arXiv subjects

Vladimir Lotoreichik

Publications and source records attributed to Vladimir Lotoreichik.

At least 73 records · Page 4Linked to original sources

arXiv2016

On the spectral properties of Dirac operators with electrostatic $δ$-shell interactions

In this paper the spectral properties of Dirac operators $A_η$ with electrostatic $δ$-shell interactions of constant strength $η$ supported on compact smooth surfaces in $\mathbb{R}^3$ are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of $A_η$ are investigated. In particular, it turns out that the discrete spectrum of $A_η$ inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of $A_η$ and the free Dirac operator $A_0$ is trace class, and in the nonrelativistic limit $A_η$ converges in the norm resolvent sense to a Schrödinger operator with an electric $δ$-potential of strength $η$.

math.SP↗

arXiv2016

Spectral transitions for Aharonov-Bohm Laplacians on conical layers

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux we establish a Hardy-type inequality. In the regime with infinite discrete spectrum we obtain sharp spectral asymptotics with refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.

math.SP↗

arXiv2016

On absence of bound states for weakly attractive $δ^\prime$-interactions supported on non-closed curves in $\mathbb{R}^2$

Let $Λ\subset\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which is either bounded with two free endpoints or unbounded with one free endpoint. Let $u_\pm|_Λ\in L^2(Λ)$ be the traces of a function $u$ in the Sobolev space $H^1({\mathbb R}^2\setminus Λ)$ onto two faces of $Λ$. We prove that for a wide class of shapes of $Λ$ the Schrödinger operator $\mathsf{H}_ω^Λ$ with $δ^\prime$-interaction supported on $Λ$ of strength $ω\in L^\infty(Λ;\mathbb{R})$ associated with the quadratic form \[ H^1(\mathbb{R}^2\setminusΛ)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u \big|^2 \mathsf{d} x - \int_Λω\big| u_+|_Λ- u_-|_Λ\big|^2 \mathsf{d} s \] has no negative spectrum provided that $ω$ is pointwise majorized by a strictly positive function explicitly expressed in terms of $Λ$. If, additionally, the domain $\mathbb{R}^2\setminusΛ$ is quasi-conical, we show that $σ(\mathsf{H}_ω^Λ) = [0,+\infty)$. For a bounded curve $Λ$ in our class and non-varying interaction strength $ω\in\mathbb{R}$ we derive existence of a constant $ω_* > 0$ such that $σ(\mathsf{H}_ω^Λ) = [0,+\infty)$ for all $ω\in (-\infty, ω_*]$; informally speaking, bound states are absent in the weak coupling regime.

math-ph↗

arXiv2016

Spectra of definite type in waveguide models

We develop an abstract method to identify spectral points of definite type in the spectrum of the operator $T_1\otimes I_2 + I_1\otimes T_2$. The method is applicable in particular for non-self-adjoint waveguide type operators with symmetries. Using the remarkable properties of the spectral points of definite type, we obtain new results on realness of weakly coupled bound states and of low lying essential spectrum in the $\mathcal{P}\mathcal{T}$-symmetric waveguide. Moreover, we show that the pseudospectrum has a normal tame behavior near the low lying essential spectrum and exclude the accumulation of non-real eigenvalues to this part of the essential spectrum. The advantage of our approach is particularly visible when the resolvent of the unperturbed operator cannot be explicitly expressed and most of the mentioned spectral conclusions are extremely hard to prove using direct methods.

math.SP↗

arXiv2015

Approximation of Schrödinger operators with $δ$-interactions supported on hypersurfaces

We show that a Schrödinger operator $A_{δ, α}$ with a $δ$-interaction of strength $α$ supported on a bounded or unbounded $C^2$-hypersurface $Σ\subset \mathbb{R}^d$, $d\ge 2$, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator $A_{δ, α}$ with a singular interaction is regarded as a self-adjoint realization of the formal differential expression $-Δ- α\langle δ_Σ, \cdot \rangle δ_Σ$, where $α\colonΣ\rightarrow \mathbb{R}$ is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.

math.SP↗

arXiv2015

Trace formulae for Schrödinger operators with singular interactions

Let $Σ\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $Ω_\pm$. In this note self-adjoint Schrödinger operators with $δ$ and $δ'$-interactions supported on $Σ$ are studied. For large enough $m\in\mathbb{N}$ the difference of $m$th powers of resolvents of such a Schrödinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(\mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(Σ)$.

math.SP↗

arXiv2015

On the Bound States of Schrödinger Operators with $δ$-interactions on Conical Surfaces

In dimension greater than or equal to three, we investigate the spectrum of a Schr{ö}dinger operator with a $δ$-interaction supported on a cone whose cross section is the sphere of co-dimension two. After decomposing into fibers, we prove that there is discrete spectrum only in dimension three and that it is generated by the axisymmetric fiber. We get that these eigenvalues are non-decreasing functions of the aperture of the cone and we exhibit the precise logarithmic accumulation of the discrete spectrum below the threshold of the essential spectrum.

math.SP↗

arXiv2015

Spectral asymptotics for resolvent differences of elliptic operators with $δ$ and $δ^\prime$-interactions on hypersurfaces

