We consider the self-adjoint Landau Hamiltonian $H_0$ in $L^2(mathbb{R}^2)$ whose spectrum consists of infinitely degenerate eigenvalues $Lambda_q$, $q in mathbb{Z}_+$, and the perturbed operator $H_upsilon = H_0 + upsilondelta_Gamma$, where $Gamma s
ubset mathbb{R}^2$ is a regular Jordan $C^{1,1}$-curve, and $upsilon in L^p(Gamma;mathbb{R})$, $p>1$, has a constant sign. We investigate ${rm Ker}(H_upsilon -Lambda_q)$, $q in mathbb{Z}_+$, and show that generically $$0 leq {rm dim , Ker}(H_upsilon -Lambda_q) - {rm dim , Ker}(T_q(upsilon delta_Gamma)) < infty,$$ where $T_q(upsilon delta_Gamma) = p_q (upsilon delta_Gamma)p_q$, is an operator of Berezin-Toeplitz type, acting in $p_q L^2(mathbb{R}^2)$, and $p_q$ is the orthogonal projection on ${rm Ker},(H_0 -Lambda_q)$. If $upsilon eq 0$ and $q = 0$, we prove that ${rm Ker},(T_0(upsilon delta_Gamma)) = {0}$. If $q geq 1$, and $Gamma = mathcal{C}_r$ is a circle of radius $r$, we show that ${rm dim , Ker} (T_q(delta_{mathcal{C}_r})) leq q$, and the set of $r in (0,infty)$ for which ${rm dim , Ker}(T_q(delta_{mathcal{C}_r})) geq 1$, is infinite and discrete.
In this note the two dimensional Dirac operator $A_eta$ with an electrostatic $delta$-shell interaction of strength $etainmathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interacti
on strengths $eta=pm 2$ the continuous spectrum of $A_eta$ inside the spectral gap of the free Dirac operator $A_0$ collapses abruptly to a single point.
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_0right) ] and conclude under $L^1$-assumptions on the differences of the coefficient
s 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.
We consider the self-adjoint Schrodinger operator in $L^2(mathbb{R}^d)$, $dgeq 2$, with a $delta$-potential supported on a hyperplane $Sigmasubseteqmathbb{R}^d$ of strength $alpha=alpha_0+alpha_1$, where $alpha_0inmathbb{R}$ is a constant and $alpha_
1in L^p(Sigma)$ is a nonnegative function. As the main result, we prove that the lowest spectral point of this operator is not smaller than that of the same operator with potential strength $alpha_0+alpha_1^*$, where $alpha_1^*$ is the symmetric decreasing rearrangement of $alpha_1$. The proof relies on the Birman-Schwinger principle and the reduction to an analogue of the P{o}lya-SzegH{o} inequality for the relativistic kinetic energy in $mathbb{R}^{d-1}$.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrodinger equation is an important and challenging problem in quantum mechanics and mathematica
l 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 Schrodinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Greens function, but not its explicit form. With this efficient general technique we are able to treat various potentials.
The compression of the resolvent of a non-self-adjoint Schrodinger operator $-Delta+V$ onto a subdomain $Omegasubsetmathbb R^n$ is expressed in a Krein-Naimark type formula, where the Dirichlet realization on $Omega$, the Dirichlet-to-Neumann maps, a
nd certain solution operators of closely related boundary value problems on $Omega$ and $mathbb R^nsetminusoverlineOmega$ 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.
In this paper we study the time dependent Schrodinger equation with all possible self-adjoint singular interactions located at the origin, which include the $delta$ and $delta$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robi
n type as particular cases. We derive an explicit representation of the time dependent Greens 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 Greens function we study the stability and oscillatory properties of the solution of the Schrodinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
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 spect
rum of a Schrodinger 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 Schrodinger operator with prescribed essential spectrum.
In this paper the spectral and scattering properties of a family of self-adjoint Dirac operators in $L^2(Omega; mathbb{C}^4)$, where $Omega subset mathbb{R}^3$ is either a bounded or an unbounded domain with a compact $C^2$-smooth boundary, are studi
ed 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 $Omega$ is unbounded, and corresponding trace formulas.
We study the two-dimensional Dirac operator with an arbitrary combination of electrostatic and Lorentz scalar $delta$-interactions of constant strengths supported on a smooth closed curve. For any combination of the coupling constants a rigorous desc
ription 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.
