R + ) whose kernels depend on the sum of variables only. Thus, aHankel operator A is defined by the formula( Af )( x ) = Z ∞ a ( x + y ) f ( y ) dy. (1.1)Of course, A is self-adjoint if a = ¯ a . If Z ∞ | a ( x ) | xdx < ∞ , then A belongs to the Hilbert-Schmidt class. This condition is satisfied if, forexample, the function a is continuous, it is not too singular at x = 0 and decayssufficiently rapidly as x → ∞ . On the contrary, if a ( x ) ∼ a x − as x → a ( x ) ∼ a ∞ x − as x → ∞ , then the operator A is no longer compact although itremains bounded. A general philosophy (see paper [4] by J. S. Howland) is thateach of these singularities gives rise to the branch [0 , a π ] or (and) [0 , a ∞ π ] of thesimple absolutely continuous spectrum.There are very few examples where the operator A can be explicitly diagonalized,that is its exact eigenfunctions can be found. The first result is due to F. Mehler[6] who considered the case a ( x ) = ( x + 2) − . He has shown that functions ψ k ( x ) = (cid:0) k tanh πk (cid:1) / P − / ik ( x + 1) , λ = π/ cosh πk, k > , (1.2)where P − / ik is the Legendre function (see [3], Chapter 3), satisfy equations Aψ k = λψ k . The functions ψ k are usually parametrized by the quasimomentum k related to λ = λ ( k ) by formula (1.2). The operator U : L ( R + ) → L ( R + ) defined Mathematics Subject Classification.
Key words and phrases.
Hankel operators, spectrum and eigenfunctions, explicit solutions,commutators. A precise definition of the operator U can be given in terms of the corresponding sesquilinearform. by the equality ( U f )( k ) = Z ∞ ψ k ( x ) f ( x ) dx (1.3)is unitary. Observe that λ ( k ) is a one-to-one mapping of R + on (0 , π ) and that( U Lf )( k ) = λ ( k ) f ( k ) which implies that the spectrum of the operator A is simple,absolutely continuous and coincides with the interval [0 , π ].Below we use the term “eigenfunction” for ψ k (although ψ k L ( R + )) suchthat Aψ k = λψ k for the spectral parameter λ from the continuous spectrum ofthe operator A . By definition , we also say that eigenfunctions ψ k of the continu-ous spectrum are orthogonal, normalized and the set of all ψ k is complete if thecorresponding operator (1.3) is unitary (if A has no point spectrum).The next result is due to W. Magnus [5] who considered the case a ( x ) = x − e − x/ . A more general result of the same type was obtained by M. Rosenblum[9] who has diagonalized the operator A with kernel a ( x ) = Γ(1 + β ) x − W − β, / ( x ) , β ∈ R , β = − , − , . . . , (1.4)where W − β, / is the Whittaker function (see [3], Chapter 6) and Γ is the gammafunction. Note that W , / ( x ) = e − x/ . The spectrum of the operator A with suchkernel is again simple and, up to a finite number of eigenvalues, it is absolutelycontinuous and coincides with the interval [0 , π ]. Its “normalized eigenfunctions”are expressed in terms of the Whittaker functions ψ k ( x ) = (2 π ) − p k | Γ(1 / − ik + β ) | sinh 2 πkx − W − β,ik ( x ) , k > . (1.5)Observe that the function a ( x ) = ( x + 2) − is singular at x = ∞ and eigenfunc-tions (1.2) decay as linear combinations of x − / ± ik as x → ∞ while function (1.4)is singular at x = 0 and eigenfunctions (1.5) behave as linear combinations of thesame functions x − / ± ik as x → a ( x ) = x − where the operator A is directly diago-nalized (see paper [2] by T. Carleman) by the Mellin transform. In this case thespectrum of A has multiplicity 2 (because of the singularities of a ( x ) both at x = 0and at x = ∞ ), it is absolutely continuous and coincides with the interval [0 , π ]. Theeigenfunctions of the Carleman operator equal x − / ± ik (up to a normalization).We emphasize a parallelism of theories of singular differential operators andHankel operators with singular kernels. Thus, the functions x − / ± ik play (bothfor x → ∞ and x →
