Uniqueness in an Inverse Boundary Problem for a Magnetic Schrödinger Operator with a Bounded Magnetic Potential

We show that the knowledge of the set of the Cauchy data on the boundary of a bounded open set in ${\mathbb{R}^n}$ , ${n \geq 3}$ , for the magnetic Schrödinger operator with L ^∞ magnetic and electric potentials, determines the magnetic field and electric potential inside the set uniquely. The proof is based on a Carleman estimate for the magnetic Schrödinger operator with a gain of two derivatives.     

Bloch Bundles, Marzari-Vanderbilt Functional and Maximally Localized Wannier Functions

We consider a periodic Schrödinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced in Marzari and Vanderbilt (Phys Rev B 56:12847–12865, 1997) and we prove some results about the existence and exponential localization of its minimizers, in dimension ${d \leq 3}$ d ≤ 3 . The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds.     

Nonlinear Instability in a Semiclassical Problem

We consider a nonlinear evolution problem with an asymptotic parameter and construct examples in which the linearized operator has spectrum uniformly bounded away from ${{\rm Re}\; z \geq 0 }$ (that is, the problem is spectrally stable), yet the nonlinear evolution blows up in short times for arbitrarily small initial data. We interpret the results in terms of semiclassical pseudospectrum of the linearized operator: despite having the spectrum in ${{\rm Re}\;z < -\gamma_0 < 0 }$ , the resolvent of the linearized operator grows very quickly in parts of the region ${{\rm Re}\;z > 0 }$ . We also illustrate the results numerically.     

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge \(c\) in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several \(\mathbb {Z}_2\) -even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension \(\Delta _\sigma = 0.518154(15)\) , and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.     

Study of the photon’s pole structure in the noncommutative Schwinger model

The photon magnetic moment for radiation propagating in magnetized vacuum is defined as a pseudotensor quantity, proportional to the external electromagnetic field tensor. After expanding the eigenvalues of the polarization operator in powers of \(k^2\) , we obtain approximate dispersion equations (cubic in \(k^2\) ), and analytic solutions for the photon magnetic moment, valid for low momentum and/or large magnetic field. The paramagnetic photon experiences a redshift, with opposite sign to the gravitational one, which differs for parallel and perpendicular polarizations. It is due to the drain of photon transverse momentum and energy by the external field. By defining an effective transverse momentum, the constancy of the speed of light orthogonal to the field is guaranteed. We conclude that the propagation of the photon non-parallel to the magnetic direction behaves as if there is a quantum compression of the vacuum or a warp of space-time in an amount depending on its angle with regard to the field.     

Inverse Problems With Partial Data for a Magnetic Schrödinger Operator in an Infinite Slab and on a Bounded Domain

In this paper we study inverse boundary value problems with partial data for the magnetic Schrödinger operator. In the case of an infinite slab in \({\mathbb{R}^n}\) , n ≥ 3, we establish that the magnetic field and the electric potential can be determined uniquely, when the Dirichlet and Neumann data are given either on the different boundary hyperplanes of the slab or on the same hyperplane. This is a generalization of the results of Li and Uhlmann (Inverse Probl Imaging 4(3):449–462, 2010), obtained for the Schrödinger operator without magnetic potentials.In the case of a bounded domain in \({\mathbb{R}^n}\) , n ≥ 3, extending the results of Ammari and Uhlmann (Indiana Univ Math J 53(1):169–183, 2004), we show the unique determination of the magnetic field and electric potential from the Dirichlet and Neumann data, given on two arbitrary open subsets of the boundary, provided that the magnetic and electric potentials are known in a neighborhood of the boundary. Generalizing the results of Isakov (Inverse Probl Imaging 1(1):95–105, 2007), we also obtain uniqueness results for the magnetic Schrödinger operator, when the Dirichlet and Neumann data are known on the same part of the boundary, assuming that the inaccessible part of the boundary is a part of a hyperplane.     

\mathcal{PT}-symmetric nonlinear metamaterials and zero-dimensional systems

A one dimensional, parity-time ( $\mathcal{PT}$ )-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken $\mathcal{PT}$ -phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern ( $\mathcal{PT}$ dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the interelement coupling. In the presence of nonlinearity, the $\mathcal{PT}$ dimer chain configuration with balanced gain and loss supports nonlinear localized modes in the form of a novel type of discrete breathers below the lower branch of the linear spectrum. These breathers that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamental gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical $\mathcal{PT}$ symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behavior is similar to a transition from the exact to the broken $\mathcal{PT}$ phase in higher-dimensional $\mathcal{PT}$ -symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of a piece-wise constant gain/loss function that allows the construction of a phase diagram with alternating stable and unstable regions.     

Aharonov and Bohm versus Welsh eigenvalues

We consider a class of two-dimensional Schrödinger operator with a singular interaction of the \(\delta \) type and a fixed strength \(\beta \) supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov–Bohm flux \(\alpha \in [0,\frac{1}{2}]\) in the center. It is shown that if \(\beta \ne 0\), there is a critical value \(\alpha _{\mathrm {crit}}\in (0,\frac{1}{2})\) such that the discrete spectrum has an accumulation point when \(\alpha <\alpha _{\mathrm {crit}}\), while for \(\alpha \ge \alpha _{\mathrm {crit}}\) the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed \(\alpha \in (0,\frac{1}{2})\) and \(|\beta |\) small enough.     

