aa r X i v : . [ m a t h . SP ] J u l HEAT KERNELS OF THE DISCRETE LAGUERRE OPERATORS
ALEKSEY KOSTENKO
Abstract.
For the discrete Laguerre operators we compute explicitly the cor-responding heat kernels by expressing them with the help of Jacobi polyno-mials. This enables us to show that the heat semigroup is ultracontractiveand to compute the corresponding norms. On the one hand, this helps us toanswer basic questions (recurrence, stochastic completeness) regarding the as-sociated Markovian semigroup. On the other hand, we prove the analogs ofthe Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbationsof the Laguerre operators. Introduction
Our main objects of study are the discrete Laguerre operators H α := α −√ α · · ·−√ α α − p α ) · · · − p α ) 5 + α . . .0 0 − p α ) . . .... ... . . . . . . , α > − , (1.1)acting in ℓ ( Z ≥ ). Explicitly, H α = (cid:0) h ( α ) n,m (cid:1) n,m ≥ with h ( α ) n,m = 0 if | n − m | > h ( α ) n,n = 2 n + 1 + α, h ( α ) n,n +1 = h ( α ) n +1 ,n = − p ( n + 1)( n + 1 + α ) , n ∈ Z ≥ . It is a special case of a self-adjoint Jacobi operator whose generalized eigenfunctionsare precisely the Laguerre polynomials L ( α ) n , explaining the name for (1.1).The operator H α features prominently in the study of nonlinear waves in (2 + 1)-dimensional noncommutative scalar field theory [1, 2, 12]. The coefficient α in (1.1)can be seen as a measure of the delocalization of the field configuration and it is re-lated to the planar angular momentum [2]. In particular, α = 0 corresponds to spher-ically symmetric waves and it has attracted further interest in [4, 18, 19, 20], where H appears as the linear part in the nonlinear Schr¨odinger equation [18, 19, 20].Thus dispersive estimates for the unitary evolution play a crucial role in the under-standing of stability of soliton manifolds appearing in these models. It turned out(see [17, 16]) that the unitary evolution e i tH α can be expressed by means of Jacobipolynomials (see Appendix A for definitions and basic facts) and this also connects Mathematics Subject Classification.
Primary 33C45, 47B36; Secondary 47D07, 81Q15.
Key words and phrases.
Schr¨odinger equation, heat equation, Jacobi polynomials.
Research supported by the Slovenian Research Agency (ARRS) under Grant No. N1-0137 andthe Austrian Science Fund (FWF) under Grant No. P28807 . dispersive estimates with uniform weighted estimates of Jacobi polynomials on theorthogonality interval (the so-called Bernstein-type inequalities).In the present article we focus on the study of the heat semigroup (e − tH α ) t> .Usually (sharp) dispersive estimates (for e i tH α these are obtained in [16, 17]) do notimply (sharp) heat kernel estimates as the example of the free Hamiltonian shows.Namely, let J be defined in ℓ ( Z ) by( J u ) n := − u n − + 2 u n − u n +1 , n ∈ Z . (1.2) J is a bounded self-adjoint operator, whose spectrum is purely absolutely contin-uous and coincides with the interval [0 , − tJ ( n, m ) = e − t I n − m (2 t ) , e i tJ ( n, m ) = e t I n − m (2i t ) , (1.3)for all n, m ∈ Z . Here I k ( z ) = i − k J k (i z ) = ∞ X n =0 n !Γ( n + k + 1) (cid:16) z (cid:17) n + k (1.4)is the modified Bessel function of the first kind [22, (10.25.2)] (we use the convention1 / Γ( m ) = 0 if m ∈ Z ≤ ). This leads to the following bounds k e i tJ k ℓ → ℓ ∞ = sup n,m ∈ Z | e i tJ ( n, m ) | = O ( | t | − / ) (1.5)as t → ∞ , however, k e − tJ k ℓ → ℓ ∞ = O ( | t | − / ) , t → + ∞ . (1.6)It is not at all surprising that the heat kernel of e − tH α is expressed by means ofJacobi polynomials (Theorem 4.1). However, now one is led to the study of Jacobipolynomials outside of the orthogonality interval. Let us now briefly outline thestructure of the paper and the main results.Section 2 is of preliminary character, where we recall the definition of H α andits basic spectral properties.In Section 3 we investigate the quadratic form t α associated with H α . Using aconvenient factorization of the matrix (1.1) (which connects H α with the spectraltheory of Krein strings, see Remark 3.2), we are able to perform a rather detailedstudy of t α (Lemma 3.1). Using the Beurling–Deny criteria, this helps us to concludethat the heat semigroup e − tH α is positivity preserving. Moreover, it is Markovianif α = 0 (that is, e − tH is also ℓ ∞ contractive). However, the string factorizationalso shows that a very simple similarity transformation (3.10) connects H α with thedifference operator e H α , which is Markovian, however, acts in a weighted ℓ space.We investigate heat semigroups e − tH α and e − t e H α in Section 4. First, we computeexplicitly the corresponding heat kernels (Theorem 4.1). On the one hand, theconnection with Jacobi polynomials enables us to obtain the on-diagonal estimatesfor the heat kernels (Theorem 4.6). On the other hand, this allows us to show thatthe continuous time random walk on Z ≥ generated by e H α is recurrent exactly when α >
0. Moreover, it is stochastically complete for all α > −
