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Konstantin Pankrashkin

Publications and source records attributed to Konstantin Pankrashkin.

At least 19 recordsLinked to original sources
arXiv2026

MIT bag model and infinite mass limit in non-smooth domains

The work is devoted to the study of Dirac operators with MIT bag boundary conditions in Euclidean domains with compact Lipschitz boundaries in arbitrary dimensions. It is shown that such operators are self-adjoint on suitable definition domains and can be recovered as the norm-resolvent limits of Dirac operators in the whole space with a large mass term outside the domain, under the assumption that an associated Robin-Laplacian eigenvalue has a prescribed asymptotic behavior with respect to a parameter in the boundary condition. This assumption is shown to hold for a class of non-smooth domains, which includes convex domains and, more generally, domains that can be "locally convexified" by suitable diffeomorphisms. To the best of our knowledge, this represents the first infinite mass interpretation for the MIT bag model in the sense of resolvent convergence for non-smooth domains. Most results are extended to the generalized MIT bag boundary conditions with the help of the recently established congruence transform.

math.AP↗
arXiv2025

Eigenvalue asymptotics for strong $δ$-interactions supported on curves with corners

Let $Γ\subset\mathbb{R}^2$ be a piecewise smooth closed curve with corners. We discuss the asymptotic behavior of the individual eigenvalues of the two-dimensional Schrödinger operator $-Δ-αδ_Γ$ for $α\to\infty$, where $δ_Γ$ is the Dirac $δ$-distribution supported by $Γ$. It is shown that the asymptotics of several first eigenvalues is determined by the corner opening only, while the main term in the asymptotic expansion for the other eigenvalues is the same as for smooth curves. Under an additional assumption on the corners of $Γ$ (which is satisfied, in particular, if $Γ$ has no acute corners), a more detailed eigenvalue asymptotics is established in terms of a one-dimensional effective operator on the boundary.

math.SP↗
arXiv2025

Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks

We study the asymptotic behavior of individual eigenvalues of the Laplacian in domains with outward peaks for large negative Robin parameters. A large class of cross-sections is allowed, and the resulting asymptotic expansions reflect both the sharpness of the peak and the geometric shape of its cross-section. The results are an extension of previous works dealing with peaks whose cross-sections are balls.

math.AP↗
arXiv2025

MIT bag in non-smooth convex domains

The Dirac operator with MIT bag boundary condition in a bounded convex domain is shown to be always self-adjoint in the $H^1$-setting. This allows one to show that such operators appear as limit of Dirac operators with large positive mass outside the domain. Similar results were previously known for smooth domains only.

math.AP↗
arXiv2025

Poisson-type problems with transmission conditions at boundaries of infinite metric trees

The paper introduces a Poisson-type problem on a mixed-dimensional structure combining a Euclidean domain and a lower-dimensional self-similar component touching a compact surface (interface). The lower-dimensional piece is a so-called infinite metric tree (one-dimensional branching structure), and the key ingredient of the study is a rigorous definition of the gluing conditions between the two components. These constructions are based on the recent concept of embedded trace maps and some abstract machineries derived from a suitable Green-type formula. The problem is then reduced to the study of Fredholm properties of a linear combination of Dirichlet-to-Neumann maps for the tree and the Euclidean domain, which yields desired existence and uniqueness results. One also shows that finite sections of tree can be used for an efficient approximation of the solutions.

math.AP↗
arXiv2024

Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters

Let $Ω\subset\mathbb{R}^n$ with $n\ge 2$ be a bounded Lipschitz domain with outer unit normal $ν$. For $α\in\mathbb{R}$ let $R_Ω^α$ be the Laplacian in $Ω$ with the Robin boundary condition $\partial_νu+αu=0$, and denote by $E(R^α_Ω)$ its principal eigenvalue. In 2017 Bucur, Freitas and Kennedy stated the following open question: Does the limit of the ratio $E(R_Ω^α)/ α^2$ for $α\to-\infty$ always exist? We give a negative answer.

math.SP↗
arXiv2024

Laplacian eigenvalues with a large negative Robin parameter on a part of the boundary