Odd Laplacians: geometrical meaning of potential and modular class

A second-order self-adjoint operator \(\Delta =S\partial ^2+U\) is uniquely defined by its principal symbol S and potential U if it acts on half-densities. We analyse the potential U as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an affine connection compensates the action of coordinate transformations on first derivatives in the first-order operator, a covariant derivative, \(\nabla =\partial +\Gamma \). Usually a potential U is derived from other geometrical constructions such as a volume form, an affine connection, or a Riemannian structure, etc. The story is different if \(\Delta \) is an odd operator on a supermanifold. In this case, the second-order potential becomes a primary object. For example, in the case of an odd symplectic supermanifold, the compensating field of the canonical odd Laplacian depends only on this symplectic structure and can be expressed by the formula obtained by K. Bering. We also study modular classes of odd Poisson manifolds via \(\Delta \)-operators, and consider an example of a non-trivial modular class which is related with the Nijenhuis bracket.     

Topological Spin Excitations Induced by an External Magnetic Field Coupled to a Surface with Rotational Symmetry

We study the Heisenberg model in an external magnetic field on curved surfaces with rotational symmetry. The Euler–Lagrange static equations, derived from the Hamiltonian, lead to the inhomogeneous double sine-Gordon equation. Nonetheless, if the magnetic field is coupled to the metric elements of the surface, and consequently to its curvature, the homogeneous double sine-Gordon equation emerges and a $2\pi $ -soliton solution is obtained. In order to satisfy the self-dual equations, surface deformations are predicted to appear at the sector where the spin direction is opposite to the magnetic field. On the basis of the model, we find the characteristic length of the $2\pi $ -soliton for three specific rotationally symmetric surfaces: the cylinder, the catenoid, and the hyperboloid. On finite surfaces, such as the sphere, torus, and barrels, fractional $2\pi $ -solitons are predicted to appear.     

On the vacuum state in quantum field theory. II

We want to construct, for every local irreducible quantum field theory which fulfils the spectrum condition, a new theory with the properties:

1. 1)
   It is physically equivalent to the given theory (in the sense ofHaag andKastler).     
   
   

Evolution of primordial magnetic fields in mean-field approximation

We study the evolution of phase-transition-generated cosmic magnetic fields coupled to the primeval cosmic plasma in the turbulent and viscous free-streaming regimes. The evolution laws for the magnetic energy density and the correlation length, both in the helical and the non-helical cases, are found by solving the autoinduction and Navier–Stokes equations in the mean-field approximation. Analytical results are derived in Minkowski spacetime and then extended to the case of a Friedmann universe with zero spatial curvature, both in the radiation- and the matter-dominated era. The three possible viscous free-streaming phases are characterized by a drag term in the Navier–Stokes equation which depends on the free-streaming properties of neutrinos, photons, or hydrogen atoms, respectively. In the case of non-helical magnetic fields, the magnetic intensity $B$ and the magnetic correlation length $\xi _B$ evolve asymptotically with the temperature, $T$ , as $B(T) \simeq \kappa _B (N_i v_i)^{\varrho _1} (T/T_i)^{\varrho _2}$ and $\xi _B(T) \simeq \kappa _\xi (N_i v_i)^{\varrho _3} (T/T_i)^{\varrho _4}$ . Here, $T_i$ , $N_i$ , and $v_i$ are, respectively, the temperature, the number of magnetic domains per horizon length, and the bulk velocity at the onset of the particular regime. The coefficients $\kappa _B$ , $\kappa _\xi $ , $\varrho _1$ , $\varrho _2$ , $\varrho _3$ , and $\varrho _4$ , depend on the index of the assumed initial power-law magnetic spectrum, $p$ , and on the particular regime, with the order-one constants $\kappa _B$ and $\kappa _\xi $ depending also on the cutoff adopted for the initial magnetic spectrum. In the helical case, the quasi-conservation of the magnetic helicity implies, apart from logarithmic corrections and a factor proportional to the initial fractional helicity, power-like evolution laws equal to those in the non-helical case, but with $p$ equal to zero.     

Schrödinger equation in multiply connected spaces and phase optics

The magnetic vector potential\(\vec A\) in a field free spaceR ₀ cannot be removed by gauge transformations in general, ifR ₀ is multiply connected.Aharonov andBohm ¹ have noticed recently that\(\vec A\) therefore should have more physical meaning than only to give the magnetic field by differentiation. They could show that\(\vec A\) inR ₀ may influence the phase ofSchrödinger'sψ-function in an observable manner. We want to point out here that this influence can be expressed in a simple, general form: “A closed magnetic field line operates uponψ like ae Φ/?-phase-shifter placed on any area bounded by the field line.” Surface like phase shifters are familiar in phase optics. There exists a narrow relationship between electron scattering at magnetic fields and some special problems of phase optics. An electron phase contrast microscope is discussed.     

