We explore the regularity of energy maximizers for the Lagrangian of a periodic dispersion managed fiber optic at fixed intensity, on a torus of length L and with vanishing average dispersion. We show that the Fourier coefficients decay at a polynomial rate, and then upgrade this to exponential decay, so that the maximizers are analytic in space for large enough L. In addition, by an asymptotic comparison to the optimizers on the real line, we prove that the solutions are non-trivial. We also consider a conjecture that the maximizer necessarily has an underlying symmetry inherent to both the energy functional and the resulting Euler-Lagrange equation. All of our results are supported with illustrative numerical experiments.

In the present paper we study two challenging problems for helium-type systems. Existence of eigenvalues at thresholds and the asymptotic behavior of the corresponding eigenfunctions. Since the usual methods for addressing these problems need a safety distance to the essential spectrum, they cannot be applied in critical cases, when an eigenvalue enters the continuum. We develop a method to address both problems and derive sharp upper and lower bounds for the asymptotic behavior of the ground state of critical helium-type systems at the threshold of the essential spectrum. This is the first proof of the precise asymptotic behavior of the ground state for this benchmark problem in quantum chemistry. Moreover, our bounds describe precisely how the asymptotic decay of the ground state changes, when the system becomes critical. In addition, we show the existence of a ground state of this quantum critical system with a finite nuclear mass. Previously this had been known only in the Born-Oppenheimer approximation of infinite nuclear mass.

In the first part of this article we present a growth condition on the potential q in the Schrödinger operator H=-Δ+ q(x) in L^2( ℝ^n) that implies Rosen inequalities for the ground state φ of H, i.e. ∀ε > 0 ∃ γ(ε) > 0 : - ln( φ(x) ) ≤ε q(x) + γ(ε). While these inequalities are not particularly interesting in themselves, they offer Logarithmic Sobolev inequalities which are absolutely essential to prove an intrinsic ultracontractivity of the associated Schrödinger semigroup e^-tH, i.e. ∀ t>0 ∃ C_t > 0 : | e^-tH u (x) | ≤ C_t φ(x) u _2 holds for every u ∈L^2( ℝ^n) almost everywhere in ℝ^n which we prove in the second part of this article. For proving Rosen inequalities we focus on solving a radial Schrödinger inequality and use Agmon's version of the comparison principle and Young's inequality for increasing functions. We follow the classic method proving intrinsic ultracontractivity of e^-tH by using weighted Sobolev function spaces, weighted Schrödinger semigroups and Logarithmic Sobolev inequalities.

This paper establishes new bounds on the maximum number of electrons N_c(Z) that an atom with nuclear charge Z can bind. Specifically, we show that N_c(Z) < 1.1185Z + O(Z^1/3) with an explicit bound on the lower order term O(Z^1/3). This result improves long–standing bounds by Lieb and Nam obtained in 1984, respectively 2012. Our bounds show the fundamental difference between fermionic and bosonic atoms for finite Z since for bosonic atoms it is known that lim N_c(Z)/Z = t_c ≈ 1.21 in the limit of large nuclear charges Z.

We study a system of three bosons interacting with short-range potentials which can move along three different lines. Two of these lines are parallel to each other within one plane. The third line is constrained to a plane perpendicular to the first one. Recently it was predicted in physics literature that such a system exhibits the so-called confinement induced Efimov effect. We prove that this prediction is not correct by showing that this system has at most finitely many bound-states.

We consider eigenvalues of the Pauli operator in R3 embedded in the continuous spectrum. In our main result we prove the absence of such eigenvalues above a threshold which depends on the asymptotic behavior of the magnetic and electric field at infinity. We show moreover that the decay conditions on the magnetic and electric field are sharp. Analogous results are obtained for purely magnetic Dirac operators.

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrödinger operators in ℝ^d, d≥ 2 . In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller–Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov–Bohm operators.

One of the crucial properties of a quantum system is the existence of bound states. While the existence of eigenvalues below zero, that is, below the essential spectrum, is well understood, the situation of zero energy bound states at the edge of the essential spectrum is far less understood. We present complementary sharp criteria for the existence and nonexistence of zero energy ground states. Our criteria give a straightforward explanation for the folklore that there is a spectral phase transition with critical dimension four, concerning the existence versus nonexistence of zero energy ground states.

We present a method to calculate the asymptotic behavior of eigenfunctions of Schrödinger operators that also works at the threshold of the essential spectrum. It can be viewed as a higher order correction to the well-known WKB method which does need a safety distance to the essential spectrum. We illustrate its usefulness on examples of quantum particles in a potential well with a long-range repulsive term outside the well.

