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In this work we study inverse spectral problems for bounded domains, smooth closed mani- folds, and semiclassical Schro ̈dinger operators, with particular concern towards the latter. A central tool in the analysis of inverse spectral problems are trace invariants, however these are not without limitations. We show that there exist pairs of non-isometric potentials for the 1D semiclassical Schro ̈dinger operator whose spectra agree up to O(h∞), and hence have the same semiclassical trace invariants, yet all corresponding eigenvalues differ no less than exponentially. This result was conjectured in the work of Guillemin and Hezari [GH12], where they prove a very similar result for the ground state eigenvalues, however cannot remove the possibility of a subsequence hk → 0 where the ground state eigenvalues may agree.

In this thesis, we will study the conductance of several models originating from condensed matter physics, including the Anderson model for random systems and the Bistritzer-MacDonald (BM) model for twisted bilayer graphene (TBG). In fact, in their time, both models exhibited unexpected conductance properties which bewildered mathematicians and physicists. The Anderson model, developed in 1958 by physicist P.W. Anderson, exhibited unexpected localization/insulating phenomena in the 1D and 2D cases, while TBG was discovered experimentally in 2018 \cite{C18} to have unconventional superconductance at certain ``magic angles'' with relatively flat bands. This thesis has two primary parts. In the first part, we prove different types of localization results in the Anderson model and other related models. In the second part, we study the BM model in various magnetic fields from spectral, semi-classical and physical perspectives; in particular, we focus on the existence and persistence of flat bands which, though mysterious, is believed to be related to the superconductance of TBG \cite{LKV21}. More specifically, for the first part, we initially provide a short non-perturbative proof of Anderson localization and dynamical localization for the 1D Anderson model with arbitrary disorder (e.g. including Bernoulli potential). After that, we derive the dynamical localization in expectation in a related random CMV model with arbitrary disorder. Finally, we work with 2D Anderson model with Bernoulli potential and prove strong dynamical localization in expectation in this setting. We start the second part by first discussing the influence of different magnetic and electric potentials on the existence/persistence of flat bands for TBG. After the general discussion, we divert our attention the strong constant magnetic fields and provide the explicit asymptotic expansion of the density of states (DOS). In particular, we point out the intrinsically different roles that chiral and anti-chiral potentials play in the magnetic response of TBG. Finally, from the expansion of the DOS, we are able to study the physical phenomena, including magnetic oscillations and quantum Hall effect of the TBG. We find that the chiral potential enhances these phenomena, while the anti-chiral potential diminishes them.

The purpose of this dissertation is to develop the spectral theory for bounded planar domains. A particularly effective method for understanding the Laplace spectrum is via the wave equation. Chapters 3-6 develop a general theory for explicitly solving the wave equation in terms of dynamical data associated to the billiard map on the underlying domain. In particular, we develop a microlocal Hadamard-Riesz type parametrix for the wave propagator near reflecting rays in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in [44] and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works of [44] and [60], is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this dissertation (in addition to the author’s previous works [69] and [68]) contain the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard-Riesz type parametrix for the wave propagator appears to be new in the literature. This is also based on the author’s prior work [69] in the special case of elliptical domains and [68] for general convex billiard tables. It allows us to circumvent the symbol calculus in [12] and [25] when computing trace formulas, which are instead derived from our explicit parametrix and a rescaling argument via Hadamard’s variational formula for the wave trace. These techniques also appear to be new in the literature. In Chapter 7, we investigate C1 isospectral deformations of the ellipse with Robin boundary conditions, allowing both the Robin function and domain to deform simultaneously. We prove that if the deformations preserve the reflectional symmetries of the ellipse, then the first variation of both the domain and Robin function must vanish. If the deformation is in fact smooth, reparametrizing allows us to show that the first variation actually vanishes to infinite order. In particular, there exist no such analytic isospectral deformations. The key ingredients are a version of Hadamard’s variational formula for variable Robin boundary conditions and an oscillatory integral representation of the wave trace variation which uses action angle coordinates for the billiard map. For the latter, we use the explicit parametrix for the wave propagator developed in the interior, microlocally near geodesic loops (see Chapter 5).

