arXiv subjects

František Štampach

Publications and source records attributed to František Štampach.

At least 19 recordsLinked to original sources

arXiv2026

Optimal fractional discrete Hardy inequalities on the half-line

We consider a Toeplitz realisation of the fractional discrete Laplacian $(-Δ)^α$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to $\ell^{2}(\mathbb{N})$. For all $α>0$, we prove that the fractional Hardy inequality $$(-Δ)^α\geq\frac{4^αΓ^2(α+1/2)}π\frac{Γ(2\,\cdot\,-1)}{Γ(2\,\cdot\,-1+2α)}$$ holds on $\ell^{2}(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of $\ell^{2}(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.

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arXiv2026

Optimal discrete $p$-Hardy-Rellich-Birman inequalities

We present a theory for constructing optimal lower bounds for the discrete half-line $p$-Laplacian of higher order $\ell\in\mathbb{N}$ and general $p>1$. The abstract framework introduces higher-order monotonicity and asymptotic constraints on a parameter sequence that determines optimal weights. As a concrete application, we specialize the parameter sequence to deduce new optimal discrete $p$-Hardy ($\ell=1$), $p$-Rellich ($\ell=2$), and $p$-Birman ($\ell\geq 3$) inequalities.

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arXiv2026

The $p$-Hardy-Rellich-Birman inequalities on the half-line

The classical discrete $p$-Hardy inequality establishes a sharp relationship between the $\ell^{p}$-norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer order $\ell \geq 1$, yielding discrete $p$-Rellich ($\ell=2$) and general $p$-Birman ($\ell \geq 3$) inequalities. As a key step in the proof, we deduce a variant of the Copson inequality with a negative exponent, which may be of independent interest. Furthermore, we demonstrate how the continuous $p$-Birman inequality can be recovered from our discrete version, providing an alternative proof of this classical result. All constants in the obtained inequalities are shown to be optimal.

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arXiv2026

Schrödinger operators on the half-line with integrable complex potentials

In our previous work, we introduced the concept of a \emph{spectral pair} for a half-line Schrödinger operator with a \emph{complex} bounded potential $q$, serving as a substitute for the spectral measure in a non-self-adjoint setting. In this paper, we study the case of $q \in L^1(\mathbb{R}_+)$. We derive explicit formulas for the spectral pair in terms of the Jost solutions of a system of two equations naturally associated with the non-self-adjoint Schrödinger operator. A key component of our work, which is of independent interest, is the existence proof and analysis of these Jost solutions.

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arXiv2025

Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients

We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients $α_{n,N}$ when the ratio $n/N$ converges as $n,N\to\infty$. First, we give a streamlined proof of ratio asymptotics for orthogonal and paraorthogonal polynomials in the case of asymptotically constant and asymptotically periodic coefficients $α_{n,N}$. Second, we determine the asymptotic zero distribution of paraorthogonal polynomials in the locally constant and locally periodic regimes. Analogous results are obtained for orthogonal polynomials under a mild additional condition on the varying coefficients.

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arXiv2025

On Lieb-Thirring inequalities for multidimensional Schrödinger operators with complex potentials

We solve the open problem by Demuth, Hansmann, and Katriel announced in [Integr. Equ. Oper. Theory 75 (2013), 1-5] by a counter-example construction. The problem concerns a possible generalisation of the Lieb-Thirring inequality for Schrödinger operators in to the case of complex-valued potentials. A counter-example has already been found for the one-dimensional case by the first and third authors in [J. Spectr. Theory 11 (2021), 1391-1413]. Here we generalise the counter-example to higher dimensions.

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arXiv2025

The Borg-Marchenko uniqueness theorem for complex potentials

We introduce and study a new theoretical concept of \textit{spectral pair} for a Schrödinger operator $H$ in $L^2(\mathbb{R}_{+})$ with a bounded \textit{complex-valued} potential. The spectral pair consists of a scalar measure and a complex-valued function. We show that in many ways, the spectral pair generalises the classical spectral measure to the non-self-adjoint case. First, extending the classical Borg-Marchenko theorem, we prove a uniqueness result: the spectral pair uniquely determines the operator $H$. Second, we derive asymptotic formulas for the spectral pair in the spirit of the classical result of Marchenko. In the case of real-valued potentials, we relate the spectral pair to the spectral measure of $H$. Lastly, we provide formulas for the spectral pair at a~simple eigenvalue of~$|H|$.

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arXiv2025

A Herglotz-Nevanlinna function from the optimal discrete $p$-Hardy weight

It was recently proved by Fischer, Keller, and Pogorzelski in [Integr. Equ. Oper. Theory, 95(24), 2023] that the classical discrete $p$-Hardy inequality admits an improvement, and the optimal $p$-Hardy weight $ω_{p}$ was determined therein. We prove that $ω_{p}$ directly corresponds to a Herglotz-Nevanlinna function, establish an integral representation for this function, and consequently confirm a slight modification of a conjecture on its absolute monotonicity from the aforementioned article.

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arXiv2024

A functional model and tridiagonalisation for symmetric anti-linear operators

We consider the class of bounded symmetric anti-linear operators $B$ with a cyclic vector. We associate with $B$ the spectral data consisting of a probability measure and a function. In terms of the spectral data of $B$, we introduce a functional model operator $\mathcal{B}$ acting on a model space. We prove an anti-linear variant of the spectral theorem demonstrating that $B$ is unitarily equivalent to $\mathcal{B}$. Next, we show that $B$ is also unitarily equivalent to an anti-linear tridiagonal operator and discuss connection with orthogonal polynomials in the anti-linear setting.

