Vladimir Lotoreichik and Jonathan Rohleder A note on optimization of the second positive Neumann eigenvalue for parallelograms
submitted. arXiv.
x
A note on optimization of the second positive Neumann eigenvalue for parallelograms (jointly with J. Rohleder)
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.
Andrii Khrabustovskyi and Vladimir Lotoreichik Homogenization of the Dirac operator with position-dependent mass
submitted. arXiv.
x
Homogenization of the Dirac operator with position-dependent mass (jointly with A. Khrabustovskyi)
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 ε-periodic square lattice. Under rather general assumptions on geometry of these inclusions we prove
that the corresponding family of Dirac operators converges as ε->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.
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 p 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.
On the Laplace operator with a weak magnetic field in exterior domains (jointly with A. Kachmar and M. Sundqvist)
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.
Spectral asymptotics of the Dirac operator in a thin shell (jointly with T. Ourmieres-Bonafos)
We investigate the spectrum of the Dirac operator with infinite mass boundary conditions posed in a
tubular neighborhood of a smooth compact hypersurface in ℝn 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.
Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain (jointly with D. Krejcirik)
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.
Isoperimetric inequalities for inner parallel curves (jointly with C. Dietze and A. Kachmar)
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.
Quasi-conical domains with embedded eigenvalues
(jointly with D. Krejcirik)
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.
Spectral analysis of the Dirac operator with a singular interaction on a broken line (jointly with D. Frymark and M. Holzmann)
We consider the one-parametric family of self-adjoint realizations of the two-dimensional massive Dirac operator with a
Lorentz scalar $\delta$-shell interaction of strength $\tau\in\RR\setminus\{-2,0,2\}$ supported on a
broken line of opening angle $2\omega$ with $\omega\in(0,\frac{\pi}{2})$.
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 $\tau < 0$, also on the strength of the interaction,
but does not depend on $\omega$.
As the main result, we prove that for any $N\in\mathbb{N}$ and strength $\tau\in(-\infty,0)\setminus\{-2\}$ 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 $\omega$ is sufficiently small. Moreover, we obtain an explicit estimate on $\omega$
sufficient for this property to hold.
For $\tau\in(0,\infty)\setminus\{2\}$, the discrete spectrum consists of at most one simple eigenvalue.
A geometric bound on the lowest magnetic Neumann eigenvalue via the torsion function (jointly with A. Kachmar)
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.
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.
Bound states of weakly deformed soft waveguides
(jointly with P. Exner and S. Kondej)
In this paper we consider the two-dimensional Schr\"odinger 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 Schr\"odinger
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 $\varepsilon > 0$.
Self-adjointness for the MIT bag model on an unbounded cone
(jointly with B. Cassano)
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 H¹-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.
Schrödinger operators with δ-potentials supported on unbounded Lipschitz hypersurfaces
(jointly with J. Behrndt and P. Schlosser)
In this note we consider the self-adjoint Schr\"odinger operator $\sfA_\aa$ in $L^2(\dR^d)$, $d\geq 2$, with a $\delta$-potential supported on a Lipschitz hypersurface $\Sigma\subseteq\mathbb{R}^d$ of strength $\aa\in L^p(\Sigma)+L^\infty(\Sigma)$.
We show the uniqueness of the ground state and, under some additional
conditions on the coefficient $\alpha$ and the hypersurface $\Sigma$, we determine the essential spectrum of $\sfA_\aa$. In the special case that $\Sigma$ is a hyperplane we
obtain a Birman-Schwinger principle with a relativistic Schr\"{o}dinger operator as
Birman-Schwinger operator. As an application we prove an optimization result for the bottom of
the spectrum of $\sfA_\aa$.
General δ-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation (jointly with B. Cassano, A. Mas
and M. Tusek)
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.
Reverse isoperimetric inequality for the lowest Robin eigenvalue of a triangle
(jointly with D. Krejcirik and T. Vu)
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
Self-adjointness of the 2D Dirac operator with singular interactions supported on star-graphs
(jointly with D. Frymark)
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. A
mong 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.
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 C²-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.