1. It is interesting tomention that the latter is a consequence of the formula for the generating functionof Meixner polynomials (see Remark 4.4 and Lemma 4.8). Let us stress in thisconnection that orthogonality relations for Meixner polynomials are equivalent tothe unitarity of e − i tH α (see [16, Remark 3.2]). HE DISCRETE LAGUERRE OPERATOR 3
In the final Section 5 we study the negative spectrum of perturbations H α,V of H α . Rank one perturbations of H α enjoy a very detailed treatment (Lemma 5.1).This has several consequences. First of all, for α ∈ ( − ,
0] this immediately impliesthat no matter how small the attractive perturbation V is, it always produces anon-empty negative spectrum (i. e., the presence of a zero energy resonance for α ∈ ( − , α >
0, we can show that for sufficiently small attractiveperturbations V , the negative spectrum of H α,V remains empty. The qualitativemeasure of “smallness” is demonstrated by two estimates (5.9) and (5.13). The lat-ter is the analog of the Bargmann bound for 1D Schr¨odinger operators. The formeris the analog of the Cwikel–Lieb–Rosenblum bound and it actually follows fromthe ultracontractivity estimate (4.18) (indeed, by theorem of Varopoulos, (4.18)is equivalent to the Sobolev-type inequality (5.10), which is known to be furtherequivalent to a CLR-type bound, [21, 9]). Let us stress that the optimal constant C ( α ) in (5.9) remains an open problem. In conclusion let us mention that all theabove results resemble a strong similarity between discrete Laguerre operators and1D radial Schr¨odinger operators (for instance, one may interpret (5.9) as a discreteanalog of the Glaser–Gr¨osse–Martin–Thirring bound [23, Theorem XIII.9(c)]). Notation. R and C have the usual meaning; R > := (0 , ∞ ), R ≥ := [0 , ∞ ), and Z ≥ a := Z ∩ [ a, ∞ ) for any a ∈ R .By Γ is denoted the classical gamma function [22, (5.2.1)]. For x ∈ C and n ∈ Z ≥ ( x ) n := x ( x +1) · · · ( x + n −
1) ( n > , ( x ) := 1; (cid:18) n + xn (cid:19) := ( x + 1) n n ! (1.7)denote the Pochhammer symbol [22, (5.2.4)] and the binomial coefficient , respec-tively. Notice that for − x / ∈ Z ≥ ( x ) n = Γ( x + n )Γ( x ) , (cid:18) n + xn (cid:19) = Γ( x + n + 1)Γ( x + 1)Γ( n + 1) . Moreover, the above formulas allow to define the Pochhammer symbol and thebinomial coefficient for noninteger x , n >
0. Finally, for − c / ∈ Z ≥ the Gausshypergeometric function [22, (15.2.1)] is defined by F (cid:18) a, bc ; z (cid:19) := ∞ X k =0 ( a ) k ( b ) k ( c ) k k ! z k ( | z | < − a or − b ∈ Z ≥ ). (1.8)For a sequence of positive reals σ = ( σ n ) n ≥ ⊂ R > and p ∈ [1 , ∞ ), we denote by ℓ p ( σ ) = ℓ p ( Z ≥ ; σ ) the usual weighted Banach space of sequences u = ( u n ) n ≥ ⊂ C such that k u k ℓ p ( σ ) = (cid:16) X n ≥ | u n | p σ n (cid:17) /p < ∞ . If p = ∞ , then the corresponding norm is given by k u k ℓ ∞ ( σ ) = sup n ≥ | u n | σ n . We shall simply write ℓ p = ℓ p ( Z ≥ ) if σ = . Finally, δ n = ( δ n,k ) k ≥ , n ∈ Z ≥ , isthe standard orthonormal basis in ℓ ( Z ≥ ), where δ n,k is Kronecker’s delta. A. KOSTENKO The discrete Laguerre operator
We start with a precise definition of the operator H α . For a sequence u = ( u n ) n ≥ we define the difference expression τ α : u τ α u by setting( τ α u ) n := − p n ( n + α ) u n − + (2 n + 1 + α ) u n − p ( n + 1)( n + 1 + α ) u n +1 , (2.1)for all n ∈ Z ≥ , where u − := 0 for notational simplicity. Then the operator H α associated with the Jacobi matrix (1.1) is defined by H α : D max → ℓ ( Z ≥ ) u τ α u , (2.2)where D max = { u ∈ ℓ ( Z ≥ ) | τ α u ∈ ℓ ( Z ≥ ) } . (2.3)Notice that D max does not depend on α , however, seems, a closed description of D max is a rather complicated task.Spectral properties of H α are well known. Let us briefly describe them. First ofall, the Carleman test (see, e.g., [3]) implies that H α is self-adjoint. Moreover, thepolynomials of the first kind for (2.1) are given by (see [26, (5.1.10)]) P α,n ( z ) := 1 σ α ( n ) L ( α ) n ( z ) , n ≥ , (2.4)where σ α ( n ) := q L ( α ) n (0) = (cid:18) n + αn (cid:19) / , n ≥ , α > − , (2.5)and L ( α ) n are the Laguerre polynomials [26, Section 5.1]: L ( α ) n ( z ) = e z z − α n ! d n dz n e − z z n + α = (cid:18) n + αn (cid:19) n X k =0 ( − n ) k ( α + 1) k k ! z k , n ≥ . (2.6)Orthogonality relations for P α,n are given by (see [26, (5.1.1)])1Γ( α + 1) Z ∞ P α,n ( λ ) P α,k ( λ )e − λ λ α dλ = δ n,k , n, k ∈ Z ≥ . (2.7)Therefore, the probability measure ρ α ( dλ ) = 1Γ( α + 1) R > ( λ )e − λ λ α dλ (2.8)is the spectral measure of H α , that is, H α is unitarily equivalent to a multiplicationoperator in L ( R > ; ρ α ). Indeed, the map F α : ℓ ( Z ≥ ) → L ( R > ; ρ α ) defined by( F α f )( λ ) := X n ≥ f n P α,n ( λ ) , λ > , (2.9)for all f ∈ ℓ c ( Z ≥ ), extends to an isometric isomorphism. Its inverse is given by( F − α F ) n = 1Γ( α + 1) Z ∞ F ( λ ) P α,n ( λ )e − λ λ α dλ, n ≥ , for every F ∈ L c ( R > ; ρ α ). Then H α = F − α M α F α , (2.10)where M α is the multiplication operator M α : F ( λ ) λF ( λ ) . HE DISCRETE LAGUERRE OPERATOR 5 acting in the Hilbert space L ( R > ; ρ α ). This in particular implies that H α is apositive operator and its spectrum σ ( H α ) coincides with [0 , ∞ ). Moreover, σ ( H α )is purely absolutely continuous of multiplicity 1.The Stieltjes transform of ρ α , which is usually called the Weyl function (or m -function) of H α , is given by m α ( z ) = 1Γ( α + 1) Z + ∞ e − λ λ α λ − z dλ = e − z E α ( − z ) , z ∈ C \ R ≥ , (2.11)where E p ( z ) := z p − Z ∞ z e − t t − p dt = z p − Γ(1 − p, z ) (2.12)denotes the principal value of the generalized exponential integral [22, (8.19.2)] andΓ( s, z ) is the incomplete Gamma function [22, (8.2.2)]. Note that m α ( −
