On 2D discrete Schrödinger operators associated with multiple orthogonal polynomials
aa r X i v : . [ m a t h . C A ] J a n ON 2D DISCRETE SCHR ¨ODINGER OPERATORS ASSOCIATEDWITH MULTIPLE ORTHOGONAL POLYNOMIALS
ALEXANDER I. APTEKAREV, MAXIM DEREVYAGIN, AND WALTER VAN ASSCHE
Abstract.
A class of cross-shaped difference operators on a two dimensionallattice is introduced. The main feature of the operators in this class is thattheir formal eigenvectors consist of multiple orthogonal polynomials. In otherwords, this scheme generalizes the classical connection between Jacobi matricesand orthogonal polynomials to the case of operators on lattices. Furthermorewe also show how to obtain 2D discrete Schr¨odinger operators out of thisconstruction and give a number of explicit examples based on known familiesof multiple orthogonal polynomials. Introduction
In this paper we introduce a class of cross-shaped difference operators actingon the lattice Z , where Z + = { , , , . . . } . Cross-shaped difference operators onlattices appear in many instances where some discrete systems are analysed. Inparticular, cross-shaped difference operators with periodic coefficients were studiedin [10] and [18]. To be more specific, the operators we are dealing with have theform(1.1) ( e ∆ f ) n,m = f n +1 ,m + f n,m +1 + q n,m f n,m + a n,m f n − ,m + b n,m f n,m − . This operator reflects the 2D interaction of the nearest neighbours on Z : • •• •• What we do can be considered as a generalization of the classical connection be-tween
Jacobi matrices and orthogonal polynomials [1]. Recall that a Jacobi matrix J is a difference operator of the form(1.2) ( Jf ) n = √ a n +1 f n +1 + q n f n + √ a n f n − , which describes the 1D interaction of the nearest neighbours on Z : • •• Date : September 30, 2018.1991
Mathematics Subject Classification.
Primary 39A70, 42C05; Secondary 47B36, 47B37,47B39, 82C20.
Key words and phrases.
Multiple orthogonal polynomials, discrete electromagnetic Schr¨odingeroperator, difference operator, operators on lattices, discrete integrable system.
However, one should not think that the generalization in question is obvious andstraightforward. Unlike the classical case of Jacobi matrices, it is not clear whetherthe corresponding eigenvalue problem(1.3) e ∆ ξ ( z ) = zξ ( z )has a solution and especially whether the entries of ξ can be chosen to be polyno-mials in the spectral variable z . To the best of our knowledge, there were only acouple of operators on lattices with polynomial eigenvectors known before and themain goal of this paper is to present a rather general method to produce such oper-ators. To this end, we construct cross-shaped difference operators e ∆ using multipleorthogonal polynomials, which are known to play a prominent role in the theory ofrandom matrices [11], [6], [14]. Thus, the operators e ∆ that we obtain in this wayhave multiple orthogonal polynomials as the entries of their eigenvectors. In orderto guarantee the existence of a polynomial solution to (1.3) we only consider specialfamilies of coefficients q n,m , a n,m , b n,m that give rise to a discrete zero curvaturecondition . This means that there is a discrete integrable system behind the scene[5].It is evident that the difference expression (1.1) defining e ∆ is not symmetric.Nevertheless, in some cases the operator e ∆ can be symmetrized like in the classicalcase of the Jacobi operator (1.2). In this case the cross-shaped difference operatorsare a subclass of a wide class of operators known in the literature as discrete electro-magnetic Schr¨odinger operators ([19], [21]). Discrete electromagnetic Schr¨odingeroperators are operators defined on the lattice Z that have the form(1.4) e ∆ s u = X k =1 m k ( V e k − a k I )( V − e k − ¯ a k I ) u + Φ u, where u is a function defined on Z , ( V e k u )( x ) := u ( x − e k ) and ( V − e k u )( x ) := u ( x + e k ) are the shift operators by e = (1 ,
0) and e = (0 , m k is the mass of the k th particle, and a , a and Φ are bounded complex-valuedfunctions on Z . The vector-valued function a = ( a , a ) can be considered asan analogue of the magnetic potential, whereas Φ is the discrete analogue of theelectric potential. If Φ and a = ( a , a ) are real-valued, then e ∆ s is a selfadjointoperator on the Hilbert space ℓ ( Z ) and it can be rewritten in the form − e ∆ s u ( x ) = a m ( u ( x + e ) + u ( x − e )) + a m ( u ( x + e ) + u ( x − e ))+ (cid:18) a m + 1 + a m + 2Φ( x ) (cid:19) u ( x ) . Actually, in this paper we consider a class of operators that are more general thanthe operator (1.4). Namely, the operators we consider here are of the form(∆ s f ) n,m = r a n +1 ,m f n +1 ,m + r b n,m +1 f n,m +1 + c n,m + d n,m f n,m + r a n,m f n − ,m + r b n,m f n,m − . D DISCRETE SCHR ¨ODINGER OPERATORS 3
