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Jussi Behrndt

Publications and source records attributed to Jussi Behrndt.

At least 37 records · Page 2Linked to original sources
arXiv2022

Sharp Boundary Trace Theory and Schrödinger Operators on Bounded Lipschitz Domains

We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian on the functions in question which, nonetheless, works perfectly with the Dirichlet and Neumann realizations of the Schrödinger differential expression $-Δ+V$. In turn, this boundary trace theory serves as a platform for developing a spectral theory for Schrödinger operators on bounded Lipschitz domains, along with their associated Weyl-Titchmarsh operators. Overall, this pushes the present state of knowledge a significant step further. For example, we succeed in extending the Dirichlet and Neumann trace operators in such a way that all self-adjoint extensions of a Schrödinger operator on a bounded Lipschitz domain may be described with explicit boundary conditions, thus providing a final answer to a problem that has been investigated for more than 60 years in the mathematical literature. Along the way, a number of other open problems are solved. The most general geometric and analytic setting in which the theory developed here yields satisfactory results is that of Lipschitz subdomains of Riemannian manifolds and for the corresponding Laplace-Beltrami operator (in place of the standard flat-space Laplacian). In particular, such an extension yields results for variable coefficient Schrödinger operators on bounded Lipschitz domains.

math.FA↗
arXiv2022

Relative oscillation theory and essential spectra of Sturm--Liouville operators

We develop relative oscillation theory for general Sturm-Liouville differential expressions of the form \[ \frac{1}{r}\left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm d}{\mathrm dx} + q\right) \] and prove perturbation results and invariance of essential spectra in terms of the real coefficients $p$, $q$, $r$. The novelty here is that we also allow perturbations of the weight function $r$ in which case the unperturbed and the perturbed operator act in different Hilbert spaces.

math.SP↗
arXiv2022

Two-dimensional Dirac operators with general $δ$-shell interactions supported on a straight line

In this paper the two-dimensional Dirac operator with a general hermitian $δ$-shell interaction supported on a straight line is introduced as a self-adjoint operator and its spectral properties are investigated in detail. In particular, it is demonstrated that the singularly continuous spectrum is always empty and that by switching a certain $δ$-shell interaction on, it is possible to generate an eigenvalue in the gap of the spectrum of the free operator or to partially or even fully close the gap. This suggests that the studied operators may serve as interesting continuum toy-models for Dirac materials. Finally, approximations by Dirac operators with regular potentials are presented.

math-ph↗
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

Spectral transition for Dirac operators with electrostatic $δ$-shell potentials supported on the straight line

In this note the two dimensional Dirac operator $A_η$ with an electrostatic $δ$-shell interaction of strength $η\in\mathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths $η=\pm 2$ the continuous spectrum of $A_η$ inside the spectral gap of the free Dirac operator $A_0$ collapses abruptly to a single point.

math.SP↗
arXiv2021

Singular Schrödinger operators with prescribed spectral properties

The paper deals with singular Schrödinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }γ_k δ(\cdot-z_k),\quad γ_k\in\mathbb{R}, \end{gather*} in $\mathsf{L}^2(\ell_-,\ell_+)$, where $(\ell_-,\ell_+)$ is a bounded interval, and $ δ(\cdot-z_k)$ is the Dirac delta-function supported at $z_k\in (\ell_-,\ell_+)$. It will be shown that the interaction strengths $γ_k$ and the points $z_k$ can be chosen in such a way that the essential spectrum and a bounded part of the discrete spectrum of this self-adjoint operator coincide with prescribed sets on a real line.

math.SP↗
arXiv2021

Perturbations of periodic Sturm--Liouville operators

We study perturbations of the self-adjoint periodic Sturm--Liouville operator \[ A_0 = \frac{1}{r_0}\left(-\frac{\mathrm d}{\mathrm dx} p_0 \frac{\mathrm d}{\mathrm dx} + q_0\right) \] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.

math.SP↗
arXiv2021

A unified approach to Schrödinger evolution of superoscillations and supershifts

Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper we give a unified approach to determine the supershift property for the solution of the time dependent Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique we are able to treat various potentials.

math.AP↗
arXiv2020

On Compressed Resolvents of Schrödinger Operators with Complex Potentials

The compression of the resolvent of a non-self-adjoint Schrödinger operator $-Δ+V$ onto a subdomain $Ω\subset\mathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $Ω$, the Dirichlet-to-Neumann maps, and certain solution operators of closely related boundary value problems on $Ω$ and $\mathbb R^n\setminus\overlineΩ$ are being used. In a more abstract operator theory framework this topic is closely connected and very much inspired by the so-called coupling method that has been developed for the self-adjoint case by Henk de Snoo and his coauthors.

math.SP↗
arXiv2020

Green's Function for the Schrödinger Equation with a Generalized Point Interaction and Stability of Superoscillations

In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the $δ$ and $δ'$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Green's function we study the stability and oscillatory properties of the solution of the Schrödinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.

math.AP↗
arXiv2019

Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains

For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expression.

math.AP↗
arXiv2019

Schrödinger evolution of superoscillations with $δ$- and $δ'$-potentials

In this paper we study the time persistence of superoscillations as the initial data of the time dependent Schrödinger equation with $δ$- and $δ'$-potentials. It is shown that the sequence of solutions converges uniformly on compact sets, whenever the initial data converges in the topology of the entire function space $A_1(\mathbb{C})$. Convolution operators acting in this space are our main tool. In particular, a general result about the existence of such operators is proven. Moreover, we provide an explicit formula as well as the large time asymptotics for the time evolution of a plane wave under $δ$- and $δ'$-potentials.

math-ph↗
arXiv2019

Construction of self-adjoint differential operators with prescribed spectral properties

In this expository article some spectral properties of self-adjoint differential operators are investigated. The main objective is to illustrate and (partly) review how one can construct domains or potentials such that the essential or discrete spectrum of a Schrödinger operator of a certain type (e.g. the Neumann Laplacian) coincides with a predefined subset of the real line. Another aim is to emphasize that the spectrum of a differential operator on a bounded domain or bounded interval is not necessarily discrete, that is, eigenvalues of infinite multiplicity, continuous spectrum, and eigenvalues embedded in the continuous spectrum may be present. This unusual spectral effect is, very roughly speaking, caused by (at least) one of the following three reasons: The bounded domain has a rough boundary, the potential is singular, or the boundary condition is nonstandard. In three separate explicit constructions we demonstrate how each of these possibilities leads to a Schrödinger operator with prescribed essential spectrum.

math.SP↗
arXiv2019

Self-adjoint Dirac operators on domains in $\mathbb{R}^3$

In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Ω; \mathbb{C}^4)$, where $Ω\subset \mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with Robin boundary conditions. Among the Dirac operators treated here is also the so-called MIT bag operator, which has been used by physicists and more recently was discussed in the mathematical literature. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman-Schwinger principle, a qualitative understanding of the scattering properties in the case that $Ω$ is unbounded, and corresponding trace formulas.

math.SP↗
arXiv2019

Two-dimensional Dirac operators with singular interactions supported on closed curves

We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $δ$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a rigorous description of the self-adjoint realizations of the operators is given and the qualitative spectral properties are described. The analysis covers also all so-called critical combinations of coupling constants, for which there is a loss of regularity in the operator domain. In this case, if the mass is non-zero, the resulting operator has an additional point in the essential spectrum, and the position of this point inside the central gap can be made arbitrary by a suitable choice of the coupling constants. The analysis is based on a combination of the extension theory of symmetric operators with a detailed study of boundary integral operators viewed as periodic pseudodifferential operators.

math.AP↗
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↗