0) := lim x ↓ m α ( − x ) = ( /α, α > , + ∞ , α ∈ ( − , . (2.13)Next, let us define the polynomials of the second kind (see [3]): Q α,n ( z ) := 1Γ( α + 1) Z ∞ P α,n ( z ) − P α,n ( λ ) z − λ e − λ λ α dλ, n ≥ , (2.14)where Q α, ( z ) ≡ Q α, ( z ) ≡ √ α +1 . Then u := ( Q α,n ( z )) n ≥ satisfies ( τ α u ) n = zu n for all n ≥
1. Notice that for all z ∈ C \ R ≥ the linear combinationΨ α,n ( z ) := Q α,n ( z ) + m α ( z ) P α,n ( z ) , n ≥ , (2.15)also known as the Weyl solution in the Jacobi operators context, satisfies(Ψ α,n ( z )) n ≥ ∈ ℓ ( Z ≥ ) (2.16)for all z ∈ C \ R ≥ . In particular, this provides us with the explicit expression ofthe resolvent of H α (actually, with its Green’s function) G α ( z ; n, m ) := (cid:10) ( H α − z ) − δ n , δ m (cid:11) ℓ = ( P α,n ( z )Ψ α,m ( z ) , n ≤ m,P α,m ( z )Ψ α,n ( z ) , n ≥ m. (2.17) Lemma 2.1.
For n, m ∈ Z ≥ , G α ( − n, m ) := lim x ↑− G α ( x ; n, m ) = α σ α (min( n,m )) σ α (max( n,m )) , α > , + ∞ , α ∈ ( − , . (2.18) Proof.
Since H α is self-adjoint, G α ( x ; n, m ) = G α ( x ; m, n ) for any x <
0, so suppose n ≤ m . By (2.17), G α ( − n, m ) = σ α ( n ) (cid:0) Q α,m (0) + σ α ( m ) m α ( − (cid:1) , (2.19) A. KOSTENKO and hence (2.13) implies (2.18) for α ∈ ( − , α >
0, we get by using (2.14), Q α,m (0) = 1Γ( α + 1) σ α ( m ) Z ∞ L ( α ) m ( λ ) − L ( α ) m (0) λ e − λ λ α dλ = 1Γ( α + 1) σ α ( m ) Z ∞ (cid:0) m X k =0 L ( α − k ( λ ) − L ( α ) m (0) (cid:1) e − λ λ α − dλ = 1Γ( α + 1) σ α ( m ) Z ∞ (cid:0) L ( α − ( λ ) − L ( α ) m (0) (cid:1) e − λ λ α − dλ = 1Γ( α + 1) σ α ( m ) Γ( α )(1 − σ α ( m ) )= 1 − σ α ( m ) ασ α ( m ) . Here in the second line we used [22, (18.18.37)] and then orthogonality of theLaguerre polynomials (2.7). It remains to plug the last expression into (2.19). (cid:3) The quadratic form
Let us consider the quadratic form corresponding to the operator H α : t α [ u ] := h H α u, u i ℓ , u ∈ dom( t α ) := dom( H α ) . (3.1)This form is positive since so is H α . Since H α is self-adjoint, t α is closable and itsclosure t α is explicitly given by t α [ u ] = k p H α u k ℓ , u ∈ dom( t α ) = dom( p H α ) , (3.2)where √ H α denotes the positive self-adjoint square root of H α . Lemma 3.1.
The domain dom( t α ) of t α does not depend on α and consists ofthose u ∈ ℓ ( Z ≥ ) for which the series X n ≥ ( n + 1) (cid:12)(cid:12) u n − u n +1 (cid:12)(cid:12) (3.3) is finite. Moreover, for every u ∈ dom( t α ) the form t α admits the representation t α [ u ] = X n ≥ (cid:12)(cid:12) √ n + α + 1 u n − √ n + 1 u n +1 (cid:12)(cid:12) . (3.4) Proof.
Observe that the matrix (1.1) admits “the string factorization” (see, e.g., [3,Appendix], [13, § § h ( α ) n,n = 1 l α ( n ) (cid:18) ω α ( n −
1) + 1 ω α ( n ) (cid:19) , h ( α ) n,n +1 = − ω α ( n ) p l α ( n ) l α ( n + 1) , (3.5)where ω α ( − := 0 and l α ( n ) = | P α,n (0) | = σ α ( n ) = ( α + 1) n n ! , (3.6) ω α ( n ) = − h ( α ) n,n +1 p l α ( n ) l α ( n + 1) = n !( α + 1) n +1 , n ≥ . (3.7)Therefore, the Jacobi matrix (1.1) can be (at least formally) written as H α = L − α ( I − S ) W − α ( I − S ∗ ) L − α , (3.8) HE DISCRETE LAGUERRE OPERATOR 7 where W α and L α are the multiplication operators L α : ( u n ) n ≥ ( σ α ( n ) u n ) n ≥ , W α : ( u n ) n ≥ ( ω α ( n ) u n ) n ≥ , (3.9) S is the shift operator S : ( u n ) n ≥ ( u n − ) n ≥ with the standard convention u − := 0, and S ∗ is the backward shift, S ∗ : ( u n ) n ≥ ( u n +1 ) n ≥ . The representa-tion (3.8) immediately implies t α [ u ] = h H α u, u i ℓ = kW − / α ( I − S ∗ ) L − α u k ℓ = X n ≥ ω α ( n ) (cid:12)(cid:12)(cid:12) u n σ α ( n ) − u n +1 σ α ( n + 1) (cid:12)(cid:12)(cid:12) = X n ≥ (cid:12)(cid:12) √ n + α + 1 u n − √ n + 1 u n +1 (cid:12)(cid:12) , for every u ∈ ℓ c ( Z ≥ ).Consider now the maximally defined form t α , i.e., t α is defined by the RHS in(3.4) on sequences u ∈ ℓ ( Z ≥ ) for which the RHS in (3.4) is finite. It is standardto show that this form is positive and closed in ℓ ( Z ≥ ). However, t α is clearly anextension of the pre-minimal form t α . However, the maximally defined operator H α is self-adjoint and hence t α admits a unique closed extension. Thus t α = t α .Finally, to show that dom( t α ) = dom( t ) for all α > − (cid:12)(cid:12) √ n + α + 1 − √ n + 1 (cid:12)(cid:12) = | α |√ n + α + 1 + √ n + 1 ≤ | α |√ n + 1for all n ≥ α > − (cid:3) Remark 3.2.