0) for Hankel operators the role of exponential functions e ± ikx for differential operators of second order. From this point of view, the Carlemanoperator plays the role of the operator − d /dx in the space L ( R ). In the author’s opinion, the reason why in the cases described above eigen-functions of a Hankel operator can be found explicitly remained unclarified. Ourapproach shows that all diagonalizable Hankel operators A commute with differen-tial operators L = − ddx ( x + γx ) ddx + αx + βx (1.6)for suitably chosen parameters α ≥ , β ∈ R and γ ≥
0. Thus, eigenfunctions ofthe operators A and L are the same which allows us to diagonalize the operator A .Hopefully the commutator method will be applied to other kernels a . In thispaper we use the commutator method to find in subs. 4.4 eigenfunctions of a new IAGONALIZATION OF HANKEL OPERATORS 3
Hankel operator with kernel a ( x ) = r x K ( √ x ) , (1.7)where K is the MacDonald function (see [3], Chapter 7). Similarly to (1.4), thisfunction decays exponentially as x → ∞ and a ( x ) ∼ x − as x →
0. An exampleof a different nature are Hankel operators with regular kernels; such operators arecompact.Note that operator (1.6) for γ = 0 and α > β ) Z ∞ ( x + y ) − W − β, / ( x + y ) y − W − β,ik ( y ) dy = π cosh πk x − W − β,ik ( x )(1.8)found earlier by H. Shanker in [10]. This identity shows that functions (1.5) areeigenfunctions of the Hankel operator with kernel (1.4). M. Rosenblum observedthat functions (1.5) are also eigenfunctions of operator (1.6) for γ = 0 and α = 1 / L are orthogonal andcomplete, the same is true for eigenfunctions of the Hankel operator A with kernel(1.4). This yields the diagonalization of this operator.Our approach is somewhat different. We prove the relation LA = AL whichshows that eigenfunctions of the operators L and A are the same. In particular, weobtain identity (1.8) without a recourse to the theory of special functions.It is well-known that the integrability of differential equations of second orderin terms of special functions has a deep group-theoretical interpretation (see, e.g.,book [12] by N. Ya. Vilenkin). As far as Hankel operators are concerned, it isevident that the diagonalization of the Carleman operator can be explained by itsinvariance with respect to the group of dilations. The relation LA = AL meansthat the operator A is invariant with respect to the group exp( − itL ). In contrastto the Carleman operator, for other Hankel operators this invariance does not lookobvious.A commutator scheme is presented in Section 2 while specific examples of kernelssingular at x = ∞ and x = 0 are discussed in Sections 3 and 4, respectively. Hankeloperators with regular kernels are considered in Section 5.2. Commutator method
For a moment, we consider the operator L defined by formula (1.6) as adifferential operator on the class C ( R + ), but later it will be defined as a self-adjointoperator in the space L ( R + ). Let the operator A be given by formula (1.1) where a ∈ C ( R + ).Let us commute the operators A and L . Suppose that f ∈ C ( R + ) and thatlim y → ( y + γy ) f ( y ) = lim y → ( y + γy ) f ′ ( y ) = 0 (2.1)as well as lim y →∞ a ′ ( x + y )( y + γy ) f ( y ) = lim y →∞ a ( x + y )( y + γy ) f ′ ( y ) = 0 (2.2)for all x ≥
0. Then integrating by parts, we find that(
ALf )( x ) = Z ∞ (cid:16) − ∂∂y (cid:0) ( y + γy ) a ′ ( x + y ) (cid:1) + a ( x + y )( αy + βy ) (cid:17) f ( y ) dy. D. R. YAFAEV
It follows that (( LA − AL ) f )( x ) = Z ∞ q ( x, y ) f ( y ) dy where q ( x, y ) = − ∂∂x (cid:0) ( x + γx ) a ′ ( x + y ) (cid:1) + ∂∂y (cid:0) ( y + γy ) a ′ ( x + y ) (cid:1) + ( αx − αy + βx − βy ) a ( x + y )=( x − y ) (cid:16) − ( z + γ ) a ′′ ( z ) − a ′ ( z ) + ( αz + β ) a ( z ) (cid:17) and z = x + y . Thus, we arrive at the following general result. Theorem 2.1.
