Magda Khalile and Vladimir Lotoreichik
Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones
arXiv.
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Spectral isoperimetric inequalities for Robin Laplacians on 2-manifolds and unbounded cones
(jointly with M. Khalile)
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_{○}≥0 and under
the constraint of fixed perimeter, the geodesic disk of constant curvature K_{○} 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.
Vladimir Lotoreichik Spectral isoperimetric inequality for the δ'-interaction on a contour arXiv.
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Spectral isoperimetric inequality for the δ'-interaction on a contour
We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional
Schrödinger operator with an attractive δ'-interaction of a fixed strength, the support of which is a C²-smooth contour.
Under the constraint of a fixed length of the contour, we prove that the lowest eigenvalue is maximized by the circle.
The proof relies on the min-max principle and the method of parallel coordinates.
Refereed publications
Jussi Behrndt, Pavel Exner, Markus Holzmann and Vladimir Lotoreichik The Landau Hamiltonian with δ-potentials supported on curves
to appear in Rev. Math. Phys.
arXiv.
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The Landau Hamiltonian with δ-potentials supported on curves
(jointly with J. Behrndt, P. Exner, and M. Holzmann)
The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian
$A_\alpha =(i \nabla + A)^2 + \alpha\delta$ in $L^2(R^2)$ with a $\delta$-potential supported on a
finite $C^{1,1}$-smooth curve $\Sigma$ are studied. Here $A = \frac{1}{2} B (-x_2, x_1)^\top$
is the vector potential, $B>0$ is the strength of the homogeneous magnetic field, and $\alpha\in L^\infty(\Sigma)$
is a position-dependent real coefficient modeling the strength of the singular interaction on the curve $\Sigma$.
After a general discussion of the qualitative spectral properties of $A_\alpha$ and its resolvent,
one of the main objectives in the present paper is a local spectral analysis of $A_\alpha$ near the Landau levels $B(2q+1)$.
Under various conditions on $\alpha$ it is shown that the perturbation smears the Landau levels into eigenvalue clusters,
and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of $\alpha$.
Furthermore, the use of Landau Hamiltonians with $\delta$-perturbations as model operators for more realistic quantum systems
is justified by showing that $A_\alpha$ can be approximated in the norm resolvent sense by a family
of Landau Hamiltonians with suitably scaled regular potentials.
Biagio Cassano and Vladimir Lotoreichik
Self-adjoint extensions of the two-valley Dirac operator with discontinuous infinite mass boundary conditions
to appear in Operators and Matrices.
arXiv.
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Self-adjoint extensions of the two-valley Dirac operator with discontinuous infinite mass boundary conditions
(jointly with B. Cassano)
We consider the four-component two-valley Dirac operator on a wedge in ℝ² 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.
Pavel Exner and Vladimir Lotoreichik Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer
to appear in Integral Equations and Operator Theory.
arXiv.
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Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer (jointly with P. Exner)
We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian H
on an unbounded, radially symmetric (generalized) parabolic layer P⊂ℝ³. It was known before that
H has an infinite number of eigenvalues below the threshold of its essential spectrum.
In the present paper, we find the discrete spectrum asymptotics for H by means of a consecutive reduction
to the analogous asymptotic problem for an effective one-dimensional Schrödinger operator
on the half-line with the potential the behaviour of which far away from the origin is determined
by the geometry of the layer P at infinity.
A sharp upper bound on the spectral gap for graphene quantum dots (jointly with T. Ourmières-Bonafos)
The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators
in two dimensional simply connected C³-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 H²(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.
Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, II: non-convex domains and higher dimensions (jointly with D. Krejcirik)
We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions.
As an improvement upon our previous work (arXiv:1608.04896, to appear in J. Convex Anal.), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk.
In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball.
The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics (jointly with D. Krejcirik and M. Znojil)
We propose a unique way how to choose a new inner product in a Hilbert space with respect to which an originally
non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint.
Our construction is based on minimising a 'Hilbert-Schmidt distance' to the original inner product among the
entire class of admissible inner products. We prove that either the minimiser exists and is unique, or it does
not exist at all. In the former case we derive a system of Euler-Lagrange equations by which the optimal inner product is determined.
A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supplied by examples
in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.
On the bound states of magnetic Laplacians on wedges (jointly with P. Exner and A. Perez-Obiol)
This note is mainly inspired by the conjecture about the existence of bound
states for magnetic Neumann Laplacians on planar wedges of any aperture
φ∈(0,π). So far, a proof was only obtained for apertures
φ≲0.511π. The conviction in the validity of this conjecture for
apertures φ≳0.511π mainly relied on numerical computations. In
this note we succeed to prove the existence of bound states for any aperture
φ ≲ 0.583π using a variational argument with suitably chosen test
functions. Employing some more involved test functions and combining a
variational argument with numerical optimisation, we extend this interval up to
any aperture φ ≲ 0.595π. Moreover, we analyse the same question
for closely related problems concerning magnetic Robin Laplacians on wedges and
for magnetic Schrödinger operators in the plane with δ-interactions
supported on broken lines.
Spectral analysis of photonic crystals made of thin rods (jointly with M. Holzmann)
In this paper we address the question how to design photonic crystals that have photonic band gaps around a finite number of
given frequencies. In such materials electromagnetic waves with these frequencies can not propagate;
this makes them interesting for a large number of applications. We focus on crystals made of periodically ordered
thin rods with high contrast dielectric properties. We show that the material parameters can be chosen in such a way that
transverse magnetic modes with given frequencies can not propagate in the crystal. At the same time, for any frequency belonging
to a predefined range there exists a transverse electric mode that can propagate in the medium.