Here one has to put f n,m = u ( x ) , f n +1 ,m = u ( x + e ) , f n − ,m = u ( x − e ) ,f n,m +1 = u ( x + e ) , f n,m − = u ( x − e ) , to see the relation of our operators to the discrete electromagnetic Schr¨odingeroperators (1.4).It is also worth mentioning that operators of the form (1.4) describe the so-calledtight binding model in solid state physics (see [13], [16] and the references giventhere), which plays a prominent role in the theory of propagation of spin waves andof waves in quasi-crystals [23, 20], in the theory of nonlinear integrable lattices [23,7], and in other places. Furthermore such symmetric difference operators appear inquantum-state transfer problems [15] and quantum computation [9].On the one hand, our construction produces many concrete examples of cross-shaped difference operators, which can be obtained from some explicitly knownfamilies of multiple orthogonal polynomials. For instance, these examples can servefor constructing certain Hamiltonians in quantum-state transfer problems and otherrelated physical problems. On the other hand, the eigenvectors of the operators inquestion consist of multiple orthogonal polynomials, whose asymptotic propertiesare well understood, and thus we have a very powerful tool for spectral analysis ofthe underlying operators. Acknowledgements.
A.I. Aptekarev was supported by the Russain Science Foun-dation (project 14-21-00025. M. Derevyagin thanks the hospitality of the Depart-ment of Mathematics of KU Leuven, where his part of the research was mainlydone while he was a postdoc there. M. Derevyagin and W. Van Assche gratefullyacknowledge the support of FWO Flanders project G.0934.13, KU Leuven researchgrant OT/12/073 and the Belgian Interuniversity Attraction Pole P07/18.2.
Multiple orthogonal polynomials
Here we briefly review a generalization of orthogonal polynomials to the casewhen we have two measures and we want our polynomials to be simultaneouslyorthogonal with respect to the given measures, see [4], [12, Chapter 23], [24].Given two positive measures µ , µ on the real line, let us consider the multi-index ( n, m ) ∈ Z . The type II multiple orthogonal polynomial is the monic poly-nomial P n,m ( x ) = x n + m + · · · of degree n + m such that the following orthogonalityrelations are satisfied: Z P n,m ( x ) x j dµ ( x ) = 0 , j = 0 , , . . . , n − , Z P n,m ( x ) x j dµ ( x ) = 0 , j = 0 , , . . . , m − . Introducing the moments s ( i ) j = Z x j dµ i ( x ) , i = 1 , , ALEXANDER I. APTEKAREV, MAXIM DEREVYAGIN, AND WALTER VAN ASSCHE and the determinant of the moment matrix(2.1) S n,m = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s (1)0 s (1)1 · · · s (1) n − s (1)1 s (1)2 · · · s (1) n ... ... · · · ... s (1) n + m − s (1) n + m · · · s (1)2 n + m − s (2)0 s (2)1 · · · s (2) m − s (2)1 s (2)2 · · · s (2) m ... ... · · · ... s (2) n + m − s (2) n + m · · · s (2) n +2 m − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , one sees that the type II multiple orthogonal polynomial can be written as P n,m ( x ) = 1 S n,m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s (1)0 s (1)1 · · · s (1) n − s (1)1 s (1)2 · · · s (1) n ... ... · · · ... s (1) n + m s (1) n + m +1 · · · s (1)2 n + m − s (2)0 s (2)1 · · · s (2) m − s (2)1 s (2)2 · · · s (2) m ... ... · · · ... s (2) n + m s (2) n + m +1 · · · s (2) n +2 m − x ... x n + m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) provided that S n,m is nonvanishing. In the latter case we say that the index ( n, m )is normal. In this paper we always assume that all multi-indices are normal and weinvestigate the nearest-neighbor recurrence relations. Theorem 2.1 ([25]) . Suppose all multi-indices ( n, m ) ∈ Z are normal. Then thetype II multiple orthogonal polynomials satisfy the system of recurrence relations P n +1 ,m ( x ) = ( x − c n,m ) P n,m ( x ) − a n,m P n − ,m ( x ) − b n,m P n,m − ( x ) , (2.2) P n,m +1 ( x ) = ( x − d n,m ) P n,m ( x ) − a n,m P n − ,m ( x ) − b n,m P n,m − ( x ) , (2.3) where the coefficients obey the following conditions (2.4) a ,m = b n, = 0 , a n, > , b ,m > , n, m > . Note that for (2.2), (2.3) to hold these relations must be consistent. As was shownin [5], this consistency can be expressed as a discrete zero curvature condition, whichalso takes the following form.