Suppose that kernel a of a Hankel operator A satisfies the differen-tial equation − ( x + γ ) a ′′ ( x ) − a ′ ( x ) + ( αx + β ) a ( x ) = 0 . (2.3) Let f ∈ C ( R + ) and let conditions (2.1) and (2.2) hold. Then ( LA − AL ) f = 0 . (2.4)Note that after a change of variables a ( x ) = ( x + γ ) − b ( x + γ ) (2.5)in (2.3), we get the Schr¨odinger equation with the Coulomb potential − b ′′ ( r ) + ( α + βr − ) b ( r ) = 0 . (2.6) In specific examples below, we are going to use Theorem 2.1 in the followingway. If L is self-adjoint and has a simple spectrum, then the equality LA = AL shows that A is a function F of L , i.e., the operators A and L have commoneigenfunctions. For a calculation of the function F , we argue as follows. Supposethat a function ψ µ satisfies conditions (2.1), (2.2) and the equation − (cid:0) ( x + γx ) ψ ′ µ ( x ) (cid:1) ′ + ( αx + βx ) ψ µ ( x ) = µψ µ ( x ) . (2.7)Then according to equality (2.4) the same equation holds for the function Aψ µ andhence, for some numbers λ = λ µ and ˇ λ = ˇ λ µ ,( Aψ µ )( x ) = λψ µ ( x ) + ˇ λ ˇ ψ µ ( x ) (2.8)where ˇ ψ µ is a solution of the equation L ˇ ψ µ = µ ˇ ψ µ linearly independent of ψ µ .Further, comparing asymptotics of the functions ψ µ ( x ), ˇ ψ µ ( x ) and ( Aψ µ )( x ) as x → x → ∞ , we see that ˇ λ = 0 and find λ = F ( µ ) as a function of µ .Finally, if ψ µ belong to the domain of some self-adjoint realization of the differentialoperator L , then, for a proper normalization of functions ψ µ , the system of all ψ µ is orthogonal and complete. In this case A = F ( L ). Note that this approach allowsone to avoid precise definitions of commutators and references to the functionalanalysis.It turns out that in all our applications F ( µ ) = π/ cosh (cid:0) π p µ − / (cid:1) , and hence A = π/ cosh (cid:0) π p L − / (cid:1) . Actually, it is somewhat more convenient to parametrize eigenfunctions by thequasimomentum k > µ and λ by the formulas µ = k + 1 / ∈ (1 / , ∞ ) , λ = π/ cosh πk ∈ (0 , π ) . (2.9) IAGONALIZATION OF HANKEL OPERATORS 5
Note that ψ k ( x ), ψ µ ( x ) and ψ λ ( x ) denote the same function provided the parame-ters k , µ and λ are related by formulas (2.9).The operator U defined by formula (1.3) is unitary and the operator U AU ∗ actsin L ( R + ) as multiplication by the function λ ( k ) = π/ cosh πk . Indeed, according tothe Fubini theorem it follows from the equation Aψ k = λ ( k ) ψ k that for g ∈ C ∞ ( R + )( AU ∗ g )( x ) = Z ∞ dkg ( k ) Z ∞ dya ( x + y ) ψ k ( y )= Z ∞ λ ( k ) ψ k ( x ) g ( k ) dk = ( U ∗ ( λg ))( x ) , or equivalently ( U Af )( k ) = λ ( k )( U f )( k ) , ∀ f ∈ L ( R + ) . (2.10)Since λ : R + → (0 , π ) is a smooth one-to-one mapping, the operator A has thesimple absolutely continuous spectrum [0 , π ].To realize this scheme, it is convenient to study the cases of singularities at x = ∞ when γ > x = 0 when γ = 0 separately.3. Singularity at infinity
Set γ = 2. We first suppose that α = β = 0. Then the function a ( x ) =( x + 2) − satisfies equation (2.3), and the corresponding operator L = − ddx p ( x ) ddx where p ( x ) = x + 2 x. Let P ν ( z ) and Q ν ( z ) be the Legendre functions (see, e.g., [3], Ch. 3) of the firstand second kinds, respectively. They are defined as solutions of the equation(1 − z ) u ′′ ( z ) − zu ′ ( z ) + ν ( ν + 1) u ( z ) = 0 , z > , satisfying the conditions P ν (1) = 1 and Q ν ( z ) = − − ln( z −
1) + c ν as z → c ν is inessential). Then the functions P − / ik ( x + 1) and Q − / ik ( x + 1) satisfy the equation Lu = ( k + 1 / u . We also note that (seeformulas (2.10.2) and (2.10.5) of [3]) P − / ik ( x + 1) = m ( k ) x − / ik + m ( k ) x − / − ik + O ( x − / ) , x → ∞ , (3.1)where m ( k ) = Γ( ik ) √ π Γ(1 / ik ) 2 ik . (3.2)The operator L is symmetric in the space L ( R + ) on the domain C ∞ ( R + ),but it is not essentially self-adjoint. Since both functions P − / ik ( x + 1) and Q − / ik ( x +1) belong to L in a neighborhood of the point x = 0, the defect indicesof the operator L are (1 , L from C ∞ ( R + ) (itwill be also denoted by L ) is defined on the domain D ( L ) consisting of functions f ( x ) from the Sobolev class H loc ( R + ) satisfying the boundary conditions ∃ lim x → f ( x ) , f ′ ( x ) = o ( x − / ) , x → , (3.3)(we call these boundary conditions regular); it is also required that f ∈ L ( R + ) and Lf ∈ L ( R + ). Actually, the direct integration by parts shows that the operator L is symmetric. Furthermore, using the appropriate Green function, we find that forall h ∈ L ( R + ) the equation ( pf ′ ) ′ = h has a solution satisfying condition (3.3). D. R. YAFAEV