It was observed by Mark Krein [13, § that spectral theory of Jacobimatrices admitting factorization (3.5) can be included into the spectral theory ofKrein strings. Indeed, setting x − := 0 , x n = x α ( n ) := n X k =0 l α ( k ) = n X k =0 ( α + 1) k k ! , n ≥ , and introducing a measure ω α on [0 , ∞ ) by ω ([0 , x )) = X x n The operator H α generates a positivity preserving semigroup e − tH α , t > . Moreover, for α = 0 , the corresponding semigroup is Markovian.Proof. By the first Beurling–Deny criterion (see [23, Theorem XIII.50]), it sufficesto notice that h H α | u | , | u |i ℓ ≤ h H α u, u i ℓ in view of (3.4). Here | u | := ( | u n | ) n ≥ . A. KOSTENKO If α = 0, then by Lemma 3.1 t [ u ] = X n ≥ ( n + 1) (cid:12)(cid:12) u n − u n +1 (cid:12)(cid:12) for all u ∈ dom( t ). Suppose additionally that u ≥ 0, that is, u n ≥ n ≥ u, ) also belongs to dom( t ) and, moreover, t [min( u, )] ≤ t [ u ] . By the second Beurling–Deny criterion (see [23, Theorem XIII.51]), e − tH , t > ℓ p for each p ∈ [1 , ∞ ]. This implies that it is Markovianand the corresponding quadratic form t is a Dirichlet form [10, § (cid:3) Remark 3.4. For α = 0 the form t α is not a Dirichlet form. Indeed, by Lemma3.1, for each non-negative u ∈ dom( t α ) = dom( t ) , we get min( u, ) ∈ dom( t ) =dom( t α ) . However, one can construct a positive u ∈ dom( t α ) such that t α [min( u, )] > t α [ u ] . Therefore, the semigroup e − tH α , t > is not Markovian if α = 0 . In fact, the form t α is closely connected with the Dirichlet form and this formwould be important in our analysis. Consider the weighted space ℓ ( Z ≥ ; σ α ). Themultiplication operator L α given by (3.9) defines an isometric isomorphism from ℓ ( Z ≥ ; σ α ) onto ℓ ( Z ≥ ). Consider the operator e H α defined on ℓ ( Z ≥ ; σ α ) by e H α = L − α H α L α . (3.10)The corresponding difference expression is given by( e τ α u ) n = − n u n − + (2 n + 1 + α ) u n − ( n + 1 + α ) u n +1 = 1 σ α ( n ) X k ≥ | k − n | =1 u n − u k ω α (min( n, k )) , n ≥ . (3.11)Then it is easy to check that the quadratic form e t α is simply given by e t α [ u ] = h e H α u, u i ℓ ( σ α ) = kW − / α ( I − S ∗ ) u k ℓ = X n ≥ ω α ( n ) | u n − u n +1 | = X n ≥ ( α + 1) n +1 n ! | u n − u n +1 | (3.12)for every u ∈ ℓ c . The closure of this form is a regular Dirichlet form in ℓ ( Z ≥ ; σ α ). Corollary 3.5. Let α > − and e H α be the operator (3.10) acting in ℓ ( Z ≥ ; σ α ) .Then e H α is Markovian, that is, the corresponding semigroup e − t e H α = L − α e − tH α L α , t > , (3.13) is positivity preserving and ℓ ∞ contractive. Remark 3.6. The first line in (3.11) shows that e H α generates a birth-and-deathprocess on Z ≥ (see [8, Chapter 17.5] ), however, the second line connects e H α witha continuous-time random walk (a simple Markov chain) on Z ≥ (see [10, 15] ).The latter is not at all surprising since their connections with the Stieltjes momentproblem and Krein–Stieltjes strings is widely known. HE DISCRETE LAGUERRE OPERATOR 9 As a by-product of the factorization (3.5) we arrive at the following continuedfraction representation of the exponential integral (2.12) and the incomplete gammafunction Γ( s, z ). We do not need this formula for our future purposes, however, itis so beautiful that we decided to include it together with a short proof. Corollary 3.7. Let α > − . Then E α ( z ) = z α Γ( − α, z ) = e − z z + α + 11 + 1 z + α + 21 + 2 z + α + 31 + 3 . . . , (3.14) which converges for all z ∈ C \ ( −∞ , .Proof. The string factorization (3.5) implies the following Stieltjes continued frac-tion representation of the Weyl function m α (see, e.g., [13, § § m α ( z ) = 1 − z l α (0) + 1 ω α (0) + 1 − z l α (1) + 1 ω α (1) + 1. . . , (3.15)which converges locally uniformly in C \ R ≥ (this follows from the self-adjointnessof H α , see, e.g., [3], [6, § α + n ) σ α ( n − σ α ( n ) = ( α + n ) ( α + 1) n − ( n − n !