arXiv subjects

Frantisek Stampach

Publications and source records attributed to Frantisek Stampach.

8 recordsLinked to original sources

arXiv2023

Criticality transition for positive powers of the discrete Laplacian on the half line

We study the criticality and subcriticality of powers $(-\Delta)^\alpha$ with $\alpha>0$ of the discrete Laplacian $-\Delta$ acting on $\ell^2(\mathbb{N})$. We prove that these positive powers of the Laplacian are critical if and only if $\alpha \ge 3/2$. We complement our analysis with Hardy type inequalities for $(-\Delta)^\alpha$ in the subcritical regimes $\alpha \in (0,3/2)$. As an illustration of the critical case, we describe the negative eigenvalues emerging by coupling the discrete bilaplacian with an arbitrarily small localized potential.

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arXiv2022

An improved discrete Rellich inequality on the half-line

Based on a new idea of factorization, we prove an improved discrete Rellich inequality and discuss its optimality. We also give a conjecture on improved higher order discrete Hardy-like inequalities and formulate an open problem for the discrete Rellich inequality on the full space.

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arXiv2021

Spectral enclosures and stability for non-self-adjoint discrete Schroedinger operators on the half-line

We make a spectral analysis of discrete Schroedinger operators on the half-line, subject to complex Robin-type boundary couplings and complex-valued potentials. First, optimal spectral enclosures are obtained for summable potentials. Second, general smallness conditions on the potentials guaranteeing a spectral stability are established. Third, a general identity which allows to generate optimal discrete Hardy inequalities for the discrete Dirichlet Laplacian on the half-line is proved.

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arXiv2021

A sharp form of the discrete Hardy inequality and the Keller-Pinchover-Pogorzelski inequality

We give a short proof of a recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.

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arXiv2019

Location of eigenvalues of non-self-adjoint discrete Dirac operators

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.

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arXiv2014

The Hahn-Exton $q$-Bessel function as the characteristic function of a Jacobi matrix

A family $\mathcal{T}^{(ν)}$, $ν\in\mathbb{R}$, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton $q$-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in $\ell^{2}(\mathbb{Z}_{+})$ are essentially self-adjoint for $|ν|\geq1$ and have deficiency indices $(1,1)$ for $|ν|<1$. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton $q$-Bessel function $J_ν(z;q)$ serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the $q$-Bessel function due to Koelink and Swarttouw.

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arXiv2014

Orthogonal polynomials associated with Coulomb wave functions

A class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials do in the theory of Bessel functions. The measure of orthogonality for this new class is described explicitly. In addition, the orthogonality measure problem is also discussed on a more general level. Apart of this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of some formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) matrix $J_{L}$ whose eigenvalues coincide with reciprocal values of the zeros of the regular Coulomb wave function $F_{L}(η,ρ)$. The spectral zeta function corresponding to the regular Coulomb wave function or, more precisely, to the respective tridiagonal matrix is studied as well.

math.CA↗

arXiv2013

Special functions and spectrum of Jacobi matrices

Several examples of Jacobi matrices with an explicitly solvable spectral problem are worked out in detail. In all discussed cases the spectrum is discrete and coincides with the set of zeros of a special function. Moreover, the components of corresponding eigenvectors are expressible in terms of special functions as well. Our approach is based on a recently developed formalism providing us with explicit expressions for the characteristic function and eigenvectors of Jacobi matrices. This is done under an assumption of a simple convergence condition on matrix entries. Among the treated special functions there are regular Coulomb wave functions, confluent hypergeometric functions, q-Bessel functions and q-confluent hypergeometric functions. In addition, in the case of q-Bessel functions, we derive several useful identities.

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