Schedule for: 22w5149 - Analytic and Geometric Aspects of Spectral Theory

We consider the D-N operator on the three-dimensional ball associated to a Schrodinger operator with a smooth potential. Its eigenvalues form clusters of size O(1/k) around the sequence of natural numbers k=1,2, … . The asymptotics of the spectral shifts in the clusters, as k tends to infinity, is given by a series of distributions on the real line, which we refer to as the band invariants. I will discuss machinery that allows one to compute the first few (and, in principle, all) band invariants. This is joint work with Salvador Pérez Esteva and Carlos Villegas Blas.

We consider boundary value problems on manifolds with boundary where the boundary exhibits singularities of fibred cusp type. The simplest (unfibred) cusp is what is sometimes called an incomplete cusp, e.g. the complement of two touching circles in the plane.

The fibred version includes the complement of two touching balls in $\mathbb{R}^n$, but our results also extend to geometries which are conformal to these incomplete cusps, for example fundamental domains of Fuchsian groups or uniformly fattened infinite cones in $\mathbb{R}^n$.

For the Laplacian on such spaces, or more general elliptic operators $P$ whose structure relates well to the geometry, we study one of the basic objects of the theory of boundary value problems: the Calderón projector, which is essentially the projection to the set of boundary values of the homogeneous equation $Pu=0$. For a smooth compact manifold with boundary, it is classical that the Calderón projector is a pseudodifferential operator (PsiDO). In the case of the Laplacian one can deduce from this that the Dirichlet-to-Neumann operator, which is fundamental to many spectral theoretic questions studied currently (like the Steklov spectrum), is a PsiDO also.

We extend these results to the case where the boundary has fibred cusp singularities: both the Calderón projector and the Dirichlet-Neumann operator are in a PsiDO calculus adapted to the geometry, the so-called phi-calculus. This yields a precise description of their integral kernels near the singularities. In the talk I will introduce the necessary background on the phi-calculus.

This is joint work with K. Fritzsch and E. Schrohe.

In this talk I shall present recently developed techniques to construct non-selfadjoint Schrödinger operators with prescribed discrete eigenvalues. The first method is used to add one more eigenvalue to the spectrum of a given Schrödinger operator. By iterating the procedure, one can construct infinitely many eigenvalues that accumulate to a prescribed point of the essential spectrum, the non-negative reals, or even accumulate at every point of the essential spectrum.

The second technique is used to perturb an eigenvalue off the essential spectrum by adding a certain compactly supported complex potential. This method has been used in a recent joint work with Jean-Claude Cuenin to construct a counterexample to the Laptev-Safronov conjecture, which stipulated that the discrete eigenvalues are bounded in modulus by the $L^p$ norm of the potential (for a certain range of $p$).

TBA

In this talk, we find the full Fourier expansion for the generalized non-holomorphic Eisenstein series for certain values of parameters. Such functions appear in the 10-dimensional type IIB superstring scattering amplitude of gravitons. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Additionally, we discuss the possible relation with the Picard-Vessiot theory. This is based on a joint work with Kim Klinger-Logan.

One way to understand the behavior of a mathematical object is the perturbative approach. In spectral theory in particular, one may try to modify the underlying operators and study how the spectra of these operators change. A natural way will be to begin with very simple perturbations, for example to add a rank one operator. If $\phi$ is a vector in a Hilbert space $H$ we may consider the operators $$ A_\alpha = A + \alpha \langle\phi,\, \cdot\, \rangle \phi $$ where $A$ is a selfadjoint operator and $\alpha$ a real number.
In this talk we will see how singular rank one perturbations can be defined when instead of a vector in the Hilbert space we consider a linear discontinuous functional $\phi$ on the domain $D(A)$ of the operator $A$. It will be shown that they correspond to selfadjoint extensions of the symmetric operator $\dot A = A|_{D_\phi}$ with deficiency indices (1, 1) where $$ D_\phi = \{ f\in D(A) : \phi(f) = 0\}. $$

We prove that some of the heat coefficients in the small-time asymptotic expansion of $e^{-t \Delta^r}$ are non-local, where $r \in (0,1)$, and $\Delta$ is a Laplace-type operator over a compact Riemannian manifold.