( α + 1) n = n for all n > 0, we arrive at (3.14). (cid:3) Remark 3.8. The continued fraction expansion (3.14) of the exponential integralis by no means new and the case α = 0 can already be found in the work of Stieltjes [25, Chapter IX] (see also [22, (8.9.2)] , [22, (8.19.17)] , [11, § and [5, (14.1.6)] ). The heat semigroup In this section we look at the one-dimensional discrete heat equation˙ ψ ( t, n ) = − H α ψ ( t, n ) , ( t, x ) ∈ R > × Z ≥ , (4.1)associated with the Laguerre operator H α , as well as at the closely related heatequation ˙ ψ ( t, n ) = − e H α ψ ( t, n ) , ( t, x ) ∈ R > × Z ≥ , (4.2)associated with the operator e H α defined in the previous section. We sete − tH α ( n, m ) := h e − tH α δ n , δ m i ℓ , ( n, m ) ∈ Z ≥ × Z ≥ , (4.3) and e − t e H α ( n, m ) := h e − t e H α δ n , δ m i ℓ ( σ α ) , ( n, m ) ∈ Z ≥ × Z ≥ . (4.4)Notice that (4.4) does not coincide with the matrix representation of e − t e H α in anorthonormal basis (if α = 0) and we defined it this way in order to write the heatkernel of e H α in the form familiar in the continuous context, that is, in the form(e − t e H α u ) n = X m ≥ e − t e H α ( n, m ) u m σ α ( m ) . (4.5)4.1. Connection with Jacobi polynomials. We begin by establishing a connec-tion between the discrete Laguerre operators and Jacobi polynomials, which followsfrom the fact that the Laplace transform of a product of two Laguerre polynomialsis expressed by means of a terminating hypergeometric series. Theorem 4.1. Let α > − . The kernel of the heat semigroup e − tH α is given by e − tH α ( n, m ) = e − tH α ( m, n )= 1(1 + t ) α (cid:18) t − t + 1 (cid:19) n (cid:18) tt + 1 (cid:19) m − n σ α ( m ) σ α ( n ) P ( α,m − n ) n (cid:18) t + 1 t − (cid:19) (4.6) for all n , m ∈ Z ≥ .Proof. Taking into account (2.10), (2.9) and then (2.4), we get h e − tH α δ n , δ m i ℓ = hF − α e − tM α F α δ n , δ m i ℓ = h e − tM α F α δ n , F α δ m i L ( ρ α ) = 1 σ α ( n ) σ α ( m )Γ( α + 1) Z ∞ e − (1+ t ) λ L ( α ) n ( λ ) L ( α ) m ( λ ) λ α dλ, for n, m ∈ Z ≥ . Thus, every element of the kernel of the operator e − tH α is theLaplace transform of a product of two Laguerre polynomials and hence we get (see[7, (4.11.35)] and [22, (15.8.7)]):e − tH α ( n, m ) σ α ( n ) σ α ( m ) = t n + m (1 + t ) n + m + α +1 2 F (cid:18) − n, − mα + 1 ; 1 t (cid:19) , (4.7)By Euler’s transformation [22, (15.8.1)], F (cid:18) − n, − mα + 1 ; 1 t (cid:19) = (cid:18) t − t (cid:19) n F (cid:18) − n, α + m + 1 α + 1 ; 11 − t (cid:19) . Hence by (A.1) and (A.2), (4.7) implies (4.6) (cid:3) Remark 4.2. Formula (4.6) can be derived from [16, Theorem 3.1] by analyticcontinuation. Namely, in [16] , it was shown that the kernel of the unitary evolution e − i tH α is given by e − i tH α ( n, m ) = e − i tH α ( m, n )= 1(1 + i t ) α (cid:18) t + i t − i (cid:19) n (cid:18) tt − i (cid:19) m − n σ α ( m ) σ α ( n ) P ( α,m − n ) n (cid:18) t − t + 1 (cid:19) (4.8) for all n , m ∈ Z ≥ . Replacing i t by t in (4.8) , we end up with (4.6) . We collect some special cases explicitly for later use. HE DISCRETE LAGUERRE OPERATOR 11 Corollary 4.3. (i) In the case n = 0 we have e − tH α (0 , m ) = σ α ( m )(1 + t ) α (cid:18) tt + 1 (cid:19) m , m ∈ Z ≥ . (4.9) (ii) In the case n = 1 we have for m ∈ Z ≥ e − tH α (1 , m ) = 1(1 + t ) α (cid:18) tt + 1 (cid:19) m − (1 + α ) t + m ( t + 1) σ α ( m ) σ α (1) (4.10) (iii) In the case n = m we have e − tH α ( m, m ) = 1(1 + t ) α (cid:18) t − t + 1 (cid:19) m P ( α, m (cid:18) t + 1 t − (cid:19) , m ∈ Z ≥ . (4.11) Proof. Just observe that P ( α,m )0 ( z ) = 1 , P ( α,m − ( z ) = − m + ( m + 1 + α ) z + 12 . (cid:3) Taking into account (3.13), one also easily derives the explicit expression for theheat kernel of (4.2). Since H α and e H α are unitarily equivalent, the matrix of e − t e H α in the orthonormal basis ( L − α δ n ) n ≥ coincides with that of e − tH α . However, ourdefinition of (4.4) is slightly different and in fact (4.3)–(4.4) givese − t e H α ( n, m ) = e − tH α ( n, m ) σ α ( n ) σ α ( m ) , ( n, m ) ∈ Z ≥ × Z ≥ . (4.12) Remark 4.4. The heat kernel can be expressed in terms of Meixner polynomials [22, (18.20.7)] : M n ( x ; β, c ) := F (cid:18) − n, − xβ ; 1 − c (cid:19) . (4.13) Thus (4.7) reads e − t e H α ( n, m ) = e − tH α ( n, m ) σ α ( n ) σ α ( m ) = 1(1 + t ) α +1 (cid:16) tt + 1 (cid:17) n + m M n (cid:16) m ; 1 + α, t t − (cid:17) . (4.14)4.2. Heat semigroup estimates. Our next aim is to obtain uniform estimateson the elements of the heat kernel. First observe the following simple bounds. Lemma 4.5. Let α > − and n, m ≥ . Then (1 + t ) α e − tH α ( n, m ) = σ α ( n ) σ α ( m ) + O ( t − ) (4.15) as t → ∞ , and e − tH α ( n, m ) = (cid:18) max( n, m )min( n, m ) (cid:19) σ α (max( n, m )) σ α (min( n, m )) t | n − m | (1 + O ( t )) (4.16) as t → +0 .Proof. Immediately follows from (A.1) and (A.6) (see also (4.7)). (cid:3) The latter indicates that one can hope for