Thus the image of the operator L coincides with L ( R + ), and hence L is self-adjoint(cf. § For a study of the operator L , it is convenient to make a standard (see,e.g., book [11] by E. C. Titchmarsh) change of variables. Set t = ω ( x ) = Z x p ( y ) − / dy and f ( x ) = ω ′ ( x ) / ˜ f ( ω ( x )) =: ( F ˜ f )( x ) . (3.4)The operator F is unitary in the space L ( R + ), and the operator e L = F − LF actsby the formula e L = − d /dt + q ( η ( t )) where q ( x ) = − − p ( x ) − p ′ ( x ) + 4 − p ′′ ( x )and η = ω − is the inverse function to ω (so that x = η ( t )).In the case p ( x ) = x + 2 x we have ω ( x ) = 2 ln (cid:0) x / + ( x + 2) / (cid:1) − ln 2 (3.5)and hence e L = − d dt + ˜ q ( t ) + 1 / , (3.6)where ˜ q ( t ) = − − (cid:0) η ( t ) + 2 η ( t ) (cid:1) − . Since ω ( x ) = (2 x ) / + O ( x ) as x → ω ( x ) = ln(2 x )+ O ( x − ) as x → ∞ , we see that η ( t ) ∼ t / t → η ( t ) ∼ e t / t → ∞ . It follows that ˜ q ( t ) ∼ − (4 t ) − as t → q ( t ) = O ( e − t ) as t → ∞ .Note that the operator e L is self-adjoint on the domain D ( e L ) consisting of functions˜ f ( t ) from the Sobolev class H loc ( R + ) satisfying the boundary conditions ∃ lim t → t − / ˜ f ( t ) , ˜ f ′ ( t ) − (2 t ) − ˜ f ( t ) = o ( t / ) , t → , (3.7)and such that ˜ f ∈ L ( R + ), e L ˜ f ∈ L ( R + ).All usual results of spectral and scattering theories can be applied to the oper-ator e L and then used for the operator L . The operator e L has a simple absolutelycontinuous spectrum coinciding with the interval [1 / , ∞ ). It does not have eigen-values because the equations e L ˜ u = µ ˜ u , or equivalently Lu = µu , for µ ∈ R donot have solutions from L ( R + ) satisfying the regular boundary conditions at zero.The diagonalization of the operator e L can be constructed (see, e.g., [11, 13]) in thefollowing way. Let ˜ u k ( t ), k >
0, be a real-valued solution of the equation e L ˜ u k = ( k + 1 / u k (3.8)satisfying boundary conditions (3.7). It has the asymptotics˜ u k ( t ) = m ( k ) e ikt + m ( k ) e − ikt + o (1) (3.9)as t → ∞ . Then the operator e U defined by the equation( e U ˜ f )( k ) = (2 π ) − / | m ( k ) | − Z ∞ ˜ u k ( t ) ˜ f ( t ) dt, (3.10)is unitary in the space L ( R + ) and ( e U e L ˜ f )( k ) = ( k + 1 / e U ˜ f )( k ).Let us now make the change of variables (3.4) and set U = F e U F − . Note that(2 π ) − / | m ( k ) | − = √ k tanh πk IAGONALIZATION OF HANKEL OPERATORS 7 for the function m ( k ) defined by equation (3.2). It follows that the operator U defined by the equation( U f )( k ) = √ k tanh πk Z ∞ P − / ik ( x + 1) f ( x ) dx, (3.11)is unitary in the space L ( R + ) and( U Lf )( k ) = ( k + 1 / U f )( k ) . (3.12) Now we return to the Hankel operator A . Observe that the function P − / ik ( x + 1) satisfies both boundary conditions (2.1) and (2.2). It follows fromTheorem 2.1 that Z ∞ ( x + y + 2) − P − / ik ( y + 1) dy = λP − / ik ( x + 1) + ˇ λQ − / ik ( x + 1) . (3.13)Considering here the limit x →
0, we see that ˇ λ = 0. Then we take the limit x → ∞ . It easily follows from (3.1) that the left-hand side of (3.13) equals2 Re (cid:16) m ( k ) Z ∞ ( x + y + 2) − y − / ik dy (cid:17) + O ( x − )= 2 Re (cid:16) m ( k ) x − / ik Z ∞ ( t + 1) − t − / ik dt (cid:17) + O ( x − ) , where we have set y = xt . Comparing this asymptotics with asymptotics (3.1) ofthe right-hand side of (3.13), we see that λ = Z ∞ ( t + 1) − t − / ik dt = π (cosh πk ) − (3.14)and hence Z ∞ ( x + y + 2) − P − / ik ( y + 1) dy = π (cosh πk ) − P − / ik ( x + 1) . (3.15)It yields equation (2.10) with the operator U defined by formula (3.11). Since theoperator U is unitary, we have recovered the result of F. Mehler [6]. Proposition 3.1.
The Hankel operator with kernel a ( x ) = ( x +2) − has the simpleabsolutely continuous spectrum coinciding with the interval [0 , π ] . Its normalizedeigenfunction corresponding to the spectral parameter λ = π (cosh πk ) − is given byformula (1.2) . We emphasize that equation (3.15) has been obtained as a direct consequence ofthe commutator method, without any use of the theory of special functions.4.