We consider a molecule in the Born–Oppenheimer approximation interacting with a plate of infinite thickness, i.e., a half-space, which is perfectly conducting or dielectric. It is well known in the physics literature that in this case the atom or molecule is attracted by the plate at sufficiently large distances. This effect is analogous to the well-known van der Waals interaction between neutral atoms or molecules. We prove that the interaction energy W of the system is given by W(r,v)=−C(v)r−3+O(r−4), where r is the distance between the molecule and the plate and v indicates their relative orientation. Moreover, C(v) is positive and continuous, thus the atom or molecule is always pulled toward the plate at sufficiently large distances, for all relative orientations v. For some specific systems, we provide sharper estimates of W(r, v). This asymptotic behavior is well known in the physics literature; however, we are not aware of any previous rigorous results, even on the existence of a ground state of the system. For pedagogical reasons, we often start with the case of a hydrogen atom and then we generalize the arguments to deal with a general molecule.

We prove local and global well–posedness results for the Gabitov–Turitsyn or dispersion managed nonlinear Schrödinger equation with a large class of nonlinearities and arbitrary average dispersion on L2(R) and H1(R) for zero and non–zero average dispersions, respectively. Moreover, when the average dispersion is non–negative, we show that the set of ground states is orbitally stable. This covers the case of non–saturated and saturated nonlinear polarizations and yields, for saturated nonlinearities, the first proof of orbital stability.

There are several proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a substantial refinement of Cwikel’s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies.

It is well-known that for usual Schrödinger operators weakly coupled bound states exist in dimensions one and two, whereas in higher dimensions the famous Cwikel–Lieb–Rozenblum bound holds. We show for a large class of Schrödinger-type operators with general kinetic energies that these two phenomena are complementary. We explicitly get a natural semi-classical type bound on the number of bound states precisely in the situation when weakly coupled bound states exist not.

We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo-relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance $|D|$ between the atoms. We rigorously compute all the terms in the binding energy up to the order $|D|^{-9}$ with error term of order $\mathcal{O}(|D|^{-10})$ . As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schr\"odinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error.

The fundamental sensitivity limit of atomic force microscopy is strongly correlated to the thermal noise of the cantilever oscillation. A method to suppress this unwanted noise is to reduce the bandwidth of the measurement, but this approach is limited by the speed of the measurement and the width of the cantilever resonance, commonly defined through the quality factor Q. However, it has been shown that optomechanical resonances in interferometers might affect the cantilever oscillations resulting in an effective quality factor Q_eff . When the laser power is sufficiently increased the cantilever oscillations might even reach the regime of self-oscillation. In this self-oscillation state, the noise of the system is partially determined by the interaction with the laser light far from equilibrium. Here, we show and discuss how tuning of the laser power leads to nonlinear optomechanical effects that can dramatically increase the effective quality factor of the cantilever leading to out-of-equilibrium noise. We model the effects using a fourth order nonlinearity of the damping coefficient.

We provide new estimates on the best constant of the Lieb-Thirring inequality for the sum of the negative eigenvalues of Schr\odinger operators, which significantly improve the so far existing bounds.

We study the one dimensional nonlinear Schrodinger equation with power nonlinearity |u|(alpha-1) u for alpha is an element of [1, 5] and initial data u(0) is an element of H-1(T) + L-2(R). We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity (alpha = 2) we obtain global well-posedness in the space C(R, H-1(T)+L-2 (R)) via Gronwall's inequality.

We report on a version of Cwikel's proof of the famous Cwikel–Lieb–Rozenblum (CLR) inequality which highlights the connection of the CLR inequality to maximal Fourier multipliers. This new approach enables us to get a constant at least ten times better than Cwikels in all dimensions. In dimensions $d\geq 5$ our results are better than all previously known ones.

We study the one dimensional nonlinear Schrodinger equation with power nonlinearity $$\\left| u \\right| ^{\\alpha - 1} u$$\n for $$\\alpha \\in [1,5]$$\n and initial data $$u_0 \\in H^1({{\\mathbb {T}}}) + L^2({{\\mathbb {R}}})$$\n . We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity (\n $$\\alpha = 2$$\n ) we obtain global well-posedness in the space $$C({{\\mathbb {R}}}, H^1({{\\mathbb {T}}}) + L^2({{\\mathbb {R}}}))$$\n via Gronwall’s inequality.

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $$\left| u \right| ^{\alpha - 1} u$$ for $$\alpha \in [1,5]$$ and initial data $$u_0 \in H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$$ . We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ( $$\alpha = 2$$ ) we obtain global well-posedness in the space $$C({{\mathbb {R}}}, H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}}))$$ via Gronwall’s inequality.