In this paper, we consider the continuity property of pseudo-differential operators with symbols whose Fourier transforms have compact support. As applications, we obtain the$L$^{$p$}-boundedness for symbols in Besov spaces and in modulation spaces.

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrdinger operator with an attractive -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.

An irreducible II_1-subfactor A⊂B is exactly 1-supertransitive if B⊖A is reducible as an A − A bimodule. We classify exactly 1-supertransitive subfactors with index at most 6+1/5, leaving aside the composite subfactors at index exactly 6 where there are severe difficulties. Previously, such subfactors were only known up to index 3+\sqrt{5} ≈ 5.23. Our work is a significant extension, and also shows that index 6 is not an insurmountable barrier.There are exactly three such subfactors with index in (3+\sqrt{5},6+1/5], all with index 3+2\sqrt{2}. One of these comes from SO(3)_q at a root of unity, while the other two appear to be closely related, and are ‘braided up to a sign’.

Let <i>X, Y</i> be compact Hausdorff spaces and <i>E, F</i> be Banach spaces. A linear map <i>T: C(X, E) C(Y, F)</i> is separating if <i>Tf, Tg</i> have disjoint cozeroes whenever <i>f, g</i> have disjoint cozeroes. We prove that a biseparating linear bijection <i>T</i> (that is, <i>T</i> and <i>T</i>^-1 are separating) is a weighted composition operator <i>Tf = h f</i> o . Here, <i>h</i> is a function from <i>Y</i> into the set of invertible linear operators from <i>E</i> onto <i>F</i>, and , is a homeomorphism from <i>Y</i> onto <i>X</i>. We also show that <i>T</i> is bounded if and only if <i>h(y)</i> is a bounded operator from <i>E</i> onto <i>F</i> for all <i>y</i> in <i>Y</i>. In this case, <i>h</i> is continuous with respect to the strong operator topology.

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 in the subcritical case. Furthermore, we show that the model then has a rich resonance structure.

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of (N−12)×(N−12) inside triangles with vertices (0,0),(0,λβ−−√) and (λ/β−−√,0) with respect to β>0, for fixed λ≥0. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice (N+σ)×(N+τ), for σ,τ>−1. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, σ=τ=−12, is a critical case.

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian \mathsf {H} on an unbounded, radially symmetric (generalized) parabolic layer \mathsf {H} . It was known before that \mathsf {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 \mathsf {H} by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrdinger 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 \mathsf {H} at infinity.

We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order <i>n</i> 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order <i>n</i> 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.

We discuss operators of the type H=-\Delta+ V (x)-lpha\delta (x-\Sigma) with an attractive interaction, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , in H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , where H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is a potential bias being a positive constant H=-\Delta+ V (x)-lpha\delta (x-\Sigma) in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) . We show that H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is then empty if the bias is supported in theexterior'region, while in the opposite case isolated eigenvalues may exist.

We analyze a family of singular Schrdinger operators describing a Neumann waveguide with a periodic array of singular traps of a delta\prime type. We show that in the limit when perpendicular size of the guide tends to zero and the delta\prime interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.