Theorem 2.2 ([25, 5]) . Suppose that all the indices ( n, m ) ∈ Z are normal. Therecurrence coefficients in the recurrence relations (2.2) – (2.3) for type II multipleorthogonal polynomials satisfy the following equations c n,m +1 = c n,m + ( a + b ) n +1 ,m − ( a + b ) n,m +1 ( c − d ) n,m , (2.5) d n,m +1 = d n,m + ( a + b ) n +1 ,m − ( a + b ) n,m +1 ( c − d ) n,m , (2.6) a n,m +1 a n,m = c n,m − d n,m c n − ,m − d n − ,m , (2.7) b n +1 ,m b n,m = c n,m − d n,m c n,m − − d n,m − . (2.8)It turns out that the consistency conditions (2.5)-(2.8), which generate an un-derlying zero curvature condition, play a central role in the theory of multipleorthogonal polynomials in the sense that the following Favard-type result holds. Theorem 2.3.
Suppose that the polynomials P n,m of degree n + m satisfy (2.2) , (2.3) and for the corresponding coefficients the consistency conditions (2.5) - (2.8) and (2.4) are fulfilled. Then there are two measures µ and µ such that the poly-nomials P n,m are multiple orthogonal polynomials with respect to µ and µ . D DISCRETE SCHR ¨ODINGER OPERATORS 5 A continuous model for multiple orthogonal polynomials
In this section we propose a way to interpret the concept of multiple orthogonalpolynomials. Let us begin by recalling that the discretization of the string equation ddM ( x ) dy ( x ) dx + q ( x ) y ( x ) = zy ( x ) , where ddM ( x ) is the derivative with respect to the measure dM ( x ), leads to spectralproblems for Jacobi matrices and thus to orthogonal polynomials (for instance see[2], [22]). In particular, if we try to solve the differential equation y ′′ ( x ) = zy ( x )by using the classical discretization scheme y ′′ ( x ) ≈ y ( x + 1) − y ( x ) + y ( x − , we get the difference equation y ( x + 1) + y ( x −
1) = ( z + 2) y ( x ) , which brings us to the context of the free Jacobi matrix and Chebyshev polynomials.Bearing the above-mentioned trick in mind, one can see that the discretizationof the following space-time dual system of generalized string equations − Ψ t ( t, x ) + ∂∂M ( x ) ∂ Ψ( t, x ) ∂x + u ( t, x )Ψ( t, x ) = z Ψ( t, x ) , − Ψ x ( t, x ) + ∂∂M ( t ) ∂ Ψ( t, x ) ∂t + v ( t, x )Ψ( t, x ) = z Ψ( t, x ) , gives a system of difference equations of the form (2.2)–(2.3). Note that suchgeneralized string equations were recently studied in [3].To get a more precise idea, let us consider the following space-time dual systemof time-dependent Schr¨odinger equations − Ψ t ( t, x ) + Ψ xx ( t, x ) + u ( t, x )Ψ( t, x ) = z Ψ( t, x ) , − Ψ x ( t, x ) + Ψ tt ( t, x ) + v ( t, x )Ψ( t, x ) = z Ψ( t, x ) , where t , x are nonnegative real numbers, z is the spectral parameter, and u , v are sufficiently good potentials. Now let us fix h >
0. Then using the followingdiscretization for the derivatives of the first orderΨ t ( t, x ) ≈ Ψ( t − h, x ) − Ψ( t, x ) − h , Ψ x ( t, x ) ≈ Ψ( t, x − h ) − Ψ( t, x ) − h , and for the derivatives of the second orderΨ xx ( t, x ) ≈ Ψ( t, x + h ) − t, x ) + Ψ( t, x − h ) h , Ψ tt ( t, x ) ≈ Ψ( t + h, x ) − t, x ) + Ψ( t − h, x ) h , we arrive at the following recurrence relationsΨ( t − h, x ) − Ψ( t, x ) h + Ψ( t, x + h ) − t, x ) + Ψ( t, x − h ) h + u ( t, x )Ψ( t, x ) = z Ψ( t, x ) , Ψ( t, x − h ) − Ψ( t, x ) h + Ψ( t + h, x ) − t, x ) + Ψ( t − h, x ) h + v ( t, x )Ψ( t, x ) = z Ψ( t, x ) . ALEXANDER I. APTEKAREV, MAXIM DEREVYAGIN, AND WALTER VAN ASSCHE