We consider the Laplacian eigenvalues for smooth planar domains with strongly attractive Robin conditions imposed on a part of the boundary and Neumann condition on the remaining boundary. The asymptotics of individual eigenvalues is described in terms of an effective operator on an interval with boundary conditions at the endpoints. For several typical geometries a more precise asymptotics in terms of the boundary curvature is obtained.

math.SP↗
arXiv2024

On Neumann-Poincaré operators and self-adjoint transmission problems

We discuss the self-adjointness in $L^2$-setting of the operators acting as $-\nabla\cdot h\nabla$, with piecewise constant functions $h$ having a jump along a Lipschitz hypersurface $Σ$, without explicit assumptions on the sign of $h$. We establish a number of sufficient conditions for the self-adjointness of the operator with $H^s$-regularity for suitable $s\in[1,\frac{3}{2}]$, in terms of the jump value and the regularity and geometric properties of $Σ$. An important intermediate step is a link with Fredholm properties of the Neumann-Poincaré operator on $Σ$, which is new for the Lipschitz setting.

math.SP↗
arXiv2023

Asymptotics of Robin eigenvalues on sharp infinite cones

Let $ω\subset\mathbb{R}^n$ be a bounded domain with Lipschitz boundary. For $\varepsilon>0$ and $n\in\mathbb{N}$ consider the infinite cone $Ω_{\varepsilon}:=\big\{(x_1,x')\in (0,\infty)\times\mathbb{R}^n: x'\in\varepsilon x_1ω\big\}\subset\mathbb{R}^{n+1}$ and the operator $Q_{\varepsilon}^α$ acting as the Laplacian $u\mapsto-Δu$ on $Ω_{\varepsilon}$ with the Robin boundary condition $\partial_νu=αu$ at $\partialΩ_\varepsilon$, where $\partial_ν$ is the outward normal derivative and $α>0$. We look at the dependence of the eigenvalues of $Q_\varepsilon^α$ on the parameter $\varepsilon$: this problem was previously addressed for $n=1$ only (in that case, the only admissible $ω$ are finite intervals). In the present work we consider arbitrary dimensions $n\ge2$ and arbitrarily shaped "cross-sections" $ω$ and look at the spectral asymptotics as $\varepsilon$ becomes small, i.e. as the cone becomes "sharp" and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity $N_ω:=\dfrac{\mathrm{Vol}_{n-1} \partialω}{\mathrm{Vol}_n ω}$. More precisely, for any fixed $j\in \mathbb{N}$ and $α>0$ the $j$th eigenvalue $E_j(Q^α_\varepsilon)$ of $Q^α_\varepsilon$ exists for all sufficiently small $\varepsilon>0$ and satisfies $E_j(Q^α_\varepsilon)=-\dfrac{N_ω^2\,α^2}{(2j+n-2)^2\,\varepsilon^2}+O\left(\dfrac{1}{\varepsilon}\right)$ as $\varepsilon\to 0^+$. The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.

math.SP↗
arXiv2023

On the self-adjointness of two-dimensional relativistic shell interactions

We study the self-adjointness of the two-dimensional Dirac operator coupled with electrostatic and Lorentz scalar shell interactions of constant strength $\varepsilon$ and $μ$ supported on a closed Lipschitz curve. Namely, we present several new explicit ranges of $\varepsilon$ and $μ$ for which there is a unique self-adjoint realization with domain included into $H^{\frac{1}{2}}$. A more precise analysis is carried out for curvilinear polygons, which allows one to take the corner openings into account. Compared to the preceding works on this topic, two new technical ingredients are employed: the explicit use of the Cauchy transform on non-smooth curves and the explicit characterization of the Fredholmness for singular integral operators.

