We consider essential self-adjointness of even order, strongly singular, homogeneous differential operators associated with differential expressions of the type \[ \tau_{2n}(c) = (-1)^n \frac{d^{2n}}{d x^{2n}} + \frac{c}{x^{2n}} \] in the Hilbert space $L^2(0,\infty)$. While the special case $n=1$ is classical and it is well-known that the above differential operator is essentially self-adjoint when defined on smooth functions with compact support if and only if $c \geq c_2:= 3/4$, the case $n \geq 2$ is far from obvious. In particular, it is not at all clear from the outset that a corresponding constant $c_n$ exists. As one of the principal results of this paper we indeed establish the existence of $c_n$. The answer is surprisingly complex and involves various aspects of the geometry and analytical theory of polynomials. Based on joint work with Fritz Gesztesy and Markus Hunziker.

We consider a class of integral functionals with Musielak-Orlicz type variable growth, which possibly are linear in some regions of the domain. This setting includes variable-exponent type integrands with $p(x) \geq 1$ as well as double-phase integrands with the lower exponent equal to one. We identify the $L^2$-subdifferential of the functional, and as an application we obtain the Euler-Lagrange equation for the variant of Rudin-Osher-Fatemi image denoising problem with variable growth regularising term. We then use the Euler-Lagrange equation to suggest a modification of the ROF scheme in order to prevent staircasing, i.e., emergence of piecewise constant structures from the noise. This talk is based on a joint work with M. Łasica and A. Matsoukas.

In the talk it will be presented a variational characterisation for a class of non linear evolution equations with constant non-negative Dirichlet boundary conditions on a bounded domain as gradient flows in the space of non-negative measures. The relevant geometry is given by the modified Wasserstein distance introduced by Figalli and Gigli that allows for a change of mass by letting the boundary act as a reservoir. We give a dynamic formulation of this distance as an action minimisation problem for curves of non-negative measures satisfying a continuity equation in the spirit of Benamou-Brenier. Then we characterise solutions to non-linear diffusion equations with Dirichlet boundary conditions as metric gradient flows of internal energy functionals in the sense of curves of maximal slope. The topic has been addressed in a joint work with Matthias Erbar.

Originating in fluid dynamics, the stability analysis of Ekman boundary layers leads to a spectral problem for a family of non-selfadjoint linear operator matrices. We present new enclosures for the point spectrum (leading to the solution of an open problem posed by L. Greenberg and M. Marletta in 2004) and investigate the number of eigenvalues. Our analysis is based on a Birman-Schwinger type argument which exploits underlying similarities to Schrödinger operators. Based on joint work with O. Ibrogimov and P. Siegl.

We investigate the spectral theory of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These periodic graphs are shown to have rather intriguing behaviour. We shall discover “partly flat bands” which are only flat for certain quasimomenta. We construct a periodic graph whose Laplacian has purely singular continuous spectrum. We prove that motion remains ballistic along at least one layer under quite general assumptions. We construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. We believe that this class of graphs can serve as a playground to better understand exotic spectra and dynamics in the future. Based on joint work with Matthias Täufer, Olaf Post and Mostafa Sabri, arxiv:2411.14965 [math.SP].

In the first part of the talk we discuss recent results on Schrödinger operators on periodic graphs which are non-standard in the sense that we allow vertices to have an infinite number of neighbours. It turns out that such non-locally finite graphs exhibit various phenomena which are absent in the locally finite setting: and this is true from a spectral as well as transport point of view. Using some explicit examples, we shall illustrate such new effects in more detail. Quite surprisingly, it turns out that one of the examples provides us with a negative answer to a question raised by Damanik et al. in a recent paper on ballistic transport (this part of talk is based on joint work with O. Post, M. Sabri, and M. Täufer). If time allows, we shall also quickly discuss spectral comparison results on discrete graphs. In recent years, various authors have derived such comparison results on Euclidean domains and quantum graphs. Our aim is to present a generalization to the discrete setting. Along the way, we also establish a so-called local Weyl law which is of independent interest (the second part of the talk is based on joint work with P. Bifulco and C. Rose).