Über eine Näherung für Gitterelektronen in äußeren Feldern

The Hamilton operator of an electron in a periodic lattice potential under influence of external electric and magnetic fields with potentialsV(r) andA(r) resp. is often replaced by an approximate operatorW ⁰ (?i?+A(r))+V(r) for one single energy bandW ⁰(k) which means a renormalization of the kinetic energy by the lattice. The validity of this replacement is examined and the magnitude of its error is roughly estimated. Neglecting other bands one obtains an error term proportional to the derivative of the electric field strengthF, if one takes a suitable position of the “raster” of the replacement operator, and to the square of the magnetic field strengthB resp. The decoupling from the other energy bands leads to error terms proportionalF ² andB ² resp. which however in the general case increase rapidly in the vicinity of overlapping energy bands.     

Molecular and cluster structures in 18O

We have studied the multi-nucleon transfer reaction ¹²C ( ⁷Li ,p) at E _lab = 44 MeV, populating states in the oxygen isotope ¹⁸O . The experiments were performed at the Tandem accelerator of the Maier-Leibniz Laboratory in Munich using the high-resolution Q3D magnetic spectrograph. States were populated up to an excitation energy of 21.2MeV with an overall energy resolution of 45keV, and 30 new states of ¹⁸O have been identified. The structure of the rotational bands observed is discussed in terms of cluster bands with the underlying cluster structures: ¹⁴C$ $\displaystyle \otimes$ $\displaystyle \alpha$ $and 12C ? 2n ? $ \alpha$ . Because of the broken intrinsic reflection symmetry in these structures the corresponding rotational bands appear as parity doublets.     

Magnetic-impurity states of electron in two-dimensional semiconductor structures

Electron states on an attractive center of small-radius r _c ? l (l = $\sqrt {\frac{{c\hbar }}{{eH}}} $ is the magnetic length) located in a two-dimensional structure are investigated in a uniform magnetic field H applied perpendicularly to the structure surface. The spectrum of magnetic-impurity (MI) particle states with an arbitrary moment projection on the direction H for Landau bands 0 ≤ N < l ²/r _c ² is derived in the approximation that mixing of Landau levels is weak. The dependence of the electron energy states on magnetic field, the layer thickness, and the impurity position are studied. It is shown that dimension lowering leads to a qualitatively different spectrum of MI states compared to the three-dimensional case [1]. A comparison of the obtained binding energy of the D ^? center with experimental data is performed.     

Strong magnetic fields,Dirichlet boundaries,and spectral gaps

We consider magnetic Schrödinger operators $$H(\lambda \vec a) = ( - i\nabla - \lambda \vec a(x))^2$$ inL ₂(R ⁿ ), where $\vec a \in C^1 (R^n ;R^n )$ and λεR. LettingM={x;B(x)=0}, whereB is the magnetic field associated with $\vec a$ , and $M_{\vec a} = \{ x;\vec a(x) = 0\}$ , we prove that $H(\lambda \vec a)$ converges to the (Dirichlet) Laplacian on the closed setM in the strong resolvent sense, as λ→∞,provided the set $M\backslash M_{\vec a}$ has measure zero. In various situations, which include the case of periodic fields, we even obtain norm resolvent convergence (again under the condition that $M\backslash M_{\vec a}$ has measure zero). As a consequence, if we are given a periodic fieldB where the regions withB=0 have non-empty interior and are enclosed by the region withB≠0, magnetic wells will be created when λ is large, opening up gaps in the spectrum of $H(\lambda \vec a)$ . We finally address the question of absolute continuity of $\vec a$ for periodic $H(\vec a)$ .     

On Essential Self-Adjointness for Magnetic Schrödinger and Pauli Operators on the Unit Disc in $${\mathbb {R}^2}$$

We study the question of magnetic confinement of quantum particles on the unit disk \({\mathbb {D}}\) in \({\mathbb {R}^2}\) , i.e. we wish to achieve confinement solely by means of the growth of the magnetic field \({B(\vec x)}\) near the boundary of the disk. In the spinless case, we show that \({B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}}\) , for \({|\vec x|}\) close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants \({\frac{\sqrt 3}{2}}\) and \({-\frac{1}{\sqrt 3}}\) are optimal. This answers, in this context, an open question from Colin de Verdière and Truc (Ann Inst Fourier 2011, Preprint, arXiv:0903.0803v3). We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles.     

Simulating spin-\frac{3}{2} particles at colliders

Support for interactions of spin- $\frac{3}{2}$ particles is implemented in the FeynRules and ALOHA packages and tested with the MadGraph 5 and CalcHEP event generators in the context of three phenomenological applications. In the first, we implement a spin- $\frac{3}{2}$ Majorana gravitino field, as in local supersymmetric models, and study gravitino and gluino pair-production. In the second, a spin- $\frac{3}{2}$ Dirac top-quark excitation, inspired from compositeness models, is implemented. We then investigate both top-quark excitation and top-quark pair-production. In the third, a general effective operator for a spin- $\frac{3}{2}$ Dirac quark excitation is implemented, followed by a calculation of the angular distribution of the s-channel production mechanism.