These results are related to the spectral properties of a weighted Laplacian and of an elliptic operator of divergence type both acting in
L^{2}(ℝ^{2}). The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of
Schroedinger operators with point interactions.
Spectral isoperimetric inequalities for singular interactions on open arcs
We consider the problem of geometric optimization for the lowest eigenvalue of the
two-dimensional Schrödinger operator with an attractive δ-interaction supported on an open arc with two free endpoints.
Under a constraint of fixed length of the arc, we prove that the maximizer is a line segment,
the respective spectral isoperimetric inequality being strict. We also show that in the optimization problem
for the same spectral quantity, but with the constraint of fixed endpoints, the optimizer is the line segment connecting them.
Furthermore, we prove that a line segment is also the maximizer in the optimization problem for the lowest eigenvalue
of the Robin Laplacian on a plane with a slit along an open arc of fixed length.
Spectral enclosures for non-self-adjoint extensions of symmetric operators (jointly with J. Behrndt, M. Langer, and J. Rohleder)
The spectral properties of non-self-adjoint extensions A_{[B]}
of a symmetric operator in a Hilbert space are studied with
the help of ordinary and quasi boundary triples and the corresponding Weyl functions.
These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric)
boundary operator B.
In the abstract part of this paper, sufficient conditions for sectoriality
and m-sectoriality as well as sufficient conditions for A_{[B]} to have a non-empty resolvent set are provided
in terms of the parameter B and the Weyl function. Special attention is paid to
Weyl functions that decay along the negative real line or inside some sector in the complex plane,
and spectral enclosures for A_{[B]} are proved in this situation. The abstract results are applied to
elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains,
to Schrödinger operators with δ-potentials of complex strengths supported on unbounded hypersurfaces or
infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.
On the spectral properties of Dirac operators with electrostatic δ-shell interactions
(jointly with J. Behrndt, P. Exner, and M. Holzmann)
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 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.
Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane (jointly with P. Exner and S. Kondej)
In this paper we consider the three-dimensional Schrödinger operator with a δ-interaction of
strength α>0 supported on an unbounded surface parametrized by the mapping
ℝ² ∋ x → (x,βf(x)), where β∈[0,∞)
and f:ℝ²→ℝ³, f≢0,
is a C²-smooth,
compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane.
It is known that the essential spectrum of this Schrödinger operator coincides with [−¼α²,+∞).
We prove that for all sufficiently small β>0 its discrete spectrum is non-empty and consists of a unique simple eigenvalue.
Moreover, we obtain an asymptotic expansion
of this eigenvalue in the limit β→0+. On a qualitative level this eigenvalue tends to -¼α²
exponentially fast as β→0+.
Optimisation of the lowest Robin eigenvalue in the exterior of a compact set (jointly with D. Krejcirik)
We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian
in the exterior of a compact planar set, subject to attractive Robin boundary conditions.
Under either a constraint of fixed perimeter or area, we show that the maximiser within
the class of exteriors of convex sets is always the exterior of a disk.
We also argue why the results fail without the convexity constraint and in higher dimensions.
Spectral transitions for Aharonov-Bohm Laplacians on conical layers (jointly with D. Krejcirik and
and T. Ourmières-Bonafos)
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.
Eigenvalue inequalities for the Laplacian with mixed boundary conditions
(jointly with J. Rohleder)
Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions
on polyhedral and more general bounded domains are established.
The eigenvalues subject to a Dirichlet boundary condition on a part
of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated
in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet
and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.
A spectral isoperimetric inequality for cones (jointly with P. Exner)
In this note we investigate three-dimensional Schrödinger operators
with δ-interactions supported on C^{2}-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle
and on the fact that circles are unique minimisers for a class of energy functionals.
Spectral and resonance properties of the Smilansky Hamiltonian (jointly with P. Exner and M. Tater)
We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics
in quantum graphs and studied further by Solomyak and others.
We derive a weak-coupling asymptotics of the ground state and add new insights
by finding the discrete spectrum numerically.
Furthermore, we show that the model has a rich resonance structure.
Master Thesis (in Russian).
Point perturbations of Schrödinger operators on Riemannian manifolds and fractal sets, adviser Prof. Dr. Igor Yu. Popov,
ITMO University, 2008.
Other publications
Pavel Exner and Vladimir Lotoreichik Optimization of the lowest eigenvalue for leaky star graphs
to appear in the proceedings of the conference "Mathematical Results in Quantum Physics (QMath13)"arXiv.
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Optimization of the lowest eigenvalue for leaky star graphs (jointly with P. Exner)
We consider the problem of geometric optimization for the lowest eigenvalue of the
two-dimensional Schrödinger operator with an attractive δ-interaction of a fixed
strength the support of which is a star graph with finitely many edges of an equal length
L ∈ (0,∞]. Under the constraint of fixed number of the edges and fixed length of them, we
prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle,
properties of the Macdonald function,
and on a geometric inequality for polygons circumscribed into the unit circle.
Vladimir Lotoreichik, Hagen Neidhardt, and Igor Yu. Popov Point contacts and boundary triples
Mathematical Results in Quantum Mechanics, Proceedings of the QMath12 Conference, P. Exner, W. König, and
H. Neidhardt (eds), World Scientific, Singapore, 2015, pp. 283--293.
arXiv.