Singularity at zero
In the first three subsections we study the Hankel operator with kernel (1.4) andin subs. 4 – with kernel (1.7). In both cases a ( x ) ∼ x − as x → a ( x ) decaysexponentially as x → ∞ . The corresponding operator L is defined by formula (1.6)where γ = 0. Note that in the case γ = 0, after a change of variables ψ ( x ) = x − ϕ ( x ) We are obliged to choose regular boundary conditions at zero since the function Q − / ik ( x +1) does not satisfy the second boundary condition (2.1) D. R. YAFAEV in (2.7), we get again (cf. equation (2.6)) the Schr¨odinger equation − ϕ ′′ ( x ) + ( α + βx − − µx − ) ϕ ( x ) = 0 (4.1)with the Coulomb potential but with a non-zero orbital term. Below we set α = 1 / W − β,p ( x ) can be defined as the solution ofequation (4.1) for µ = 1 / − p such that W − β,p ( x ) = x − β e − x/ (1 + O ( x − )) (4.2)as x → ∞ . Of course, W − β, − p ( x ) = W − β,p ( x ). In particular, the function b ( x ) = W − β, / ( x ) satisfies equation (2.6) (where α = 1 / x → § W − β,ik ( x ) = m ( k ) x / ik + m ( k ) x / − ik + O ( x / ) , k > , x → , (4.3)where m ( k ) = Γ( − ik )Γ − (1 / − ik + β ) . (4.4)If p ≥ − / p + β = − , − , . . . , we have as x → W − β,p ( x ) ∼ Γ(2 p )Γ(1 / p + β ) − x / − p , p > ,W − β, ( x ) ∼ − Γ(1 / β ) x / ln x. (4.5)If − / p + β = − n where n = 1 , , . . . , then taking into account formulas(6.9.4) and (6.9.36) of [3], we can express the Whittaker functions in terms of theLaguerre polynomials: W − β,p ( x ) = ( − n − ( n − e − x/ x p +1 / L pn − ( x ) . (4.6)If γ = 0 and α = 1 /
4, then L = − ddx x ddx + x / βx. (4.7)We emphasize that the coefficient β may be arbitrary. It turns out that the strongdegeneracy of the function x at x = 0 gives rise to a branch of the absolutelycontinuous spectrum of the operator L .First, let us define L as a self-adjoint operator in the space L ( R + ). We willcheck that the operator L is essentially self-adjoint on the domain C ∞ ( R + ). Let ( F ˜ f )( x ) = x − / ˜ f (ln x ). Then the transformation F : L ( R ) → L ( R + ) is unitaryand the operator e L = F − LF acts by formula (3.6) where ˜ q ( t ) = e t / βe t .This is already a standard Sturm-Liouville operator in the space e H = L ( R ). Thepotential ˜ q ( t ) tends to 0 as t → −∞ and to + ∞ as t → + ∞ . In particular, e L isessentially self-adjoint on C ∞ ( R ) which implies that L is essentially self-adjoint on C ∞ ( R + ) in the space L ( R + ). Thus, a boundary condition at the point x = 0 isunnecessary. Since ˜ q ( t ) → ∞ as t → ∞ , a quantum particle can evade to −∞ only.This ensures that the spectrum of the operator e L is simple. The expansion over eigenfunctions of the operator e L can be performedby the following standard procedure (see, e.g., [13], § / , ∞ ), for β < − / e L has a finite number of simple eigenvalues µ , . . . , µ N , N = N ( β ), lying below the point 1 /
4. We denote by e H ( p ) the subspace Note that the integral in (3.4) diverges for p ( x ) = x and hence the definition of the operator F should be changed. IAGONALIZATION OF HANKEL OPERATORS 9 spanned by the corresponding eigenfunctions. Let ˜ u k ( t ), k >
0, be a real-valuedsolution of equation (3.8) belonging to L ( R + ). It has asymptotics (3.9) as t → −∞ with a function m ( k ) which will be calculated later. Then the operator e U : e H → L ( R + ) defined by the equation (cf. (3.10))( e U ˜ f )( k ) = (2 π ) − / | m ( k ) | − Z ∞−∞ ˜ u k ( t ) ˜ f ( t ) dt, (4.8)is bounded, e U (cid:12)(cid:12) e H ( p ) = 0, the mapping e U : e H ⊖ e H ( p ) → L ( R + ) is unitary andequation ( e U e L ˜ f )( k ) = ( k + 1 / e U ˜ f )( k )holds.The functions u k ( x ) = x − / ˜ u k (ln x ), k >
0, satisfy the equation − ( x u ′ k ( x )) ′ + 4 − x u k ( x ) + βxu k ( x ) = ( k + 1 / u k ( x ) (4.9)and can be expressed in terms of Whittaker functions: u k ( x ) = x − W − β,ik ( x ) . (4.10)It follows from (4.3) that the function ˜ u k ( t ) = e t/ u k ( e t ) has as t → −∞ asymp-totics (3.9) with the function m ( k ) defined by (4.4). Calculating | m ( k ) | and makingin (4.4) the change of variables t = ln x , we find that the operator U = F e U F − isgiven by the equation( U f )( k ) = π − √ k sinh 2 πk | Γ(1 / − ik + β ) | Z ∞ x − W − β,ik ( x ) f ( x ) dx. It is bounded, U (cid:12)(cid:12) H ( p ) = 0, the mapping U : H ⊖ H ( p ) → L ( R + ) is unitary andequation (3.12) holds. Here H ( p ) is the subspace spanned by the eigenfunctions ψ , . . . , ψ N of the operator L .Let us calculate these functions. The function u p ( x ) = x − W − β,p ( x ) for p ≥ k is played by − p . In view of (4.2) itbelongs to L at infinity. However, it follows from asymptotics (4.5) that it doesnot belong to L in a neighborhood of the point x = 0 unless − / p + β = − n where n = 1 , , . . . . Moreover, in view of (4.6) for all β = − / u L . Thus, if β ≥ − /
2, the operator L is purely absolutely continuous. If β < − /