Choosing x = t = 0, h = 1 and setting P n,m ( z ) = Ψ( nt, mh ) , u n,m = u ( nt, mh ) − , v n,m = v ( nt, mh ) − , m, n ∈ Z + we get from the latter recurrence relations that P n,m +1 ( z ) + u n,m P n,m ( z ) + P n − ,m ( z ) + P n,m − ( z ) = zP n,m ( z ) ,P n +1 ,m ( z ) + v n,m P n,m ( z ) + P n − ,m ( z ) + P n,m − ( z ) = zP n,m ( z ) , which are the nearest neighbour recurrence relations for multiple orthogonal poly-nomials. However, as we already learned, the coefficients of the nearest neighbourrecurrence relations cannot be arbitrary. As a consequence, the fact that a fewcoefficients in the just obtained relations are equal to 1 makes our choice for therest trivial. Indeed, it is easy to see that only for the case of constant coefficients u n,m = u , and v n,m = v , = u the consistency conditions (2.5)–(2.8) are satis-fied. 4. The underlying pairs of operators
In analogy with the case of orthogonal polynomials on the real line, we introducetwo difference operators on Z associated with the recurrence relations (2.2)–(2.3),whose coefficients obey the discrete integrable system (2.5)–(2.8),(4.1) ( H f ) n,m = f n +1 ,m + c n,m f n,m + a n,m f n − ,m + b n,m f n,m − , (4.2) ( H f ) n,m = f n,m +1 + d n,m f n,m + a n,m f n − ,m + b n,m f n,m − , where f = (cid:0) f n,m (cid:1) is a sequence defined on Z . Then it is clear that (2.2)–(2.3) canbe rewritten as the eigenvector problems H π ( z ) = zπ ( z ) ,H π ( z ) = zπ ( z ) , (4.3)where π ( z ) = (cid:0) P n,m ( z ) (cid:1) is a table of multiple orthogonal polynomials. In order toimagine what these relations represent, let us rewrite the operators H and H inthe following way:( H f ) n,m = f n +1 ,m + ( c n,m + b n,m ) f n,m + a n,m f n − ,m + b n,m ( f n,m − − f n,m ) , ( H f ) n,m = f n,m +1 + ( d n,m + a n,m ) f n,m + b n,m f n,m − + a n,m ( f n − ,m − f n,m ) . Now, recalling the interpretation from Section 3 with time-dependent Schr¨odingerequations, one can think of (4.3) as the relations that describe two coexisting evolu-tions: one is the transformation of the vector ( P ,m , P ,m , . . . ) in the discrete time m and the other one is the progression of the vector ( P n, , P n, , . . . ) in the discretetime n . In other words, one could visualize this as two waves going from the bound-aries Z + and i Z + = (0 , m ), m ∈ Z + , to infinity along i Z + and Z + , respectively.These ideas suggest that the boundary data play a crucial role for the theory. Forthis reason we introduce two monic classical Jacobi matrices( H f ) n, = f n +1 , + c n, f n, + a n, f n − , , ( H f ) ,m = f ,m +1 + d ,m f ,m + b ,m f ,m − , a n, , b n, > , which, due to the fact that a ,m = a n, = 0 for n, m ∈ Z + , are restrictions of H and H to the subspaces spanned by functions defined on Z + and i Z + , respectively.Observe that the entire initial information about multiple orthogonal polynomialsis encrypted in the matrices H and H . D DISCRETE SCHR ¨ODINGER OPERATORS 7
Theorem 4.1.
The Jacobi matrices H and H (and as a consequence the operators H and H ) determine the measures µ and µ , respectively. In other words, thesolution of the discrete integrable system (2.5) – (2.8) can be reconstructed from theboundary data.Proof. The proof of this statement is straightforward and it is enough to noticethat the polynomials P n, and P ,m are orthogonal polynomials associated with H and H . Thus, it remains to apply the classical Favard theorem (see [12, Section2.5]) to determine the measures µ and µ . (cid:3) Remark.