the following uniform estimatesup n,m ∈ Z ≥ | e − tH α ( n, m ) | σ α ( n ) σ α ( m ) ≤ C (1 + t ) α (4.17)for all positive t > 0, where C = C ( α ) > α . The next statementconfirms the desired bound for α ≥ Theorem 4.6. If α ≥ , then k e − tH α k ℓ ( σ α ) → ℓ ∞ ( σ − α ) = 1(1 + t ) α , t > . (4.18) If α ∈ ( − , , then k e − tH α k ℓ ( σ α ) → ℓ ∞ ( σ − α ) ≥ t ) α , t > . (4.19) Proof. By definition, for t > k e − tH α k ℓ ( σ α ) → ℓ ∞ ( σ − α ) = sup n,m ∈ Z ≥ | e − tH α ( n, m ) | σ α ( n ) σ α ( m ) = k e − t e H α k ℓ ( σ α ) → ℓ ∞ . Using (4.9), we get k e − tH α k ℓ ( σ α ) → ℓ ∞ ( σ − α ) ≥ e − tH α (0 , 0) = 1(1 + t ) α for all α > − t > 0. Thus, it remains to show that k e − tH α k ℓ ( σ α ) → ℓ ∞ ( σ − α ) ≤ t ) α , t > , when α ≥ 0. By Corollary 3.5, e − t e H α is positivity preserving and ℓ ∞ contractiveand hence 0 < e − t e H α ( n, n ) ≤ n ≥ t > 0. Therefore, | e − t e H α ( n, m ) | ≤ e − t e H α ( n, n ) · e − t e H α ( m, m ) ≤ max (cid:16) e − t e H α ( n, n ) , e − t e H α ( m, m ) (cid:17) , which immediately implies k e − tH α k ℓ ( σ α ) → ℓ ∞ ( σ − α ) = sup n ≥ e − tH α ( n, n ) σ α ( n ) = sup n ≥ e − t e H α ( n, n ) . Thus, (4.11) implies that it suffices to prove the inequality (cid:18) t − t + 1 (cid:19) n P ( α, n (cid:18) t + 1 t − (cid:19) ≤ (cid:18) n + αn (cid:19) for all n ≥ t > 0. First, using the Rodrigues formula (A.3), we get (cid:18) t − t + 1 (cid:19) n P ( α, n (cid:18) t + 1 t − (cid:19) = (cid:18) t − t + 1 (cid:19) n n X k =0 (cid:18) n + αn − k (cid:19)(cid:18) nk (cid:19) (cid:18) t − (cid:19) k (cid:18) t t − (cid:19) n − k = 1( t + 1) n n X k =0 (cid:18) n + αn − k (cid:19)(cid:18) nk (cid:19) t n − k ) = 1( t + 1) n n X k =0 (cid:18) n + αk (cid:19)(cid:18) nk (cid:19) t k . Since ( t + 1) n = n X k =0 (cid:18) nk (cid:19) t k > n X k =0 (cid:18) n k (cid:19) t k HE DISCRETE LAGUERRE OPERATOR 13 for all t > n ∈ Z ≥ , it suffices to show that (cid:18) n + αk (cid:19)(cid:18) nk (cid:19) ≤ (cid:18) n + αn (cid:19)(cid:18) n k (cid:19) (4.20)for all n, k ∈ Z ≥ with k < n . Using (1.7), it is easy to observe that (4.20) isequivalent to the following inequality k − Y j =0 n + α − jk − j ≤ n − Y j =0 n + α − jn − j k − Y j =0 n − j − k − j − . However, the latter holds exactly when k − Y j =0 k − j − k − j ≤ k − Y j =0 n − j − n − j n − Y j = k n + α − jn − j . Since k < n and α ≥ 0, this inequality clearly holds and hence we arrive at thedesired inequality, which finishes the proof of (4.18). (cid:3) Let us also state explicitly the following result. Corollary 4.7. If α ≥ , then k e − t e H α k ℓ ( σ α ) → ℓ ∞ = 1(1 + t ) α , t > . (4.21) If α ∈ ( − , , then k e − tH α k ℓ ( σ α ) → ℓ ∞ ≥ t ) α , t > . (4.22)4.3. Transience and stochastic completeness. The operator e H α generates arandom walk on Z ≥ (see Remark 3.6). Explicit form of the heat kernel enables usto characterize basic properties of the corresponding random walk. Lemma 4.8. The Markovian semigroup (e − t e H α ) t> is transient if and only if α > .Moreover, it is stochastically complete (conservative) for all α > − .Proof. The first claim can be deduced either from Lemma 2.1 or Lemma 4.5 (see[10, Lemma 1.5.1]).Recall that (e − t e H α ) t> is called stochastically complete (conservative) ife − t e H α = (4.23)for some (and hence for all) t > 0. Taking into account Remark 4.4, (4.23) followsfrom [22, (18.23.3)]. Indeed, the generating function for Meixner polynomials isgiven by X n ≥ ( β ) n n ! M n ( x ; β, c ) z n = (cid:16) − zc (cid:1) x (1 − z ) − x − β , where x ∈ Z ≥ and | z | < 1. However, by (4.5) and (4.14), we get(e − t e H α ) n = X m ≥ e − t e H α ( n, m ) σ α ( m ) = 1(1 + t ) α +1 (cid:16) tt + 1 (cid:17) n X m ≥ ( α + 1) m m ! M n (cid:16) m ; 1 + α, t t − (cid:17)(cid:16) tt + 1 (cid:17) m = 1(1 + t ) α +1 (cid:16) tt + 1 (cid:17) n X m ≥ ( α + 1) m m ! M m (cid:16) n ; 1 + α, t t − (cid:17)(cid:16) tt + 1 (cid:17) m = 1(1 + t ) α +1 (cid:16) tt + 1 (cid:17) n (cid:16) − t − t (cid:17) n (cid:16) − tt + 1 (cid:17) − n − α − = 1 . (cid:3) Remark 4.9. A few remarks are in order.(i) For large classes of graphs there are rather transparent geometric criteriafor stochastic completeness (e.g., via volume growth). In particular, apply-ing [14, Theorem 5] (see also [15, Chapter 9] ) to (3.11) , we get X n ≥ σ α ( n ) ω α ( n ) = X n ≥ n + α + 1 = ∞ , which implies stochastic completeness. Taking into account (4.23) this pro-vides another derivation of the generating function for Meixner polynomi-als.