2, it also has the eigenvalues µ n = 1 / − ( | β | + 1 / − n ) where n = 1 , , . . . and n < | β | + 1 / p > ψ n ( x ) = e − x/ x p − / L pn − ( x ) , p = | β | + 1 / − n. (4.11) Now we return to the Hankel operator A with kernel (1.4). It follows from(4.2) that a ( x ) exponentially decays as x → ∞ , and it follows from the first formula(4.5) for p = 1 / a ( x ) ∼ x − as x →
0. Observe that in view of asymptotics(4.2) and (4.3), function (4.10) satisfies both boundary conditions (2.1) and (2.2).Hence it follows from Theorem 2.1 that Z ∞ a ( x + y ) y − W − β,ik ( y ) dy = λ ( k ) x − W − β,ik ( x ) + ˇ λ ( k ) x − M − β,ik ( x ) (4.12)where the Whittaker function M − β,ik is the solution of equation (2.6) exponentiallygrowing as x → ∞ . Therefore considering the limit x → ∞ in (4.12), we see that necessarily ˇ λ ( k ) = 0. Then we take the limit x → a ( x ) ∼ x − as x →
0, we have Z ∞ a ( x + y ) y − W − β,ik ( y ) dy =2 Re (cid:16) m ( k ) Z ∞ ( x + y ) − y − / ik dy (cid:17) + O ( x / )=2 λ ( k ) Re (cid:0) m ( k ) x − / ik (cid:1) + O ( x / )where λ ( k ) is again given by formula (3.14). This yields equation (1.8).It remains to calculate eigenvalues λ , . . . , λ N of the operator A . The correspond-ing eigenfunctions are given by formula (4.11). We proceed again from equation(4.12) where the role of ik is played by p = | β | + 1 / − n . As before considering thelimit x → ∞ , we see that ˇ λ n = 0 and hence Z ∞ a ( x + y ) e − y/ y p − / L pn − ( y ) dy = λ n e − x/ x p − / L pn − ( x ) . (4.13)It follows from (1.4) and (4.2) that the left-hand side here equalsΓ(1 + β ) x − β − e − x/ Z ∞ e − y y p − / L pn − ( y ) dy (cid:0) O ( x − ) (cid:1) , x → ∞ . Putting together formulas (2.8.46) and (10.12.33) of [3], we see that( n − Z ∞ e − y y p − / L pn − ( y ) dy = Γ( p + n − / . (4.14)Recall also that L pn − ( x ) is a polynomial of degree n − − n − / ( n − x n − . Hence it follows from relation (4.13) that λ n = ( − n π/ sin πβ, n = 1 , , . . . , n < | β | + 1 / . (4.15)Since eigenfunctions of the operator L are orthogonal and complete, we haverecovered the result of M. Rosenblum [9]. Proposition 4.1.
The Hankel operator A with kernel (1.4) has the simple abso-lutely continuous spectrum coinciding with the interval [0 , π ] . Its normalized eigen-function corresponding to a point λ = π (cosh πk ) − from the continuous spectrumis given by the formula ψ k ( x ) = π − √ k sinh 2 πk | Γ(1 / − ik + β ) | x − W − β,ik ( x ) , k > . Moreover, if β < − / , then the operator A has eigenvalues (4.15) with the corre-sponding eigenfunctions defined by (4.11) . Next, we turn to the Hankel operator with singular kernel (1.7) whichprobably was not considered in the literature. Recall that the MacDonald functionis defined by the relation K p ( z ) = 2 − ie πip/ H (1) p ( iz ) where H (1) p is the Hankelfunction. Now the function b ( x ) = xa ( x ) satisfies the Schr¨odinger equation (2.3)for the zero energy α = 0 and the coupling constant β = 2. Of course, we couldhave taken arbitrary β >
0, but we have to exclude negative β since in this casethe function b ( x ) grows as x → ∞ .It follows from the well-known properties of H (1) p that function (1.7) has asymp-totics a ( x ) = 4 π / x − / e −√ x (1 + O ( x − / )) (4.16)as x → ∞ and a ( x ) ∼ x − as x → IAGONALIZATION OF HANKEL OPERATORS 11
The corresponding operator L = − ddx x ddx + 2 x can be studied quite similarly to operator (4.7). For example, the operator e L = F − LF acts by formula (3.6) where ˜ q ( t ) = 2 e t . A solution of equation (2.7) where µ = k + 1 / L at infinity can be expressed again in terms of theMacDonald function u k ( x ) = x − / K ik ( √ x ) . According to formulas (7.2.12) and (7.2.13) of [3] we have u k ( x ) = m ( k ) x − / ik + m ( k ) x − / − ik + O ( x / ) , x → , where m ( k ) = iπ − ik (cid:0) Γ(1 + 2 ik ) sinh 2 πk (cid:1) − . Calculating | m ( k ) | and using (4.8), we see that formula (1.3) now looks as( U f )( k ) = 2 π − √ k sinh 2 πk Z ∞ x − / K ik ( √ x ) f ( x ) dx. (4.17)The operator L does not have eigenvalues because the functions x − / K p ( √ x )for p ≥ L in a neighborhood of the point x = 0. Thus, similarlyto subs. 4.2, we see that the operator U defined by formula (4.17) is unitary in thespace L ( R + ) and the operator L has the simple absolutely continuous spectrum[1 / , ∞ ).Theorem 2.1 implies that Z ∞ a ( x + y ) y − / K ik ( p y ) dy = λ ( k ) x − / K ik ( √ x ) + ˇ λ ( k ) x − / H (2)2 ik ( i √ x )(4.18)(the Hankel function H (2)2 ik ( iz ) exponentially increases as z → ∞ ) for some constants λ ( k ) and ˇ λ ( k ). Since the integral in (4.18) (exponentially) decays as x → ∞ ,necessarily ˇ λ ( k ) = 0. Comparing the asymptotics of the left- and right-hand sidesof (4.18) as x → a ( x ) ∼ x − as x →
0, we find that the constant λ ( k ) is again given by formula (3.14). Thus, similarly to the previous subsection,we obtain Proposition 4.2.
