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Vladimir Lotoreichik

Publications and source records attributed to Vladimir Lotoreichik.

At least 19 recordsLinked to original sources
arXiv2026

On the discrete spectrum of Dirac operators with Lorentz-scalar $δ$-shell interactions supported on unbounded curves

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $δ$-shell interaction of strength $τ\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $Σ\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $Σ$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $Σ$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $τ$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.

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arXiv2026

Isoperimetric inequalities for the lowest magnetic Steklov eigenvalue

This paper studies the optimization of the lowest eigenvalue of the magnetic Steklov problem on planar domains. In the bounded domain setting and for magnetic fields of moderate strengths, we prove that among all simply-connected smooth domains of given area, the disk maximises the lowest magnetic Steklov eigenvalue. For exterior domains, we establish a similar isoperimetric inequality for magnetic fields of moderate strength under fixed perimeter constraint and additional geometric and symmetry assumptions. The proofs rely on the method of torsion-type trial functions in the bounded domain case and on the method of trial functions dependent only on the distance to the boundary in the exterior domain case.

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arXiv2026

Convergence of Schrödinger operators on domains with scaled resonant potentials

We consider Schrödinger operators on a bounded, smooth domain of dimension $d \ge 2$ with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to analyse the convergence of these operators as the scaling parameter tends to zero. If the scaled potential is resonant, the limit in strong resolvent sense is a Robin Laplacian with boundary coefficient expressed in terms of the mean curvature of the boundary. A counterexample shows that norm resolvent convergence cannot hold in general in this setting. If the scaled potential is non-resonant and satisfies an explicit assumption on the smallness of the negative part, the limit in strong resolvent sense is the Dirichlet Laplacian. We conjecture that we can drop this additional assumption in the non-resonant case.

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arXiv2026

Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane

We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. We show that the essential spectrum of this operator is $[\frac14,\infty)$ and that, under convexity assumption on the domain, there exist discrete eigenvalues below $\frac14$ if, and only if, the Robin parameter is below a non-positive critical constant, which depends on the shape of the domain. As the main result, we prove that the lowest Robin eigenvalue for the exterior of a bounded geodesically convex domain $Ω$ in the hyperbolic plane does not exceed such an eigenvalue for the exterior of the geodesic disk, whose geodesic curvature of the boundary is not smaller than the averaged geodesic curvature of the boundary of $Ω$. This result implies as a consequence that under fixed area or fixed perimeter constraints the exterior of the geodesic disk maximises the lowest Robin eigenvalue among exteriors of bounded geodesically convex domains. Moreover, we obtain under the same geometric constraints a reverse inequality between the critical constants.

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arXiv2025

On shape optimization with large magnetic fields in two dimensions

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this conclusion. Our main result implies that if, for a bounded simply-connected planar domain, the n-th eigenvalue of the magnetic Dirichlet Laplacian with uniform magnetic field is smaller than the corresponding eigenvalue for a disk of the same area, then the Fraenkel asymmetry of that domain tends to zero in the strong magnetic field limit. Comparable results are also derived for the magnetic Dirichlet Laplacian on rectangles, as well as the magnetic Dirac operator with infinite mass boundary conditions on smooth domains. As part of our analysis, we additionally provide a new estimate for the torsion function on rectangles.

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arXiv2025

A note on optimization of the second positive Neumann eigenvalue for parallelograms

It has recently been conjectured by Bogosel, Henrot, and Michetti that the second positive eigenvalue of the Neumann Laplacian is maximized, among all planar convex domains of fixed perimeter, by the rectangle with one edge length equal to twice the other. In this note we prove that this conjecture is true within the class of parallelogram domains.

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arXiv2024

On the Laplace operator with a weak magnetic field in exterior domains

We study the magnetic Laplacian in a two-dimensional exterior domain with Neumann boundary condition and uniform magnetic field. For the exterior of the disk we establish accurate asymptotics of the low-lying eigenvalues in the weak magnetic field limit. For the exterior of a star-shaped domain, we obtain an asymptotic upper bound on the lowest eigenvalue in the weak field limit, involving the $4$-moment, and optimal for the case of the disk. Moreover, we prove that, for moderate magnetic fields, the exterior of the disk is a local maximizer for the lowest eigenvalue under a $p$-moment constraint.

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arXiv2024

Quasi-conical domains with embedded eigenvalues

The spectrum of the Dirichlet Laplacian on any quasi-conical open set coincides with the non-negative semi-axis. We show that there is a connected quasi-conical open set such that the respective Dirichlet Laplacian has a positive (embedded) eigenvalue. This open set is constructed as the tower of cubes of growing size connected by windows of vanishing size. Moreover, we show that the sizes of the windows in this construction can be chosen so that the absolutely continuous spectrum of the Dirichlet Laplacian is empty.

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arXiv2024

Inequalities between Dirichlet and Neumann eigenvalues of the magnetic Laplacian

We consider the magnetic Laplacian with the homogeneous magnetic field in two and three dimensions. We prove that the $(k+1)$-th magnetic Neumann eigenvalue of a bounded convex planar domain is not larger than its $k$-th magnetic Dirichlet eigenvalue. In three dimensions, we restrict our attention to convex domains, which are invariant under rotation by an angle of $π$ around an axis parallel to the magnetic field. For such domains, we prove that the $(k+2)$-th magnetic Neumann eigenvalue is not larger than the $k$-th magnetic Dirichlet eigenvalue provided that this Dirichlet eigenvalue is simple. The proofs rely on a modification of the strategy due to Levine and Weinberger.