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arXiv2024

Optimal discrete Hardy-Rellich-Birman inequalities

We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power $\ell$ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit optimal discrete Rellich ($\ell=2$) and Birman ($\ell\geq3$) weights. For $\ell=1$, we rediscover the optimal Hardy weight of Keller-Pinchover-Pogorzelski. For $\ell=2$, we improve upon the best known Rellich weights due to Gerhat-Krejčiřík-Štampach and Huang-Ye. For $\ell\geq3$, our main result proves a conjecture by Gerhat-Krejčiřík-Štampach and improves the discrete analogue of the classical Birman weight due to Huang-Ye to the optimal.

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arXiv2023

On the ground state of lattice Schrödinger operators

We prove necessary and sufficient conditions for lattice Schrödinger operators to have a zero energy bound state in arbitrary dimension. The two criteria are sharp, complementary, and depend crucially on both the dimension and asymptotic behaviour of the potential. The method relies on a discrete variant of Agmon's comparison principle which is also proven. Our results represent a discrete variant of the recent criteria obtained in the continuous setting by D. Hundertmark, M. Jex, and M. Lange [$\textit{Forum Mathematics, Sigma}$ $\textbf{11}$ (2023)].

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arXiv2023

An inverse spectral problem for non-self-adjoint Jacobi matrices

We consider the class of bounded symmetric Jacobi matrices $J$ with positive off-diagonal elements and complex diagonal elements. With each matrix $J$ from this class, we associate the spectral data, which consists of a pair $(ν,ψ)$. Here $ν$ is the spectral measure of $|J|=\sqrt{J^*J}$ and $ψ$ is a $\textit{phase function}$ on the real line satisfying $|ψ|\leq1$ almost everywhere with respect to the measure $ν$. Our main result is that the map from $J$ to the pair $(ν,ψ)$ is a bijection between our class of Jacobi matrices and the set of all spectral data.

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arXiv2022

Non-self-adjoint Toeplitz matrices whose principal submatrices have real spectrum

We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to $\infty$. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in $\mathbb{C}\setminus\{0\}$. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the submatrices have to be real. The latter claim is also proven for matrices given by a more general symbol. Further, the limiting eigenvalue distribution of a real banded Toeplitz matrix is related to the solution of a determinate Hamburger moment problem. We use this to derive a formula for the limiting measure using a parametrization of the Jordan curve. We also describe a Jacobi operator, whose spectral measure coincides with the limiting measure. We show that this Jacobi operator is a compact perturbation of a tridiagonal Toeplitz matrix. Our main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically. Update: The proof of Theorem 8 contains an error. An erratum is attached in the end

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arXiv2022

Asymptotic root distribution of Charlier polynomials with large negative parameter

We analyze the asymptotic distribution of roots of Charlier polynomials with negative parameter depending linearly on the index. The roots cluster on curves in the complex plane. We determine implicit equations for these curves and deduce the limiting density of the root distribution supported on these curves. The proof is based on a determination of the limiting Cauchy transform in a specific region and a careful application of the saddle point method. The obtained result represents a solvable example of a more general open problem.

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arXiv2022

Asymptotic spectral properties of the Hilbert $L$-matrix

We study asymptotic spectral properties of the generalized Hilbert $L$-matrix \[ L_{n}(ν)=\left(\frac{1}{\max(i,j)+ν}\right)_{i,j=0}^{n-1}, \] for large order $n$. First, for general $ν\neq0,-1,-2,\dots$, we deduce the asymptotic distribution of eigenvalues of $L_{n}(ν)$ outside the origin. Second, for $ν>0$, asymptotic formulas for small eigenvalues of $L_{n}(ν)$ are derived. Third, in the classical case $ν=1$, we also prove asymptotic formulas for large eigenvalues of $L_{n}\equiv L_{n}(1)$. I particular, we obtain an asymptotic expansion of $\|L_{n}\|$ improving Wilf's formula for the best constant in truncated Hardy's inequality.

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arXiv2022

The Hilbert $L$-matrix

We analyze spectral properties of the Hilbert $L$-matrix $$\left(\frac{1}{\max(m,n)+ν}\right)_{m,n=0}^{\infty}$$ regarded as an operator $L_ν$ acting on $\ell^{2}(\mathbb{N}_{0})$, for $ν\in\mathbb{R}$, $ν\neq0,-1,-2,\dots$. The approach is based on a spectral analysis of the inverse of $L_ν$, which is an unbounded Jacobi operator whose spectral properties are deducible in terms of the unit argument ${}_{3}F_{2}$-hypergeometric functions. In particular, we give answers to two open problems concerning the operator norm of $L_ν$ published by L. Bouthat and J. Mashreghi in [Oper. Matrices 15, No. 1 (2021), 47--58]. In addition, several general aspects concerning the definition of an $L$-operator, its positivity, and Fredholm determinants are also discussed.

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arXiv2021

On diagonalizable quantum weighted Hankel matrices

A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on $\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator commutes with a diagonalizable Jacobi operator corresponding to Al-Salam-Chihara orthogonal polynomials. Yet another weighted Hankel matrix, which commutes with a Jacobi operator associated with the continuous $q$-Laguerre polynomials, is diagonalized. As an application, several new integral formulas for selected quantum orthogonal polynomials are deduced. In addition, an open research problem concerning a quantum Hilbert matrix is also mentioned.

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arXiv2021

The asymptotic zero distribution of Lommel polynomials as polynomials of the order with a variable complex argument

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence of the weak limit of corresponding root-counting measures and deduce formulas for the supporting curves and density. The obtained result represents a solvable example of a more general problem which is still open. Numerical illustrations of the main result are also involved.

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