In this paper we introduce and study the algebra of almost strongly fuzzy bounded operators on fuzzy normed linear spaces. Some connections with the algebra of strongly fuzzy bounded operators are presented. The completeness of these algebras is also established. A characterization of bounded elements of the algebra of almost strongly fuzzy bounded operators is given. We also introduce two fuzzy spectral radii for almost strongly fuzzy bounded operators. For bounded elements of the considered algebra the equality between these fuzzy spectral radii is proved.

For a Schr\"odinger operator on the plane $\mathbb{R}^2$ with electric potential $V$ and Aharonov--Bohm magnetic field we obtain an upper bound on the number of its negative eigenvalues in terms of the $L^1(\mathbb{R}^2)$-norm of $V$. Similar to Calogero's bound in one dimension, the result is true under monotonicity assumptions on $V$. Our proof method relies on a generalisation of Calogero's bound to operator-valued potentials. We also establish a similar bound for the Schr\"odinger operator (without magnetic field) on the half-plane when a Dirchlet boundary condition is imposed and on the whole plane when restricted to antisymmetric functions., Comment: 14 pages

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds cannot be established.

We consider the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. In this situation it may happen that there is no existence and uniqueness theorem for balanced solutions of a given initial value problem. We describe the set of solutions the equation does have and establish that the adjoint of the minimal operator is still the maximal operator, even though unique continuation of balanced solutions fails.

This chapter deals with eigenvalues and eigenvectors of matrices. The material here is a sequel to Chapter 2 dealing with the solution of linear equations. Eigenvalues are very important since many engineering problems naturally lead to eigenvalue problems. When the size of a matrix is large special numerical methods are necessary for obtaining eigenvalues and eigenvectors.

In spectral bisection, a Fielder vector is used for partitioning a graph into two connected subgraphs according to its sign pattern. In this article, we investigate graphs having Fiedler vectors with unbalanced sign patterns such that a partition can result in two connected subgraphs that are distinctly different in size. We present a characterization of graphs having a Fiedler vector with exactly one negative component, and discuss some classes of such graphs. We also establish an analogous result for regular graphs with a Fiedler vector with exactly two negative components. In particular, we examine the circumstances under which any Fiedler vector has unbalanced sign pattern according to the number of vertices with minimum degree.

We present a general lower bound for the fundamental tone for the p-Laplacian on Riemannian manifolds carrying a special kind of function. We then apply our result to the cases of negatively curved simply connected manifolds, a class of warped product manifolds and for a class of Riemannian submersions.

We analyze data to build a quantitative understanding of the world. Linear algebra is the foundation to algorithms, dating back one hundred years, for extracting structure from data. Modern technologies provide an abundance of multi-dimensional data, in which multiple variables or factors can be compared simultaneously. To organize and analyze such datasets we can use a tensor, the higher order analogue of a matrix. However, many theoretical and practical challenges arise in extending linear algebra to the setting of tensors. In the first part of this thesis, I study and develop the algebraic theory of tensors. Tensors of low real rank, as well as singular vectors and singular values of tensors, parametrize semi-algebraic sets, defined by polynomial equations and inequalities. I give exact algebraic characterizations of sets of tensors of interest, using real algebraic geometry, polyhedral geometry and computer algebra. I obtain a membership test for the set of real rank two tensors, I describe the variety of singular vectors of orthogonally decomposable tensors, and I obtain inequalities relating the singular values of the flattenings of a tensor. I show that rank and symmetric rank coincide for tensors of low symmetric rank, bounded by seven. I conclude by describing tensor hypernetworks, a flexible framework for decomposing and approximating tensors in application-specific contexts. Throughout these results, an in-depth study of small tensors is used to set up the general theory. The theoretical results explain pitfalls of existing tensor algorithms, and also suggest new approaches for finding structure in a tensor. In the second part of this thesis, I present three algorithms for tensor data. The algorithms use algebraic and geometric structure to give guarantees of optimality. Tensors have a close connection to multivariate distributions in statistics. I obtain the first non-trivial instance of an exact maximum likelihood estimate for a model with hidden variables. I give a numerical algorithm to recover paths from their third order signature tensors, an inverse problem from stochastic analysis. I also give an algorithm to cluster multi-dimensional data, with structured clusters encoded via algebraic constraints in a tensor. The structure facilitates the interpretation of clusters in a dataset of cancer cell lines.