To sum up what we have so far we note that Theorem 4.1 says thatstarting with the nearest neighbour recurrence relations one can reconstruct theunderlying measures and, consequently, the corresponding sequences of moments.If we go in the opposite direction then we start with the moments. Next, we findthe multiple orthogonal polynomials and after that we end up with the recurrencecoefficients. Hence, we know how to solve inverse and direct problems for H and H . However, one should not think that any two measures can be put into thisscheme. As a matter of fact, any two measures define two Jacobi matrices, i.e.,two 1D discrete Schr¨odinger operators on Z + . In the standard way, these twooperators can be glued into one operator on Z (e.g, one can take a 2 × Z + ∪ i Z + . Still,it remains unclear whether this operator on Z + ∪ i Z + can be extended to Z inone way or another (or, perhaps, our two initial 1D operators can be glued into one2D operator). One can see that the problem of the existence of a table of multipleorthogonal polynomials is thus equivalent to the possibility of extending the 1Ddiscrete Schr¨odinger operator on Z + ∪ i Z + to a 2D discrete Schr¨odinger operatoron Z . It is worth mentioning that this extension can be done if and only if thesystem ( µ , µ ) is a perfect system (see [5] for further details).5. Cross-shaped difference operators on Z It is obvious that for a general cross-shaped difference operator e ∆ of the form(5.1) ( e ∆ f ) n,m = 12 f n +1 ,m + 12 f n,m +1 + q n,m f n,m + a n,m f n − ,m + b n,m f n,m − it is not at all clear whether the eigenvalue problem(5.2) e ∆ ξ ( z ) = zξ ( z )has a polynomial solution.If one supposes that (5.2) has a polynomial solution then one gets that therelation 12 f , + 12 f , + q , f , + a , f − , + b , f , − = zf , , with the initial conditions f − , = 0 , f , − = 0 , f , = 1 , must define two linear monic polynomials f , and f , . This basically means thatthere exists a representation q , = c , d , ALEXANDER I. APTEKAREV, MAXIM DEREVYAGIN, AND WALTER VAN ASSCHE such that f , = z − d , , f , = z − c , . In general, we see that the relation12 f n +1 ,m + 12 f n,m +1 + q n,m f n,m + a n,m f n − ,m + b n,m f n,m − = zf n,m determines two polynomials f n +1 ,m and f n +1 ,m . In other words, there are tworepresentations f n +1 ,m + q ′ n,m f n,m + a ′ n,m f n − ,m + b ′ n,m f n,m − = zf n,m ,f n,m +1 + q ′′ n,m f n,m + a ′′ n,m f n − ,m + b ′′ n,m f n,m − = zf n,m , that must be consistent on Z . Schematically, what we do here is to try to split the2D interaction in question as the arithmetic mean of the following two interactions: • •• • •• •• Since it is a very difficult problem to characterize all such representations, we usethe following multiple polynomial Ansatz to proceed: q ′ n,m = c n,m , q ′′ n,m = d n,m , a ′ n,m = a ′′ n,m = a n,m , b ′ n,m = b ′′ n,m = b n,m . In order to be able to present a class of operators for which the eigenvalueproblem has a polynomial solution, we suppose that(5.3) D n,m = − (cid:18) a n +1 ,m +1 a n +1 ,m + b n +1 ,m +1 b n,m +1 (cid:19) + 8 = 0 . Then we introduce the matrices(5.4) L n,m = z − q n,m + 4 q n +1 ,m D n,m − q n +1 ,m D n,m − a n,m − b n,m (cid:16) q n +1 ,m − D n,m − − q n +1 ,m − D n,m − (cid:17) , (5.5) M n,m = z − q n,m − q n +1 ,m D n,m + 4 q n +1 ,m D n,m − a n,m − b n,m − (cid:16) q n,m D n − ,m − q n,m D n − ,m (cid:17)
01 0 0 . We associate these transition matrices L n,m and M n,m with the operator e ∆. Thesematrices allow us to define a vector wave function Ψ n,m ( z ) on Z :Ψ n +1 ,m ( z ) = L n,m ( z )Ψ n,m ( z ) , Ψ n,m +1 ( z ) = M n,m ( z )Ψ n,m ( z ) . If this function is correctly defined, then by choosing the initial stateΨ , ( z ) = D DISCRETE SCHR ¨ODINGER OPERATORS 9 we arrive at the polynomial solutionΨ n,m = P n,m ( z ) P n − ,m ( z ) P n,m − ( z ) , which consists of multiple orthogonal polynomials and at the same time gives apolynomial solution to the eigenvalue problem for e ∆. Theorem 5.1.
