arXiv subjects

Vladimir Lotoreichik

Publications and source records attributed to Vladimir Lotoreichik.

At least 37 records · Page 2Linked to original sources

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.

math.SP↗

arXiv2022

Bound states of weakly deformed soft waveguides

In this paper we consider the two-dimensional Schrödinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the function $\mathbb{R}\ni x \mapsto d+\varepsilon f(x)$, where $d > 0$ is a constant, $\varepsilon > 0$ is a small parameter, and $f$ is a compactly supported continuous function. We prove that if $\int_{\mathbb{R}} f \,\mathsf{d} x > 0$, then the respective Schrödinger operator has a unique simple eigenvalue below the threshold of the essential spectrum for all sufficiently small $\varepsilon >0$ and we obtain the asymptotic expansion of this eigenvalue in the regime $\varepsilon\rightarrow 0$. An asymptotic expansion of the respective eigenfunction as $\varepsilon\rightarrow 0$ is also obtained. In the case that $\int_{\mathbb{R}} f \,\mathsf{d} x < 0$ we prove that the discrete spectrum is empty for all sufficiently small $\varepsilon > 0$. In the critical case $\int_{\mathbb{R}} f \,\mathsf{d} x = 0$, we derive a sufficient condition for the existence of a unique bound state for all sufficiently small $\varepsilon > 0$.

math.SP↗

arXiv2022

An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain

In the present paper we introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth open set. The considered perturbation is of lower order and corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if, and only if, the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the additional assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius of the disk for this property to hold.

math.SP↗

arXiv2022

Self-adjointness of the 2D Dirac operator with singular interactions supported on star-graphs

We consider the two-dimensional Dirac operator with Lorentz-scalar $δ$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum of half-line Dirac operators with off-diagonal Coulomb potentials. This decomposition reduces the computation of the deficiency indices to determining the number of eigenvalues of a one-dimensional spin-orbit operator in the interval $(-1/2,1/2)$. If the number of edges of the star graph is two or three, these deficiency indices can then be analytically determined for a range of parameters. For higher numbers of edges, it is possible to numerically calculate the deficiency indices. Among others, examples are given where the strength of the Lorentz-scalar interactions directly change the deficiency indices while other parameters are all fixed and where the deficiency indices are $(2,2)$, neither of which have been observed in the literature to the best knowledge of the authors. For those Dirac operators which are not already self-adjoint and do not have $0$ in the spectrum of the associated spin-orbit operator, the distinguished self-adjoint extension is also characterized.

math.SP↗

arXiv2022

Spectral optimization for Robin Laplacian on domains admitting parallel coordinates

In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $Ω$ corresponding to a negative Robin parameter does not exceed the analogous quantity for a disk whose boundary has a curvature larger than or equal to the maximum of that for $\partialΩ$.

math.SP↗

arXiv2022

On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter

We consider the magnetic Robin Laplacian with a negative boundary parameter. Among a certain class of domains, we prove that the disk maximizes the ground state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all convex centrally symmetric domains.

math.SP↗

arXiv2022

Schrödinger operators with $δ$-potentials supported on unbounded Lipschitz hypersurfaces

In this note we consider the self-adjoint Schrödinger operator $\mathsf{A}_α$ in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $δ$-potential supported on a Lipschitz hypersurface $Σ\subseteq\mathbb{R}^d$ of strength $α\in L^p(Σ)+L^\infty(Σ)$. We show the uniqueness of the ground state and, under some additional conditions on the coefficient $α$ and the hypersurface $Σ$, we determine the essential spectrum of $\mathsf{A}_α$. In the special case that $Σ$ is a hyperplane we obtain a Birman-Schwinger principle with a relativistic Schrödinger operator as Birman-Schwinger operator. As an application we prove an optimization result for the bottom of the spectrum of $\mathsf{A}_α$.

math.SP↗

arXiv2021

The fate of Landau levels under $δ$-interactions

We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(\mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Λ_q$, $q \in \mathbb{Z}_+$, and the perturbed operator $H_\upsilon = H_0 + \upsilonδ_Γ$, where $Γ\subset \mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $\upsilon \in L^p(Γ;\mathbb{R})$, $p>1$, has a constant sign. We investigate ${\rm Ker}(H_\upsilon -Λ_q)$, $q \in \mathbb{Z}_+$, and show that generically $$0 \leq {\rm dim \, Ker}(H_\upsilon -Λ_q) - {\rm dim \, Ker}(T_q(\upsilon δ_Γ)) < \infty,$$ where $T_q(\upsilon δ_Γ) = p_q (\upsilon δ_Γ)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q L^2(\mathbb{R}^2)$, and $p_q$ is the orthogonal projection on ${\rm Ker}\,(H_0 -Λ_q)$. If $\upsilon \neq 0$ and $q = 0$, we prove that ${\rm Ker}\,(T_0(\upsilon δ_Γ)) = \{0\}$. If $q \geq 1$, and $Γ= \mathcal{C}_r$ is a circle of radius $r$, we show that ${\rm dim \, Ker} (T_q(δ_{\mathcal{C}_r})) \leq q$, and the set of $r \in (0,\infty)$ for which ${\rm dim \, Ker}(T_q(δ_{\mathcal{C}_r})) \geq 1$, is infinite and discrete.