Let B ( n , q ) be the class of block graphs on n vertices having all their blocks of the same size. We prove that if G ∈ B ( n , q ) has at most three pairwise adjacent cut vertices then the minimum spectral radius ρ ( G ) is attained at a unique graph. In addition, we present a lower bound for ρ ( G ) when G ∈ B ( n , q ) .

This paper aims to investigate the fourth-order boundary value problems with distributional potentials. We first prove that the operators associated with the problems are self-adjoint and the corresponding eigenvalues are real. Then we obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problems: the boundary conditions, the coefficient functions and the endpoints. Moreover, we find the differential expressions for each parameter.

We give a unified proof of the interlacing properties of eigenvalues of principle submatrices of totally positive matrices.

For a graph G , the signless Laplacian matrix Q ( G ) is defined as Q ( G ) = D ( G ) + A ( G ) , where A ( G ) is the adjacency matrix of G and D ( G ) the diagonal matrix whose main entries are the degrees of the vertices in G . The Q -spectrum of G is that of Q ( G ) . In the present paper, we are interested in the minimum values of the second largest signless Laplacian eigenvalue q 2 ( G ) of a connected graph G . We find the five smallest values of q 2 ( G ) over the set of connected graphs G with given order n . We also characterize the corresponding extremal graphs.

[EN] In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lame operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient. B. C. is member of GNAMPA (INDAM) and he was supported by Fondo Sociale Europeo Programma Operativo Nazionale Ricerca e Innovazione 2014-2020, progetto PON: progetto AIM1892920-attivita 2, linea 2.1. L. C. acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.

Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms. When suitably scaled, graph Laplacians approach limiting continuum operators in the large data limit. Studying these limiting operators, therefore, sheds light on learning algorithms. This paper is devoted to the study of a parameterized family of divergence form elliptic operators that arise as the large data limit of graph Laplacians. The link between a three-parameter family of graph Laplacians and a three-parameter family of differential operators is explained. The spectral properties of these differential operators are analyzed in the situation where the data comprises of two nearly separated clusters, in a sense which is made precise. In particular, we investigate how the spectral gap depends on the three parameters entering the graph Laplacian, and on a parameter measuring the size of the perturbation from the perfectly clustered case. Numerical results are presented which exemplify the analysis and which extend it in the following ways: the computations study situations in which there are two nearly separated clusters, but which violate the assumptions used in our theory; situations in which more than two clusters are present, also going beyond our theory; and situations which demonstrate the relevance of our studies of differential operators for the understanding of finite data problems via the graph Laplacian. The findings provide insight into parameter choices made in learning algorithms which are based on weighted adjacency matrices; they also provide the basis for analysis of the consistency of various unsupervised and semi-supervised learning algorithms, in the large data limit.

In this paper, a new M-eigenvalue inclusion set for a partially symmetric tensor is provided. It is proved that the new set is tighter than some existing M-eigenvalue inclusion sets. Based on the obtained results, an upper bound of the largest M-eigenvalue is given and a modified WQZ-algorithm is established which guarantees the generated converges to the largest M-eigenvalue of the tensor faster.

We show that an approximate lattice in a nilpotent Lie group admits a relatively dense subset of central $(1-\epsilon)$-Bragg peaks for every $\epsilon > 0$. For the Heisenberg group we deduce that the union of horizontal and vertical $(1-\epsilon)$-Bragg peaks is relatively dense in the unitary dual. More generally we study uniform approximate lattices in extensions of lcsc groups. We obtain necesary and sufficient conditions for the existence of a continuous horizontal factor of the associated hull-dynamical system, and study the spectral theory of the hull-dynamical system relative to this horizontal factor., Comment: 53 pages, Comments welcome!