math.SP↗
arXiv2023

Embedded trace operator for infinite metric trees

We consider a class of infinite weighted metric trees obtained as perturbations of self-similar regular trees. Possible definitions of the boundary traces of functions in the Sobolev space on such a structure are discussed by using identifications of the tree boundary with a surface. Our approach unifies some constructions proposed by Maury, Salort, Vannier (2009) for dyadic discrete weighted trees (expansion in orthogonal bases of harmonic functions on the graph and using Haar-type bases on the domain representing the boundary), and by Nicaise, Semin (2018) and Joly, Kachanovska, Semin (2019) for fractal metric trees (approximation by finite sections and identification of the boundary with a interval): we show that both machineries give the same trace map, and for a range of parameters we establish the precise Sobolev regularity of the traces. In addition, we introduce new geometric ingredients by proposing an identification with arbitrary Riemannian manifolds. It is shown that any compact manifold admits a suitable multiscale decomposition and, therefore, can be identified with a metric tree boundary in the context of trace theorems.

math-ph↗
arXiv2022

Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions

We discuss the spectral properties of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported by a compact smooth surface. It turns out that the criticality of the interaction may result in a new interval of essential spectrum. The position and the length of the interval are explicitly controlled by the coupling constants and the principal curvatures of the surface. This effect is completely new compared to lower dimensional critical situations or special geometries considered up to now, in which only a single new point in the essential spectrum was observed.

math.SP↗
arXiv2022

On Schrödinger operators with $δ'$-potentials supported on star graphs

The spectral properties of two-dimensional Schrödinger operators with $δ'$-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete spectrum is non-trivial but finite. A more detailed study is presented for the case of a star graph with two branches, in particular, the small angle asymptotics for the eigenvalues is obtained.

math.SP↗
arXiv2022

Precise option pricing by the COS method--How to choose the truncation range

The Fourier cosine expansion (COS) method is used for pricing European options numerically very fast. To apply the COS method, a truncation range for the density of the log-returns need to be provided. Using Markov's inequality, we derive a new formula to obtain the truncation range and prove that the range is large enough to ensure convergence of the COS method within a predefined error tolerance. We also show by several examples that the classical approach to determine the truncation range by cumulants may lead to serious mispricing. Usually, the computational time of the COS method is of similar magnitude in both cases.

q-fin.CP↗
arXiv2020

Spectral analysis of the multi-dimensional diffusion operator with random jumps from the boundary

We develop a Hilbert-space approach to the diffusion process of the Brownian motion in a bounded domain with random jumps from the boundary introduced by Ben-Ari and Pinsky in 2007. The generator of the process is introduced by a diffusion elliptic differential operator in the space of square-integrable functions, subject to non-self-adjoint and non-local boundary conditions expressed through a probability measure on the domain. We obtain an expression for the difference between the resolvent of the operator and that of its Dirichlet realization. We prove that the numerical range is the whole complex plane, despite the fact that the spectrum is purely discrete and is contained in a half-plane. Furthermore, for the class of absolutely continuous probability measures with square-integrable densities we characterise the adjoint operator and prove that the system of root vectors is complete. Finally, under certain assumptions on the densities, we obtain enclosures for the non-real spectrum and find a sufficient condition for the non-zero eigenvalue with the smallest real part to be real. The latter supports the conjecture of Ben-Ari and Pinsky that this eigenvalue is always real.

math.SP↗
arXiv2020

An eigenvalue estimate for a Robin $p$-Laplacian in $C^1$ domains

Let $Ω\subset \mathbb{R}^n$ be a bounded $C^1$ domain and $p>1$. For $α>0$, define the quantity \[ Λ(α)=\inf_{u\in W^{1,p}(Ω),\, u\not\equiv 0} \Big(\int_Ω|\nabla u|^p\,\mathrm{d}x - α\int_{\partialΩ} |u|^p \,\mathrm{d} s\Big)\Big/ \int_Ω|u|^p\,\mathrm{d} x \] with $\mathrm{d} s$ being the hypersurface measure, which is the lowest eigenvalue of the $p$-laplacian in $Ω$ with a non-linear $α$-dependent Robin boundary condition. We show the asymptotics $Λ(α) =(1-p)α^{p/(p-1)}+o(α^{p/(p-1)})$ as $α$ tends to $+\infty$. The result was only known for the linear case $p=2$ or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for $C^{1,λ}$ domains.

math.AP↗