We consider self-adjoint realizations of a second-order elliptic differential expression on ${\mathbb R}^n$ with singular interactions of $δ$ and $δ^\prime$-type supported on a compact closed smooth hypersurface in ${\mathbb R}^n$. In our main results we prove spectral asymptotics formulae with refined remainder estimates for the singular values of the resolvent difference between the standard self-adjoint realizations and the operators with a $δ$ and $δ^\prime$-interaction, respectively. Our technique makes use of general pseudodifferential methods, classical results on spectral asymptotics of $ψ$do's on closed manifolds and Krein-type resolvent formulae.

math.SP↗

arXiv2014

An eigenvalue inequality for Schrödinger operators with $δ$ and $δ'$-interactions supported on hypersurfaces

We consider self-adjoint Schrödinger operators in $L^2 (\mathbb{R}^d)$ with a $δ$-interaction of strength $α$ and a $δ'$-interaction of strength $β$, respectively, supported on a hypersurface, where $α$ and $β^{-1}$ are bounded, real-valued functions. It is known that the inequality $0 < β\leq 4/α$ implies inequality of the eigenvalues of these two operators below the bottoms of the essential spectra. We show that this eigenvalue inequality is strict whenever $β< 4 / α$ on a nonempty, open subset of the hypersurface. Moreover, we point out special geometries of the interaction support, such as broken lines or infinite cones, for which strict inequality of the eigenvalues even holds in the borderline case $β= 4 / α$.

math.SP↗

arXiv2014

Spectral analysis of the half-line Kronig-Penney model with Wigner-von Neumann perturbations

The spectrum of the self-adjoint Schrödinger operator associated with the Kronig-Penney model on the half-line has a band-gap structure: its absolutely continuous spectrum consists of intervals (bands) separated by gaps. We show that if one changes strengths of interactions or locations of interaction centers by adding an oscillating and slowly decaying sequence which resembles the classical Wigner-von Neumann potential, then this structure of the absolutely continuous spectrum is preserved. At the same time in each spectral band precisely two critical points appear. At these points "instable" embedded eigenvalues may exist. We obtain locations of the critical points and discuss for each of them the possibility of an embedded eigenvalue to appear. We also show that the spectrum in gaps remains discrete.

math.SP↗

arXiv2014

Point contacts and boundary triples

We suggest an abstract approach for point contact problems in the framework of boundary triples. Using this approach we obtain the perturbation series for a simple eigenvalue in the discrete spectrum of the model self-adjoint extension with weak point coupling. An example of a two-level quantum model is provided.

math-ph↗

arXiv2014

Schrödinger operators with $δ$-interactions supported on conical surfaces

We investigate the spectral properties of self-adjoint Schrödinger operators with attractive $δ$-interactions of constant strength $α> 0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum is given by $[-α^2/4,+\infty)$ and that the discrete spectrum is infinite and accumulates to $-α^2/4$. Furthermore, an asymptotic estimate of these eigenvalues is obtained.

math.SP↗

arXiv2014

Weakly coupled bound state of 2D Schrödinger operator with potential-measure

We consider a self-adjoint two-dimensional Schrödinger operator $H_{αμ}$, which corresponds to the formal differential expression \[ -Δ- αμ, \] where $μ$ is a finite compactly supported positive Radon measure on ${\mathbb R}^2$ from the generalized Kato class and $α>0$ is the coupling constant. It was proven earlier that $σ_{\rm ess}(H_{αμ}) = [0,+\infty)$. We show that for sufficiently small $α$ the condition $\sharpσ_{\rm d}(H_{αμ}) = 1$ holds and that the corresponding unique eigenvalue has the asymptotic expansion $$ λ(α) = -(C_μ+ o(1))\exp\Big(-\tfrac{4π}{αμ({\mathbb R}^2)}\Big), \qquad α\rightarrow 0+, $$ with a certain constant $C_μ> 0$. We obtain also the formula for the computation of $C_μ$. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend Simon's results, see \cite{Si76}, to the case of potentials-measures. Also for regular potentials our results are partially new.

math.SP↗

arXiv2013

Schrödinger operators with δ- and δ'-interactions on Lipschitz surfaces and chromatic numbers of associated partitions

We investigate Schrödinger operators with δ- and δ'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schrödinger operators with δ- and δ'-interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ'-interactions.

math-ph↗

arXiv2013

Lower bounds on the norms of extension operators for Lipschitz domains

Let $Ω\subset\dR^d$ be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space $H^1(Ω)$ up to functions in the Sobolev space $H^1(\dR^d)$ via a linear operator. The minimal possible norm of such an operator is estimated from below in terms of spectral properties of self-adjoint Robin Laplacians on domains $Ω$ and $\dR^d\setminus\ovΩ$. Another estimate of this norm is also given, where spectral properties of Schrödinger operators with the $δ$-interaction supported on the hypersurface $\partialΩ$ are involved. General results are illustrated with examples.

math.SP↗

arXiv2013

Note on 2D Schrödinger operators with delta-interactions on angles and crossing lines

In this note we sharpen the lower bound from [LLP10] on the spectrum of the 2D Schroedinger operator with a delta-interaction supported on a planar angle. Using the same method we obtain the lower bound on the spectrum of the 2D Schroedinger operator with a delta-interaction supported on crossing straight lines. The latter operators arise in the three-body quantum problem with delta-interactions between particles.

math-ph↗

arXiv2013

Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $Ω\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(Ω)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partialΩ)$.

math.SP↗

arXiv2012

Schrödinger operators with delta and delta'-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with $δ$ and $δ'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman--Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten--von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with $δ$ and $δ'$-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

math.SP↗