Let e ∆ be a cross-shaped difference operator of the form (5.1) suchthat the condition (5.3) is satisfied. Then the eigenvalue problem for e ∆ has a familyof multiple orthogonal polynomials as its solution if and only if the following discretezero curvature condition holds (5.6) L n,m +1 M n,m − M n +1 ,m L n,m = 0 , n, m ∈ Z , where L n,m and M n,m are the matrices in (5.4) – (5.5) .Proof. If we have a family of multiple orthogonal polynomials then we can introducethe following cross-shaped difference operator∆ = 12 ( H + H ) . It is easy to see that the action of ∆ on f can be described by the following(∆ f ) n,m = 12 f n +1 ,m + 12 f n,m +1 + d n,m + c n,m f n,m + a n,m f n − ,m + b n,m f n,m − . Thus, from (4.3) we get that ∆ π ( z ) = zπ ( z ) . To prove this statement we first have to show how to reconstruct the coefficients c n,m , d n,m from the a n,m , b n,m , and q n,m = c n,m + d n,m . To begin, we observe that the above relation gives the following three equations c n,m + d n,m = 2 q n,m ,c n +1 ,m + d n +1 ,m = 2 q n +1 ,m ,c n,m +1 + d n,m +1 = 2 q n,m +1 , (5.7)where c n,m , d n,m , c n +1 ,m , d n +1 ,m , c n,m +1 , d n,m +1 are six unknowns. Next, therelations (2.5), (2.7), and (2.8) give c n,m − d n,m + d n +1 ,m − c n,m +1 = 0 ,a n +1 ,m +1 a n +1 ,m c n,m − a n +1 ,m +1 a n +1 ,m d n,m − c n +1 ,m + d n +1 ,m = 0 ,b n +1 ,m +1 b n,m +1 c n,m − b n +1 ,m +1 b n,m +1 d n,m − c n,m +1 + d n,m +1 = 0 . (5.8) Therefore, we arrive at the system (5.7)–(5.8) of six linear equations with six un-knowns. The determinant of (5.7)–(5.8) is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − a n +1 ,m +1 a n +1 ,m − a n +1 ,m +1 a n +1 ,m − b n +1 ,m +1 b n,m +1 − b n +1 ,m +1 b n,m +1 − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − (cid:18) a n +1 ,m +1 a n +1 ,m + b n +1 ,m +1 b n,m +1 (cid:19) + 8 . Hence, due to (5.3) we see that one can solve (5.7), (5.8) and find c n,m , d n,m provided that a n,m , b n,m , q n,m are given: c n,m = q n,m − q n +1 ,m D n,m + 4 q n +1 ,m D n,m , d n,m = q n,m + 4 q n +1 ,m D n,m − q n +1 ,m D n,m . The latter relations show that the relation (5.6) is equivalent to the consistencyconditions (2.5)–(2.8), which are in turn the necessary and sufficient conditions fora family of multiple orthogonal polynomials to exist according to Theorem 2.3. (cid:3)
Remark.
As one can see from the example constructed in Section 3, if the condition(5.3) is not satisfied then the operators H , H can not be uniquely determined. Analgorithm for computing the coefficients a n,m , b n,m , c n,m , d n,m from the coefficientsof the operators H and H is given in [8].6.
2D discrete Schr¨odinger operators
In a natural way one can introduce the space ℓ ( Z ) of square summable familieson Z . Moreover, as in the case of Jacobi matrices, the difference expression ∆generates an operator in ℓ ( Z ), which will also be denoted by ∆. It is clear that∆ is not symmetric. However, mimicking the idea of the transformation of monicJacobi matrices to symmetric ones, we can symmetrize ∆ by making use of theconsistency conditions (2.5)–(2.8). Theorem 6.1.
