Abstract
For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.
Highlights
Our main objects of study are the discrete Laguerre operators ⎛ 1+α √ − 1+α · · ·⎞ Hα := ⎜⎜⎜⎜⎜⎜⎜⎜⎝−√100 + α 3+α √ − 2(2 + α) √ − 2(2
It turned out that the unitary evolution eit Hα can be expressed by means of Jacobi polynomials
It is not at all surprising that the heat kernel of e−t Hα is expressed by means of Jacobi polynomials (Theorem 4.1)
Summary
Dispersive estimates for the unitary evolution play a crucial role in the understanding of stability of soliton manifolds appearing in these models It turned out (see [19,20]) that the unitary evolution eit Hα can be expressed by means of Jacobi polynomials It is not at all surprising that the heat kernel of e−t Hα is expressed by means of Jacobi polynomials (Theorem 4.1). Using the Beurling–Deny criteria, this helps us to conclude that the heat semigroup e−t Hα is positivity preserving. We investigate heat semigroups e−t Hα and e−t Hα in Sect. The connection with Jacobi polynomials enables us to obtain the on-diagonal estimates for the heat kernels (Theorem 4.6). For α > 0, we can show that for sufficiently small attractive perturbations V , the negative spectrum of
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