Since Kac's famous article, “Can we hear the shape of a drum?”, Isospectrality has been a widely studied notion. It is a fruitful subject with still many open questions and several bifurcations. Motivated by the work of Morassi [1] and Bilotta et al [2] we consider the notion of "quasi-isospectrality". Two operators are "quasi-isospectral" if their spectra differs only in a finite number of eigenvalues. In this short talk I will discuss existence of families of quasi-isospectral Schrödinger operators on an interval and some of their properties. This is based in joint work with Clara Aldana [3].

[1] A. Bilotta and A. Morassi and E. Turco. Quasi-isospectral sturm-liouville operators and applications to system identification. Procedia Engineering, 199:1050–1055, 2017. X International Conference on Structural Dynamics, EURODYN 2017
[2] A. Morassi. Constructing rods with given natural frequencies. Mechanical Systems and Signal Processing, 40(1):288–300, 2013. [3] C.L. Aldana and C. Perez (2022). On Quasi-isospectral potentials. arXiv:2202.06110. Preprint.

A linear operator $A$ on a Hilbert space $H$ is called dissipative if $\mathrm{Im}\langle f,Af\rangle\geq 0$ for all $f$ belonging to the domain of $A$. In a series of papers, C. Fischbacher studied dissipative extensions of operators of the form $A=S+iV$ where $S$ is symmetric and $V$ is bounded and nonnegative. In this talk, dissipative extensions of the Schrödinger operator on the half-line with different perturbations will be presented, and some ideas for the Schrödinger operator on the interval will be discussed.

Spectral information of non-selfadjoint differential operators is of great help to solve problems that have been appearing in recent applications such as quantu m graphs. Due to these modern applications, Cuenin and Tretter developed a complete theory of non-symmetric perturbations for self-adjoint operators. In particular, they established stability results for spectral gaps, essential s pectrum gaps, estimates for the resolvent, and stability results for infinities spectral gaps. In addition, they generalized and improved classical perturbation results.
In this talk, I wil present some new results for normal operators. I will speak about the effect of relatively bounded perturbations on the spectru m of normal operators and present stability results for spectral gaps considerin g the spectrum of the unperturbated operator is close to the real axis or it is contained in a sector symmetric to the real axis.

We model a non-Hermitian quantum mechanics by a Sturm-Liouville operator with complex potentials. In contrast to the real case, the Weyl dichotomy in limit point and limit circle case on complex potentials has not yet been totally characterized. Our talk will be about the second order differential equations associated to the Sturm-Liouville eigenvalue problem and the $L^2$ properties of its solutions.

We present a short sketch of the main facts and ideas developed in [Mic09] about Mourre Theory applied to the compact perturbation $H:=\Delta +L + V$ of the discrete Laplacian operator $\Delta$ on $\ell^2(\mathbb Z)$, where $L$ is a barrier potential and $V$ is an anisotropic potential. By the construction of Weyl sequences for the operator $H$ we know that $\sigma_{\text{ess}}(H)= (\sigma_{\text{ess}}(\Delta)+\{l_{-}\}) \cup (\sigma_{\text{ess}}(\Delta)+\{l_{+}\})$ for some $l_{\pm} \in \mathbb R$, but this behaviour is not true in general for $\sigma_{\text{ac}}$, $\sigma_{\text{sc}}$ and $\sigma_{pp}$. The construction of a conjugate operator for $H$ in the sense of Mourre estimate will allow us to study spectral properties of this operator, including the absence of singular continuous spectrum.

[Mic09] N. Michaelis. Spectral theory of anisotropic discrete Schrödinger operators in dimension 1. PhD thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2009.