(ii) Stochastic completeness implies uniqueness of the Cauchy problem for (4.2) in ℓ ∞ (respectively, for (4.1) in ℓ ∞ ( σ α ) ), as well as non-existence of pos-itive bounded λ -harmonic functions. We do not plan to discuss this issuehere and only refer for further details to, e.g., [10] , [14] , [15, Chapter 7] . Eigenvalue estimates Consider the perturbed operator H α,V := H α − V, (5.1)where V is a multiplication operator on ℓ ( Z ) given by( V u ) n := v n u n , n ∈ Z ≥ . (5.2)We shall always assume that ( v n ) n ≥ is a real sequence. If V is unbounded (i.e.,( v n ) n ≥ is unbounded), we define the operator H α,V as the maximal operator (anal-ogous to H α ) and this operator is self-adjoint according to the Carleman test (see,e.g., [3]). We shall denote the total multiplicity of the negative spectrum of H α,V by κ − ( H α,V ). Notice that κ − ( H α,V ) is the number (counting multiplicities) of negativeeigenvalues of H α,V if the negative spectrum of H α,V is discrete.For a real sequence v = ( v n ) n ≥ ⊂ R , denote v ± := ( | v | ± v ) / 2. Let also V ± be the corresponding multiplication operators. Since V = V + − V − , the min-maxprinciple implies the following standard estimate κ − ( H α,V ) ≤ κ − ( H α,V + ) . (5.3) HE DISCRETE LAGUERRE OPERATOR 15 Rank one perturbations. We begin with the simplest possible case, whichhowever demonstrates several important features. Let us consider the operator H α ( v n ) := H α − v n h· , δ n i δ n , v n > . (5.4)Thus, H α ( v n ) is a rank one perturbation of the operator H α (the correspondingmatrix coincides with (1.1) except the coefficient h ( α ) n,n replaced by h ( α ) n,n − v n ). Lemma 5.1. Let v n > . If α ∈ ( − , , then κ − ( H α ( v n )) = 1 . (5.5) If α > , then κ − ( H α ( v n )) = ( , v n ∈ (0 , α ) , , v n > α. (5.6) Proof. Since H α is a positive operator, κ − ( H α ( v n )) ≤ 1. Suppose that E < H α ( v n ), that is, there exists f ∈ ℓ ( Z ≥ ) such that H α ( v n ) f = Ef. Therefore, we get ( H α − E ) f = v n h f, δ n i δ n , which shows that f = v n h f, δ n i ( H α − E ) − δ n . Hence by (2.17) 1 v n = (cid:10) ( H α − E ) − δ n , δ n (cid:11) = G α ( E ; n, n ) . (5.7)Since H α is positive, G α ( · ; n, n ) is increasing on ( −∞ , G α ( E ; n, n ) → E → −∞ . Therefore, by (2.1) G α ( · ; n.n ) maps ( −∞ , 0) onto R > if α ∈ ( − , , α − ) if α > 0, which implies that (5.7) has a solution exactly when v n ∈ ( (0 , ∞ ) , α ∈ (0 , , ( α, ∞ ) , α > . This immediately proves the desired claim. (cid:3) Remark 5.2. Notice that in the case n = 0 the corresponding eigenvalue λ ( v ) isexplicitly given by λ ( v ) = m − α (1 /v ) < , (5.8) where m α is the Weyl function (2.11) . For α = 0 this case was considered in [19] . The general case. We begin with the following extension of Lemma 5.1. Lemma 5.3. Let α ∈ ( − , . If v = v + , then κ − ( H α,V ) ≥ .Proof. The proof is immediate from Lemma 5.1 and the min-max principle. (cid:3) Our main aim is to extend the second claim in Lemma 5.1 to more generalpotentials. We begin with the following result, which may be seen as the analog ofthe famous Cwikel–Lieb–Rozenblum bound (see [9], [21], [23, Theorem XIII.12]). Theorem 5.4. Let α > and v + ∈ ℓ α ( Z ≥ ; σ α ) . Then the operator H α,V isbounded from below, its negative spectrum is discrete and, moreover, there is aconstant C = C ( α ) > (independent of u ) such that κ − ( H α,V ) ≤ C ( α ) X n ≥ ( v + n ) α ( α + 1) n n ! . (5.9) Proof. By (5.3), we can assume that V = V + , that is, v n = v + n ≥ n ≥ e H α is Markovian, by Varopoulos’s theorem(see [27, Theorem II.5.2]), (4.21) is equivalent to the validity of the following Sobolev-type inequality (cid:16) X n ≥ | u n | α ( α + 1) n n ! (cid:17) α α +2 ≤ C ( α ) X n ≥ ( α + 1) n +1 n ! | u n − u n +1 | , (5.10)for all u ∈ dom( e t α ) and the constant C ( α ) depends only on α . By Theorem 1.2 from[21], the latter implies that the operator e H α,V = e H α − V is bounded from below,its negative spectrum is discrete and, moreover, there is a constant C = C ( α ) > κ − ( e H α,V ) ≤ C ( α ) X n ≥ v αn ( α + 1) n n ! . (5.11)It remains to notice that the operators e H α,V and H α,V are unitarily equivalentsince V commutes with L α and thus κ − ( H α,V ) = κ − ( H α − V ) = κ − ( L α ( e H α − V ) L − α ) = κ − ( e H α − V ) = κ − ( e H α,V ) . (cid:3) Remark 5.5. Taking into account (4.21) , C ( α ) ≥ α (see [9, Remark 2.1] ).Moreover, the constants C ( α ) and C ( α ) satisfy (see [9, Theorem 2.1] ) C ( α ) α ≤ C ( α ) ≤ e α C ( α ) α . (5.12) The optimal constants in (5.9) and (5.10) remain an open problem. We finish this section with another estimate, which can be seen as the analog ofthe Bargmann bound (see, e.g., [23, Theorem XIII.9(a)]. Theorem 5.6. If α > , then κ − ( H α,V ) ≤ α X n ≥ v + n ( α + 1) n n ! . (5.13) Proof. Again, by (5.3) it suffices to prove (5.13) for V = V + . Let ε > 0. It is thestandard Birman–Schwinger argument (see [24]) that − E < H α,εV if and only if ε − is the eigenvalue of V / ( H α + E ) − V / . Therefore, if theoperator V / H − α V / extends to a bounded operator on ℓ ( Z ≥ ) (notice that H − α is densely defined on ℓ ( Z ≥ ) since 0 is not an eigenvalue of H α ), then the numberof negative eigenvalues of H α,V equals the number of eigenvalues of V / H − α V / which are greater than 1. And hence κ − ( H α,V ) ≤ tr ( V / H − α V / ) . Taking into account (3.8), we (at least formally) get (however, see Remark 5.7below) V / H − α V / = V / L α ( I − S ) − W α ( I − S ∗ ) − L α V / , HE DISCRETE LAGUERRE OPERATOR 17 and hence the trace of this operator is given explicitly bytr( V / H − α V / ) = X n ≥ h V / H − α V / δ n , δ n i ℓ = X n ≥ kW / α ( I − S ∗ ) − L α V / δ n k ℓ = X n ≥ v n ( α + 1) n n ! kW / α ( I − S ∗ ) − δ n k ℓ = X n ≥ v n ( α + 1) n n ! n X k =0 k !( α + 1) k +1 . Thus we get κ − ( H α,V ) ≤ X n ≥ v n ( α + 1) n n ! n X k =0 k !( α + 1) k +1 . (5.14)Now take into account that by [22, (15.4.20)] X k ≥ k !( α + 1) k +1 = 1 α + 1 X k ≥ k !( α + 2) k = 1 α + 1 F (cid:18) , α + 2 ; 1 (cid:19) = 1 α . Combining the latter with (5.14), we arrive at the desired estimate. (cid:3) Remark 5.7. A few remarks are in order.(i) Using the string factorization (3.8) , it follows from [13, § that the oper-ator V / H − α V / with α > is compact exactly when n X k =0 v k l α ( k ) X k>n ω α ( k ) = n X k =0 v k ( α + 1) k k ! X k>n k !( α + 1) k +1 = o (1) as n → ∞ . In particular, the inclusion v ∈ ℓ ( Z ≥ ) would imply compact-ness of V / H − α V / if α > .(ii) Clearly, ℓ ( σ α ) is contained in ℓ α ( σ α ) if α > and hence (5.9) appliesto a wider class of potentials than (5.13) . However, this embedding is notcontinuous. Moreover, we do not know the optimal C ( α ) in (5.9) .(iii) The case α ∈ ( − , requires different considerations and it will be consid-ered elsewhere. However, let us mention that using the Birman–Schwingerprinciple and applying the commutation to the string factorization, onecan show that for α ∈ ( − , , κ − ( H α,V ) ≤ X n ≥ n !( α + 1) n +1 X k ≥ n v + n ( α + 1) k k ! , (5.15) where the second summand on the RHS in (5.15) is dual to the RHS (5.14) .(iv) The study of spectral types (ac-spectrum, sc-spectrum etc.) of a positivespectrum of H α,V is beyond the scope of the present paper, however, seethe recent preprint [28] . Appendix A. Jacobi polynomials For α , β > − 1, let w ( α,β ) ( x ) = (1 − x ) α (1 + x ) β for x ∈ ( − , 1) be a Jacobiweight. The corresponding orthogonal polynomials P ( α,β ) n , normalized by P ( α,β ) n (1) = (cid:18) n + αn (cid:19) = ( α + 1) n n ! (A.1)for all n ≥ Jacobi polynomials . They are expressed as (terminating)Gauss hypergeometric series (1.8) by [26, (4.21.2)] P ( α,β ) n ( x ) P ( α,β ) n (1) = F (cid:18) − n, n + α + β + 1 α + 1 ; 1 − x (cid:19) . (A.2)They also satisfy Rodrigues’ formula [26, (4.3.1), (4.3.2)] P ( α,β ) n ( x ) = n X k =0 (cid:18) n + αn − k (cid:19)(cid:18) n + βk (cid:19) (cid:18) x − (cid:19) k (cid:18) x + 12 (cid:19) n − k (A.3)= ( − n n n ! (1 − x ) − α (1 + x ) − β d n dx n (cid:8) (1 − x ) α + n (1 + x ) β + n (cid:9) . (A.4)This formula immediately implies P ( α,β ) n ( − x ) = ( − n P ( β,α ) n ( x ) , (A.5)and hence P ( α,β ) n ( − 1) = ( − n (cid:18) n + βn (cid:19) = ( − n ( β + 1) n n ! . (A.6)Jacobi polynomials include the Chebyshev polynomials, the ultraspherical (Gegen-bauer) polynomials, and the Legendre polynomials (see [22], [26] for further details). Acknowledgments. I am grateful to Noema Nicolussi for useful discussions. References [1] C. Acatrinei, Noncommutative radial waves , J. Phys. A: Math. Theor. , 215401 (2008).[2] C. Acatrinei, Discrete nonlocal waves , JHEP , 057 (2013).[3] N. I. 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