math.SP↗

arXiv2021

General $δ$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation

In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.

math.AP↗

arXiv2020

Trace Hardy inequality for the Euclidean space with a cut and its applications

We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Σ\subset\mathbb R^d$, $d \ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in terms of the geodesic distance to the boundary of $Σ$. We provide its applications to the heat equation on $\mathbb R^d$ with an insulating cut at $Σ$ and to the Schrödinger operator with a $δ'$-interaction supported on $Σ$. We also obtain generalizations of this trace Hardy inequality for a class of unbounded cuts.

math.AP↗

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

Optimization of the lowest eigenvalue of a soft quantum ring

We consider the self-adjoint two-dimensional Schrödinger operator $H_μ$ associated with the differential expression $-Δ-μ$ describing a particle exposed to an attractive interaction given by a measure $μ$ supported in a closed curvilinear strip and having fixed transversal one-dimensional profile measure $μ_\bot$. This operator has nonempty negative discrete spectrum and we obtain two optimization results for its lowest eigenvalue. For the first one, we fix $μ_\bot$ and maximize the lowest eigenvalue with respect to shape of the curvilinear strip the optimizer in the first problem turns out to be the annulus. We also generalize this result to the situation which involves an additional perturbation of $H_μ$ in the form of a positive multiple of the characteristic function of the domain surrounded by the curvilinear strip. Secondly, we fix the shape of the curvilinear strip and minimize the lowest eigenvalue with respect to variation of $μ_\bot$, under the constraint that the total profile measure $α>0$ is fixed. The optimizer in this problem is $μ_\bot$ given by the product of $α$ and the Dirac $δ$-function supported at an optimal position.

math-ph↗

arXiv2020

Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions

We obtain new Faber-Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber-Krahn inequality for the Schrödinger operator with point interaction: the optimiser is the ball with the point interaction supported at its centre. Next, we establish three-dimensional Faber-Krahn inequalities for one- and two-body Schrödinger operator with attractive Coulomb interactions, the optimiser being given in terms of Coulomb attraction at the centre of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.

math-ph↗

arXiv2020

A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of $\mathbb{R}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegö type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.

math.SP↗

arXiv2019

Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones

We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_\circ \ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_\circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.

math.SP↗

arXiv2019

Self-adjoint extensions of the two-valley Dirac operator with discontinuous infinite mass boundary conditions

We consider the four-component two-valley Dirac operator on a wedge in $\mathbb{R}^2$ with infinite mass boundary conditions, which enjoy a flip at the vertex. We show that it has deficiency indices $(1,1)$ and we parametrize all its self-adjoint extensions, relying on the fact that the underlying two-component Dirac operator is symmetric with deficiency indices $(0,1)$. The respective defect element is computed explicitly. We observe that there exists no self-adjoint extension, which can be decomposed into an orthogonal sum of two two-component operators. In physics, this effect is called mixing the valleys.

math-ph↗

arXiv2019

A sharp upper bound on the spectral gap for graphene quantum dots

The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected $C^3$-domains with infinite mass boundary conditions. This bound is given in terms of a conformal variation, explicit geometric quantities and of the first eigenvalue for the disk. Its proof relies on the min-max principle applied to the squares of these Dirac operators. A suitable test function is constructed by means of a conformal map. This general upper bound involves the norm of the derivative of the underlying conformal map in the Hardy space $\mathcal{H}^2(\mathbb{D})$. Then, we apply known estimates of this norm for convex and for nearly circular, star-shaped domains in order to get explicit geometric upper bounds on the eigenvalue. These bounds can be re-interpreted as reverse Faber-Krahn-type inequalities under adequate geometric constraints.

math.SP↗

arXiv2019

On Dirac operators in $\mathbb{R}^3$ with electrostatic and Lorentz scalar $δ$-shell interactions

In this article Dirac operators $A_{η, τ}$ coupled with combinations of electrostatic and Lorentz scalar $δ$-shell interactions of constant strength $η$ and $τ$, respectively, supported on compact surfaces $Σ\subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $δ$-potentials are modelled by coupling conditions at $Σ$. In the proof of the self-adjointness of $A_{η, τ}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{η, τ}$ is possible. In particular, the essential spectrum of $A_{η, τ}$ is determined, it is shown that at most finitely many discrete eigenvalues can appear, and several symmetry relations in the point spectrum are obtained. Moreover, the nonrelativistic limit of $A_{η, τ}$ is computed and it is discussed that for some special interaction strengths $A_{η, τ}$ is decoupled to two operators acting in the domains with the common boundary $Σ$.

math.SP↗