Suppose that in addition to the curvature condition (2.5) – (2.8) , thecoefficients c n,m and d n,m verify the condition (6.1) c n +1 ,m − d n +1 ,m = c n,m +1 − d n,m +1 . Then there exists a family h n,m = 0 defined on Z such that the diagonal operator ( Df ) n,m = h n,m f n,m symmetrizes the operator ∆ by means of a similarity transformation, that is, theoperator ∆ s = D − ∆ D is symmetric in ℓ ( Z ) .Proof. Note that if a difference expression of the form( b ∆ f ) n,m = α n,m f n +1 ,m + β n,m f n,m +1 + γ n,m f n,m + δ n,m f n − ,m + ε n,m f n,m − is symmetric in ℓ ( Z ) then(6.2) α n,m = δ n +1 ,m , β n,m = ε n,m +1 , D DISCRETE SCHR ¨ODINGER OPERATORS 11 which can be obtained from the relation (cid:16) b ∆ f, g (cid:17) ℓ ( Z ) = (cid:16) f, b ∆ g (cid:17) ℓ ( Z ) on the standard basis ( e n,m ), where ( e n,m ) is understood as a table of numberswith 1 on the position ( n, m ) and the rest of the elements are zeros. Since we wantthe operator(∆ s f ) n,m = h n +1 ,m h n,m f n +1 ,m + h n,m +1 h n,m f n,m +1 + c n,m + d n,m f n,m + a n,m h n − ,m h n,m f n − ,m + b n,m h n,m − h n,m f n,m − to be symmetric, it must obey the relation(6.3) h n +1 ,m h n,m = a n +1 ,m h n,m h n +1 ,m , h n,m +1 h n,m = b n,m +1 h n,m h n,m +1 , which can be rewritten as follows h n +1 ,m = 2 a n +1 ,m h n,m , h n,m +1 = 2 b n,m +1 h n,m . Now, we see that for the existence of the family h n,m = 0 the following compatibilitycondition must be satisfied a n +1 ,m b n +1 ,m +1 = b n,m +1 a n +1 ,m +1 . The latter relation can be obtained from the consistency relations (2.7)–(2.8). In-deed, it easily follows from (2.7), (2.8), and (6.1) that a n +1 ,m +1 a n +1 ,m = c n +1 ,m − d n +1 ,m c n,m − d n,m = c n,m +1 − d n,m +1 c n,m − d n,m = b n +1 ,m +1 b n,m +1 , which is exactly what we need. Hence, the table h n,m = 0 can be constructed, say,by the initialization h , = 1. Finally, noticing that p a n,m = h n,m h n − ,m , p b n,m = h n,m h n,m − , we get the operator(∆ s f ) n,m = r a n +1 ,m f n +1 ,m + r b n,m +1 f n,m +1 + c n,m + d n,m f n,m ++ r a n,m f n − ,m + r b n,m f n,m − , which can be represented as a sum∆ s = J + J of two symmetric Jacobi “matrices” of the following form( J f ) n,m = r a n +1 ,m f n +1 ,m + c n,m f n,m + r a n,m f n − ,m , ( J f ) n,m = r b n,m +1 f n,m +1 + d n,m f n,m + r b n,m f n,m − . Therefore, ∆ s is symmetric. (cid:3) Remark.
It should also be noted that the underlying eigenvalue problem reducesto the following one ∆ s π ( s ) ( z ) = zπ ( s ) ( z ) , where we set ∆ s = D − ∆ D, π ( s ) ( z ) = D − π ( z ) . In other words, the multiple orthogonal polynomials P ( s ) n,m corresponding to thesymmetric operator ∆ s and the multiple orthogonal polynomials P n,m correspond-ing to the monic operator ∆ are related in the following manner P ( s ) n,m ( z ) = 1 h n,m P n,m ( z ) . Note that we chose h , = 1 in order to have P ( s )0 , = 1.It is also easy to give sufficient conditions for a 2D discrete Schr¨odinger operatorto have eigenfunctions that consist of multiple orthogonal polynomials. Theorem 6.2.
Let e ∆ s have the form ( e ∆ s f ) n,m = r a n +1 ,m f n +1 ,m + r b n,m +1 f n,m +1 + q n,m f n,m + r a n,m f n − ,m + r b n,m f n,m − . Suppose that there exist two sets of numbers c n,m and d n,m such that (6.4) q n,m = c n,m + d n,m , and the coefficients a n,m , b n,m , c n,m , d n,m satisfy the consistency conditions (2.5) – (2.8) together with (6.1) . Also assume that (6.5) a n +1 ,m +1 a n +1 ,m + b n +1 ,m +1 b n,m +1 = 2 . Then the families a n,m , b n,m , q n,m determine the sets c n,m , d n,m uniquely. In otherwords, the operator e ∆ s , the consistency conditions (2.5) – (2.8) , and (6.1) uniquelydefine the operators H and H of the form (4.1) – (4.2) . Thus, in this case, theoperator e ∆ s generates a family of multiple orthogonal polynomials (cid:0) P n,m (cid:1) ∞ n,m =0 . Some examples
In this section we recall a few examples of multiple orthogonal polynomials givenin [17], [25] to illustrate our method.7.1.
Multiple Hermite polynomials.