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arXiv2024

Homogenization of the Dirac operator with position-dependent mass

We address the homogenization of the two-dimensional Dirac operator with position-dependent mass. The mass is piecewise constant and supported on small pairwise disjoint inclusions evenly distributed along an $\varepsilon$-periodic square lattice. Under rather general assumptions on geometry of these inclusions we prove that the corresponding family of Dirac operators converges as $\varepsilon\to 0$ in the norm resolvent sense to the Dirac operator with a constant effective mass provided the masses in the inclusions are adjusted to the scaling of the geometry. We also estimate the speed of this convergence in terms of the scaling rates.

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arXiv2023

A geometric bound on the lowest magnetic Neumann eigenvalue via the torsion function

We obtain an upper bound on the lowest magnetic Neumann eigenvalue of a bounded, convex, smooth, planar domain with moderate intensity of the homogeneous magnetic field. This bound is given as a product of a purely geometric factor expressed in terms of the torsion function and of the lowest magnetic Neumann eigenvalue of the disk having the same maximal value of the torsion function as the domain. The bound is sharp in the sense that equality is attained for disks. Furthermore, we derive from our upper bound that the lowest magnetic Neumann eigenvalue with the homogeneous magnetic field is maximized by the disk among all ellipses of fixed area provided that the intensity of the magnetic field does not exceed an explicit constant dependent only on the fixed area.

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arXiv2023

Isoperimetric inequalities for inner parallel curves

We prove weighted isoperimetric inequalities for smooth, bounded, and simply connected domains. More precisely, we show that the moment of inertia of inner parallel curves for domains with fixed perimeter attains its maximum for a disk. This inequality, which was previously only known for convex domains, allows us to extend an isoperimetric inequality for the magnetic Robin Laplacian to non-convex centrally symmetric domains. Furthermore, we extend our isoperimetric inequality for moments of inertia, which are second moments, to $p$-th moments for all $p$ smaller than or equal to two. We also show that the disk is a strict local maximiser in the nearly circular, centrally symmetric case for all $p$ strictly less than three, and that the inequality fails for all $p$ strictly bigger than three.

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arXiv2023

Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain

We consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric.

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arXiv2023

Spectral asymptotics of the Dirac operator in a thin shell

We investigate the spectrum of the Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a smooth compact hypersurface in $\mathbb{R}^n$ without boundary. We prove that when the tubular neighborhood shrinks to the hypersurface, the asymptotic behavior of the eigenvalues is driven by a Schrödinger operator involving electric and Yang-Mills potentials, both of geometric nature.

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arXiv2023

Reverse isoperimetric inequality for the lowest Robin eigenvalue of a triangle

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.

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arXiv2023

Self-adjointness for the MIT bag model on an unbounded cone

We consider the massless Dirac operator with the MIT bag boundary conditions on an unbounded three-dimensional circular cone. For convex cones, we prove that this operator is self-adjoint defined on four-component $H^1$--functions satisfying the MIT bag boundary conditions. The proof of this result relies on separation of variables and spectral estimates for one-dimensional fiber Dirac-type operators. Furthermore, we provide a numerical evidence for the self-adjointness on the same domain also for non-convex cones. Moreover, we prove a Hardy-type inequality for such a Dirac operator on convex cones, which, in particular, yields stability of self-adjointness under perturbations by a class of unbounded potentials. Further extensions of our results to Dirac operators with quantum dot boundary conditions are also discussed.

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arXiv2023

Spectral analysis of the Dirac operator with a singular interaction on a broken line

We consider the one-parametric family of self-adjoint realizations of the two-dimensional massive Dirac operator with a Lorentz scalar $δ$-shell interaction of strength $τ\in\mathbb{R}\setminus\{-2,0,2\}$ supported on a broken line of opening angle $2ω$ with $ω\in(0,\fracπ{2})$. The essential spectrum of any such self-adjoint realization is symmetric with respect to the origin with a gap around zero whose size depends on the mass and, for $τ< 0$, also on the strength of the interaction, but does not depend on $ω$. As the main result, we prove that for any $N\in\mathbb{N}$ and strength $τ\in(-\infty,0)\setminus\{-2\}$ the discrete spectrum of any such self-adjoint realization has at least $N$ discrete eigenvalues, with multiplicities taken into account, in the gap of the essential spectrum provided that $ω$ is sufficiently small. Moreover, we obtain an explicit estimate on $ω$ sufficient for this property to hold. For $τ\in(0,\infty)\setminus\{2\}$, the discrete spectrum consists of at most one simple eigenvalue.

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arXiv2023

Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator

We prove that the $(k+d)$-th Neumann eigenvalue of the biharmonic operator on a bounded connected $d$-dimensional $(d\ge2)$ Lipschitz domain is not larger than its $k$-th Dirichlet eigenvalue for all $k\in\mathbb{N}$. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the $(k+d+1)$-th Neumann eigenvalue of the biharmonic operator does not exceed its $k$-th Dirichlet eigenvalue for all $k\in\mathbb{N}$. In particular, in two dimensions, this special class consists of domains having an axis of symmetry.

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