The fate of Landau levels under δ-interactions
(jointly with J. Behrndt, M. Holzman, and G. Raikov)
We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(\mathbb{R}^d)$ whose spectrum consists of
infinitely degenerate eigenvalues $\Lambda_q$, $q \in \mathbb{Z}_+$, and the
perturbed operator $H_\upsilon = H_0 + \upsilon\delta_\Gamma$, where $\Gamma \subset \mathbb{R}^d$
is a regular Jordan $C^{1,1}$-curve, and $\upsilon \in L^p(\Gamma;\mathbb{R})$, $p>1$, has a constant sign. We investigate
$\Ker(H_\upsilon -\Lambda_q)$, $q \in \mathbb{Z}_+$, and show that generically
$$0 \leq \dim\Ker(H_\upsilon -\Lambda_q) - \dim\Ker(T_q(\upsilon\delta_\Gamma) < \infty,$$ where
$T_q(\upsilon\delta_\Gamma = p_q (\upsilon \delta_\Gamma)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q L^2(\mathbb{R}^d)$,
and $p_q$ is the orthogonal projection on $\Ker\,(H_0 -\Lambda_q)$.
If $\upsilon \neq 0$ and $q = 0$, we prove that $\Ker\,(T_0(\upsilon\delta_\Gamma)) = \{0\}$.
If $q \geq 1$, and $\Gamma = \mathcal{C}_r$ is a circle of radius $r$,
we show that $\dim\Ker(T_q(\upsilon\delta_{\mathcal{C}_r})) \leq q$, and the set of $r \in (0,\infty)$
for which $\dim\Ker(T_q(\upsilon\delta_{\mathcal{C}_r})) \geq 1$, is infinite and discrete.
On the isoperimetric inequality for the magnetic Robin Laplacian with negative boundary parameter
(jointly with A. Kachmar)
We consider the magnetic Robin Laplacian with a negative boundary parameter on a bounded,
planar C²-smooth domain. The respective magnetic field is homogeneous.
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 centrally symmetric domains.
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.
Spectral optimization for Robin Laplacian in domains admitting parallel coordinates
(jointly with P. Exner)
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 ∂Ω.
Trace Hardy inequality for the Euclidean space with a cut and its applications (jointly with M. Dauge and M. Jex)
We obtain a trace Hardy inequality for the Euclidean space with a bounded cut Σ⊂ℝd, d≥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 ℝd 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.
Optimization of the lowest eigenvalue of a soft quantum ring (jointly with P. Exner)
We consider the self-adjoint two-dimensional Schrödinger operator
Hμ 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⟂. This operator has nonempty negative discrete spectrum and we obtain two optimization results for its lowest eigenvalue.
For the first one, we fix μ⟂ 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 Hμ 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⟂, under the constraint that the total profile measure α>0 is fixed. The optimizer in this problem is μ⟂
given by the product of α and the Dirac δ-function supported at an optimal position.
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.
Faber-Krahn inequalities for Schrödinger operators with point and with Coulomb interactions
(jointly with A. Michelangeli)
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.
A variational formulation for Dirac operators in bounded domains.
Applications to spectral geometric inequalities
(jointly with P. Antunes, R. Benguria, and T. Ourmieres-Bonafos)
We investigate spectral features of the Dirac operator with infinite mass
boundary conditions in a smooth bounded domain of ℝ².
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.
(jointly with D. Krejcirik, K. Pankrashkin, and M. Tusek)
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 as the Laplace 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 the Dirichlet Laplacian. We prove that for a large class of measures the numerical range is the whole complex plane,
despite the fact that the spectrum is purely discrete. 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.
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.
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.
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.
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.
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 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.
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
L2(ℝ2). The proofs rely on perturbation theory of linear operators, Floquet-Bloch analysis, and properties of
Schroedinger operators with point interactions.
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 Aη
with electrostatic δ-shell interactions of constant strength η
supported on compact smooth surfaces in R3 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 Aη
are investigated. In particular, it turns out that the discrete spectrum of Aη
inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of
Aη and the free Dirac operator A0 is trace class, and in the nonrelativistic
limit Aη 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.
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 C2-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
in Proceedings of the conference "Mathematical Results in Quantum Physics" (QMath13, Atlanta 2016; F. Bonetto, D. Borthwick, E. Harrell, M. Loss, eds.),
Contemporary Mathematics, AMS, Providence, R.I. 2018; 187–196
arXiv.
x
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.