Multiple Hermite polynomials H n,m aremonic polynomials of degree n + m that satisfy the following orthogonality condi-tions Z ∞−∞ x k H n,m ( x ) e − x + c x dx = 0 , k = 0 , , . . . , n − , Z ∞−∞ x k H n,m ( x ) e − x + c x dx = 0 , k = 0 , , . . . , m − , D DISCRETE SCHR ¨ODINGER OPERATORS 13 where c = c . The corresponding recurrence relations are explicitly given as xH n,m ( x ) = H n +1 ,m ( x ) + c H n,m ( x ) + n H n − ,m ( x ) + m H n,m − ( x ) ,xH n,m ( x ) = H n,m +1 ( x ) + c H n,m ( x ) + n H n − ,m ( x ) + m H n,m − ( x ) . So, in this case we have that a n,m = n , b n,m = m , c n,m = c , d n,m = c . Hence, the relation (6.1) is obviously satisfied and, therefore, we have the symmetricoperator (∆ s f ) n,m = √ n + 12 f n +1 ,m + √ m + 12 f n,m +1 + c + c f n,m + √ n f n − ,m + √ m f n,m − . (7.1)From the form of this operator it is clear that one cannot uniquely reconstruct thecorresponding multiple orthogonal polynomials. Indeed, we end up with the oper-ator (7.1) if we start with any multiple Hermite polynomials H n,m = H n,m ( c ′ , c ′ )such that c ′ + c ′ = c + c .7.2. Multiple Laguerre polynomials of the first kind.
These polynomials aregiven by the orthogonality relations Z ∞ x k L n,m ( x ) x α e − x dx = 0 , k = 0 , , . . . , n − , Z ∞ x k L n,m ( x ) x α e − x dx = 0 , k = 0 , , . . . , m − , where α , α > − α − α / ∈ Z . For multiple Laguerre polynomials of the firstkind it is known that c n,m = 2 n + m + α + 1 , d n,m = n + 2 m + α + 1 , which do not satisfy (6.1). Thus, the underlying cross-shaped operator cannot besymmetrized.7.3. Multiple Meixner polynomials of the first kind.
The multiple Meixnerpolynomials of the first kind M (1) n,m are the monic polynomials of degree n + m forwhich ∞ X k =0 M (1) n,m ( k ) k ℓ ( β ) k ( c ) k k ! = 0 , ℓ = 0 , , . . . , n − , ∞ X k =0 M (1) n,m ( k ) k ℓ ( β ) k ( c ) k k ! = 0 , ℓ = 0 , , . . . , m − , where β > < c = c <
1. In this case, the nearest neighbour recurrencerelation takes the form xM (1) n,m ( x ) = M (1) n +1 ,m ( x ) + ( β + n + m ) c − c + n − c + m − c ! M (1) n,m ( x )+ c n (1 − c ) ( β + n + m − M (1) n − ,m ( x ) + c m (1 − c ) ( β + n + m − M (1) n,m − ( x ) , xM (1) n,m ( x ) = M (1) n,m +1 ( x ) + ( β + n + m ) c − c + n − c + m − c ! M (1) n,m ( x )+ c n (1 − c ) ( β + n + m − M (1) n − ,m ( x ) + c m (1 − c ) ( β + n + m − M (1) n,m − ( x ) . Since the relation (6.1) is true in this case, we obtain the following 2D Schr¨odingeroperator(∆ s f ) n,m = p c ( n + 1)( n + m + β ) √ − c ) f n +1 ,m + p c ( m + 1)( n + m + β ) √ − c ) f n,m +1 + (cid:18) ( β + n + m )2 (cid:18) c − c + c − c (cid:19) + n − c + m − c (cid:19) f n,m + p c n ( n + m + β − √ − c ) f n − ,m + p c m ( n + m + β − √ − c ) f n,m − . (7.2)Also, the recurrence coefficients satisfy (6.5), that is a n +1 ,m +1 a n +1 ,m + b n +1 ,m +1 b n,m +1 = 2 β + n + m + 1 β + n + m = 2 (cid:18) β + n + m (cid:19) > . This means that the 2D Schr¨odinger operator (7.2) determines the correspondingmultiple orthogonal polynomials uniquely.
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Alexander I. Aptekarev, Keldysh Institute for Applied Mathematics, Russian Acad-emy of Sciences, Miusskaya pl. 4, 125047 Moscow, RUSSIAMaxim Derevyagin, University of Mississippi, Department of Mathematics, Hume Hall305, P. O. Box 1848, University, MS 38677-1848, USA
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