The Hankel operator A with kernel (1.7) has the simple abso-lutely continuous spectrum coinciding with the interval [0 , π ] . Its normalized eigen-function corresponding to a spectral point λ = π (cosh πk ) − is given by the formula ψ k ( x ) = 2 π − √ k sinh 2 πkx − / K ik ( √ x ) , k > . As a by-product of our considerations, we obtain the equation x / Z ∞ ( x + y ) − / K ( √ x + y ) y − / K ik ( √ y ) dy = π (cosh πk ) − K ik ( √ x ) . We have not found this equation in the literature on special functions. Note, how-ever, that it can formally be deduced from the Shanker equation (1.8) if one usesthe relation (formula (6.9.19) of [3])lim β →∞ Γ( β + 1) W − β,m ( x/β ) = 2 x / K m (2 x / ) . The Carleman operator A trivially fits into the scheme exposed above. Nowthe operator A commutes with operator (1.6) for α = β = γ = 0. This operatorhas the absolutely continuous spectrum of multiplicity 2 coinciding with [1 / , ∞ ).It has eigenfunctions x − / ik for all k ∈ R which are also eigenfunctions of theoperator A . The relation between the spectral parameters λ and k is again givenby formula (3.14) so that the operator A has the absolutely continuous spectrumof multiplicity 2 coinciding with [0 , π ].5. Regular kernels
Let us here consider kernels a ( x ) which decay rapidly as x → ∞ and havefinite limits as x →
0. We set γ = 2 and distinguish the cases α > β is arbitraryand α = 0, β >
0. Let α = 1 / β = 2 in the first and second cases, respectively.If α = 1 /
4, then the solution of equation (2.6) is given (see subs. 4.1) by the formula b ( r ) = W − β, / ( r ) where W − β, / is the Whittaker function. If α = 0 and β = 2,then the solution of (2.6) equals b ( r ) = r / K ( √ r ) where K is the MacDonaldfunction (see subs. 4.4). The corresponding functions (2.5) decay exponentially atinfinity and have finite limits as x →
0. It follows that the operators A are compact.Let the function ω ( x ) be defined by formula (3.5), η = ω − and˜ q ( t ) = − − (cid:0) η ( t ) + 2 η ( t ) (cid:1) − + αη ( t ) + βη ( t ) . (5.1)Since η ( t ) ∼ e t /
2, the potential ˜ q ( t ) → + ∞ as t → ∞ . It follows that the operators e L and hence L have now discrete spectra. We point out that these operatorsare again defined by formulas (3.6) and (1.6) on functions satisfying boundaryconditions (3.7) and (3.3), respectively.Theorem 2.1 implies that the operators A and L have common eigenfunctions.Apparently, eigenfunctions of the operator L cannot be expressed in terms ofstandard special functions. However, in their terms we can calculate eigenvalues λ , λ , . . . of the operator A . Indeed, suppose that ψ µ ∈ D ( L ) and Lψ µ = µψ µ .Then the function ˜ ψ µ ( t ) = ( F − ψ µ )( t ) satisfies the equation˜ ψ ′′ µ ( t ) + ˜ q ( t ) ˜ ψ µ ( t ) = ( µ − /
4) ˜ ψ µ ( t )where ˜ q ( t ) is function (5.1). For a suitable normalization, asymptotics of ˜ ψ µ ( t ) as t → ∞ is given (see, e.g., book [7]) by the semiclassical formula˜ ψ µ ( t ) ∼ ˜ q ( t ) − / exp (cid:16) − Z t ˜ q ( s ) / ds (cid:17) . It follows that ˜ ψ µ ( t ) ∼ e − t/ exp( − − e t − βt ) in the first case and ˜ ψ µ ( t ) ∼ e − t/ exp( − / e t/ ) in the second case. Returning to the eigenfuctions ψ µ ( x ),we find that ψ µ ( x ) ∼ x − − β e − x/ and ψ µ ( x ) ∼ x − / e −√ x , x → ∞ , (5.2)in the first and second cases, respectively.On the other hand, using asymptotics (4.2) and (4.16) for function (1.7), we seethat ( Aψ µ )( x ) ∼ x − − β e − x/ Z ∞ e − y/ ψ µ ( y ) dy and ( Aψ µ )( x ) ∼ √ πx − / e −√ x Z ∞ ψ µ ( y ) dy IAGONALIZATION OF HANKEL OPERATORS 13 as x → ∞ in the first and second cases, respectively. Comparing these relationswith relations (5.2) and using the equation Aψ µ = λ µ ψ µ , we get expressions foreigenvalues of the operators A : λ µ = Z ∞ e − y/ ψ µ ( y ) dy and λ µ = √ π Z ∞ ψ µ ( y ) dy (5.3)in the first and second cases, respectively.Thus we have obtained the following results. Proposition 5.1.