We investigate the spectrum of the Sturm-Liouville difference equation on the half-line with self-adjoint operator coefficients in an infinite dimensional Hilbert space together with the Dirichlet boundary condition. We find the Jost solution and examine its analytical and asymptotical properties. Using these properties, we obtain the continuous and point spectrum of the discrete operator generated by the Sturm-Liouville difference equation with self-adjoint operator coefficients. We also show that this operator has a finite number of eigenvalues with finite multiplicities under a certain condition on the operator coefficients.

This work, which can be considered as a small monograph, is devoted to the study of twoand three-dimensional boundary-value problems for eigenvalues of the Laplace operator with frequently alternating type of boundary conditions. The main goal is to construct asymptotic expansions of the eigenvalues and eigenfunctions of the considered problems. Asymptotic expansions are constructed on the basis of original combinations of asymptotic analysis methods: the method of matching asymptotic expansions, the boundary layer method and the multi-scale method. We perform the analysis of the coefficients of the formally constructed asymptotic series. For strictly periodic and locally periodic alternation of the boundary conditions, the described approach allows one to construct complete asymptotic expansions of the eigenvalues and eigenfunctions. In the case of nonperiodic alternation and the averaged third boundary condition, sufficiently weak conditions on the alternation structure are obtained, under which it is possible to construct the first corrections in the asymptotics for the eigenvalues and eigenfunctions. These conditions include in consideration a wide class of different cases of nonperiodic alternation. With further, very serious weakening of the conditions on the structure of alternation, it is possible to obtain two-sided estimates for the rate of convergence of the eigenvalues of the perturbed problem. It is shown that these estimates are unimprovable in order. For the corresponding eigenfunctions, we also obtain unimprovable in order estimates for the rate of convergence.

In this paper, relations between discrete Sturm--Liouville problem with nonlocal integral boundary condition characteristics (poles, critical points, spectrum curves) and graphs characteristics (vertices, edges and faces) were found. The previous article was devoted to the Sturm--Liouville problem in the case two-points nonlocal boundary conditions.

We discuss the question of classifying the connected simple graphsHfor which the second largest eigenvalue of the signless LaplacianQ(H) is ≤ 4. We discover that the question is inextricable linked to a knapsack problem with infinitely many allowed weights. We take the first few steps towards the general solution. We prove that this class of graphs is minor closed.

A matrix-weighted graph is an undirected graph with a $k\times k$ positive semidefinite matrix assigned to each edge. There are natural generalizations of the Laplacian and adjacency matrices for such graphs. These matrices can be used to define and control expansion for matrix-weighted graphs. In particular, an analogue of the expander mixing lemma and one half of a Cheeger-type inequality hold for matrix-weighted graphs. A new definition of a matrix-weighted expander graph suggests the tantalizing possibility of families of matrix-weighted graphs with better-than-Ramanujan expansion., 21 pages, 2 figures

With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green’s function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L2ω,q((ω0, a); E).

For n ≥ 2 , we obtain Liouville type theorems for the minimal surface equation in half space R + n with constant Neumann boundary value or linear Dirichlet boundary value.

We consider the equation $-\Delta u= |x|^{\alpha}|u|^{p-1}u$ for any $\alpha\geq 0$, either in $\mathbb R^2$ or in the unit ball $B$ of $\mathbb R^2$ centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as $p\rightarrow +\infty$ of all the radial solutions to these problems. In particular, we show that there is no uniform a priori bound (in $p$) for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the recently shown fact that positive solutions have uniform a priori bounds for $\alpha=0$ and Dirichlet boundary conditions., Comment: 6 figures, 1 table, and 43 pages

This paper is concerned with the existence and multiplicity of solutions for p(x)-Laplacian equations with Robin boundary condition. Our technical approach is based on variational methods.