We explain in which sense Schrödinger operators with complex potentials appear to violate locality (or Weyl's asymptotics), and we pose three open problems related to this phenomenon.
We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendency theorem due to Siegel and Shidlovskii.
We show that there exist limit-periodic Schrödinger operators such that the associated integrated density of states is Lipschitz continuous. These operators arise in the inverse spectral theoretic KAM approach of Pöschel.
I present a discussion of the hierarchy of Toda flows that gives center stage to the associated cocycles and the maps they induce on the $m$ functions. In the second part, these ideas are then applied to canonical systems; an important feature of this discussion will be my proposal that the role of the shift on Jacobi matrices should now be taken over by the more general class of twisted shifts.
General Hardy-Carleman type inequalities for Dirac operators are proved. New inequalities are derived involving particular traditionally used weight functions. In particular, a version of the Agmon inequality and Treve type inequalities are established. The case of a Dirac particle in a (potential) magnetic field is also considered. The methods used are direct and based on quadratic form techniques.
Boundary value problems on hedgehog-type graphs for Sturm-Liouville differential operators with general matching conditions are studied. We investigate inverse spectral problems of recovering the coefficients of the differential equation from the spectral data. For this inverse problem we prove a uniqueness theorem and provide a procedure for constructing its solution.
In this work we investigate the spectral statistics of random Schrödinger operators $H^ω=-Δ+\sum_{n\in\mathbb{Z}^d}(1+|n|^α)q_n(ω)|δ_n\rangle\langleδ_n|$, $α>0$ acting on $\ell^2(\mathbb{Z}^d)$ where $\{q_n\}_{n\in\mathbb{Z}^d}$ are i.i.d random variables distributed uniformly on $[0,1]$.
In this article, we study the zeta function $ζ_q$ associated to the Laplace operator $Δ_q$ acting on the space of the smooth $(0,q)$-forms with $q=0,\ldots,n$ on the complex projective space $\mathbb{P}^n(\mathbb{C})$ endowed with its Fubini-Study metric. In particular, we show that the values of $ζ_q$ at non-positive integers are rational. Moreover, we give a formula for $ \sum_{q\geq 0}(-1)^{q+1}qζ_q'(0),$ the associated holomorphic analytic torsion.
Rahayu Kusdarwati, Muhammad Yohan Firmansyah, Yudi Cahyoko
Abstract Artemia is an important live feed in the hatchery. Quality of Artemia can not be separated from the feed quality that given. The quality and quantity of feed in the waters constitute factors that determine the growth rate and nutrition contentent the Artemia. This study aims to determine effect of different live feed type to the growth rate and nutritional content on Artemia sp.. The research method used was experimental with Completely Randomized Design (CRD) using four treatments and five replications. The treatments used were: silage fish (A), Skeletonema sp. (B), Chaetoceros sp. (C) and Tetraselmis sp. (D). Analysis of data uses Anova. To know the difference among the treatments were done by Duncan Multiple range test. The results showed that difference of natural feed influence highly significant (p<0,05) on the rate of growth in absolute length and significant influence (p <0,05) on average daily growth weight of Artemia sp. Absolute length growth rate was highest in treatment D (3,92mm), then a row followed by treatment C (3,275mm), A (1,89mm) and B (1,775mm). The daily growth rate of weight was highest in treatment D (25,43%), then a row followed by treatment C (21,91%), B (19,24%) and A (18,77%). . Artemia that given live feed produces highest nutritional value of D (protein 44,96%; carbohydrate 18,47% and fat 26,91%) wasted Tetraselmis sp. and the lowest obtained by treatment A (protein 41,21%; carbohydrate 8,88% and fat 29,1%) wasted silage fish. Water quality during Artemia cultivation was temperature 28-320C, pH 7, dissolved oxygen 5-8 mg/L, salinity 31 ppt and ammonia 00,25 mg/L.
We show that a large class of limit-periodic Schrödinger operators has purely absolutely continuous spectrum in arbitrary dimensions. This result was previously known only in dimension one. The proof proceeds through the non-perturbative construction of limit-periodic extended states. An essential step is a new estimate of the probability (in quasi-momentum) that the Floquet Bloch operators have only simple eigenvalues.
We study the low-energy asymptotics of the spectral shift function for Schrödinger operators with potentials decaying like $O(\frac{1}{|x|^2})$. We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue and zero resonance.
We study the solution of the Toda lattice Cauchy problem with steplike initial data. The initial data are supposed to tend to zero as $n \to +\infty$. By the inverse scattering transform method formulas allowing us to find solution of the Toda lattice are obtained.
In this paper, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with $k$ cut vertices (respectively, $k$ cut edges) with the minimal distance spectral radius.
We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we use these bounds in order to obtain some results concerning the $L^p$ spectrum.
We give the Jordan form and the Singular Value Decomposition for an integral operator ${\cal N}$ with a non-symmetric kernel $N(y,z)$. This is used to give solutions of Fredholm equations for non-symmetric kernels, and to determine the behaviour of ${\cal N}^n$ and $({\cal N}{\cal N^*})^n$ for large $n$.
Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of Pólya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-Pólya-Weinberger conjecture proved by Ashbaugh and Benguria.