The Hankel operator A with kernel a ( x ) = ( x + 2) − W − β, / ( x + 2) and the differential operator (1.6) for γ = 2 and α = 1 / have common eigen-functions. If ψ µ ∈ D ( L ) , Lψ µ = µψ µ and ψ µ ( x ) has the first asymptotics (5.2) as x → ∞ , then Aψ µ = λ µ ψ µ where λ µ is determined by the first formula (5.3) . Proposition 5.2.
The Hankel operator A with kernel a ( x ) = ( x + 2) − / K ( p x + 2)) and the differential operator (1.6) for γ = 2 , α = 0 and β = 2 have commoneigenfunctions. If ψ µ ∈ D ( L ) , Lψ µ = µψ µ and ψ µ ( x ) has the second asymptotics (5.2) as x → ∞ , then Aψ µ = λ µ ψ µ where λ µ is determined by the first secondformula (5.3) . Finally, we consider kernel (1.4) for exceptional values β = − l where l = 1 , , . . . . To be more precise, we now set a ( x ) = ( − l − ( l − − x − W l, / ( x ) = e − x/ L l − ( x ) , l = 1 , , . . . , (5.4)(here we have taken formula (4.6) into account). The Hankel operator A with thiskernel has rank l . Here we show how this simple example fits into the schemeexposed above.The spectral analysis of the corresponding operator (4.7) remains the same as insubs. 4.2. In addition to the absolutely continuous spectrum [1 / , ∞ ), the operator L has eigenvalues µ n = 1 / − ( l + 1 / − n ) where n = 1 , . . . , l .However, instead of the absolutely continuous spectrum, the operator A has thezero eigenvalue of infinite multiplicity. Indeed, as in subs. 4.3, Theorem 2.1 yieldsequation (4.12) where again ˇ λ ( k ) = 0. Observe that a ( x ) and hence in view of (4.3)the integral in the left-hand side of (4.12) have finite limits as x →
0. Therefore itfollows from (4.3) that necessarily λ ( k ) = 0 for all k >
0. Hence the kernel of theoperator A is spanned by the functions x − W l,ik ( x ), k > ψ n ( x ) corresponding to non-zero eigenvalues λ n of the operator A are defined by formula (4.11) where p = l + 1 / − n , n < l + 1 /
2, and λ n can befound from equation (4.13): Z ∞ L l − ( x + y ) e − y y p − / L pn − ( y ) dy = λ n x p − / L pn − ( x ) . (5.5)Recall that L αp ( x ) is a polynomial of degree p with the coefficient ( − p /p ! at x p .Comparing coefficients at the highest power x l − in the left- and right-hand sidesof (5.5) and taking into account formula (4.14), we find that λ n = ( − n − l ( n − l − Z ∞ e − y y p − / L pn − ( y ) dy = ( − n − l . Thus, we have obtained the following result.
Proposition 5.3.
The Hankel operator A with kernel (5.4) has rank l . Its non-zero eigenvalues are given by the formula λ n = ( − n − l where n = 1 , . . . , l , andthe corresponding eigenfunctions ψ n ( x ) are defined by equality (4.11) where p = l + 1 / − n . References [1] N. I. Akhieser and I. M. Glasman,
The theory of linear operators in Hilbert space , vols. I, II,Ungar, New York, 1961.[2] T. Carleman,
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I, II, Indiana Univ. Math. J. (1992), no. 2, 409–426 and 427–434.[5] W. Magnus, On the spectrum of Hilbert’s matrix , Amer. J. Math. (1950), 405-412.[6] F. G. Mehler, Math. Ann. (1881), 161-194.[7] F. W. J. Olver, Asymptotics and special functions , Academic Press, 1974.[8] V. V. Peller,
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On the Hilbert matrix , I, II, Proc. Amer. Math. Soc. (1958), 137-140,581-585.[10] H. Shanker, An integral equation for Whittaker’s confluent hypergeometric function , Proc.Cambridge Philos. Soc. (1949), 482-483.[11] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equa-tions , Vol. 1, Oxford, 1946.[12] N. Ya. Vilenkin,
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Mathematical scattering theory. Analytic theory , American Mathematical So-ciety, Providence, RI, 2010.
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