I discuss a simple toy problem for the Dirichlet Laplacian in a sequence of domains where the contribution of the boundary to the spectral asymptotics is of the same order as the contribution from the interior

We establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. In particular, we correct erroneous results published by Liu (J Geom Anal 31:10164–10193, 2021).

This volume is dedicated to the memory of Mikhail (Misha) Shubin who passed away on May 13, 2020. The articles that are presented in the volume are written by people who knew Misha well; some of them were Misha's students or collaborators. The diversity of topics reflects the diversity of Misha's interests. One of the authors of this preface (LF) knew Misha from 1974 when he took his pseudodifferential operators class as an undergraduate student. The second one (MB) was Misha's colleague and collaborator.

The paper by G. Liu [arxiv:2109.02561] contains an error. In this note, I give a brief review of the problem and indicate what the error is.

decided to become a mathematician.He was admitted to mech-mat of MSU in 1961 and became a pupil of a renowned expert in PDEs, M. I. Vishik.He completed his PhD in 1969 with a focus on the theory (including index) of matrix-valued Wiener-Hopf operators.In 1981, he defended his Doctor of Science degree (an analog to the Habilitation of Germany), based on his work in the theory of operators with almost periodic coefficients, where he was one of the leaders.Since then, M. Shubin ventured successfully into a variety of areas of mathematics, demonstrating a remarkable breadth of interests and expertise.In his papers and books (totaling around 140), he made major influential contributions in a variety of areas, including (but not limited to) operator theory, spectral theory of differential and pseudodifferential operators (especially operators with almost periodic and random coefficients), the theory and applications of pseudodifferential operators and their discrete analogs, microlocal analysis, geometric analysis and analysis on manifolds (e.g., spectral theory on noncompact manifolds and Lie groups, index theory, general Riemann-Roch theorems, and invariants of manifolds), integrable systems, nonstandard analysis, etc.Besides his many solo publications, he coauthored works with his students and younger colleagues, as well as with prominent experts, such as M. Gromov, V. Kondratiev, V. Maz'ya, S. P. Novikov, T. Sunada, and others.For more details, one can consult the memorial article [2].Besides his many significant research papers, Misha coauthored several important survey articles.His book Pseudodifferential Operators and Spectral Theory [4], first published in Russian in 1978, went through several English editions and still remains a popular source for studying microlocal analysis.Another remarkable book The Schrödinger Equation [1], which he coauthored with F. Berezin, is also a well-known and in some ways unique source.

The article is dedicated to thye memory of a distinguished mathematician Professor Misha Shubin

Let $A$ be an elliptic pseudodifferential operator of positive order on a compact closed manifold, and let $T$ be a pseudodifferential operator of negative order such that $T^m$ is of trace class. We compute $\log\det(A(I+T))-\log\det A-\log\det_m (I+T)$ where first two determinants are zeta function regularized, and the last one is a regularized Fredholm determinant.

For a compact, connected metric graphs with a boundary that consists of k vertices, we prove that an arbitrary symmetric k× k matrix with real entries can be realized as the Dirichlet-to-Neumann operator for the Laplacian plus a constant.

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric complex associated with a gradient like vector field and show how differential forms can be integrated on its unstable/stable sets. Integration leads to a morphism between the de Rham complex and the geometric complex.

In this paper the authors explain, in great detail, how to equip the compactified (un)stable sets and trajectory spaces of a gradient-like vector field with the structure of a smooth manifold with corners, in a canonical way. This is done for vector fields which are gradient-like with respect to a proper Morse function, satisfy the Smale transversality condition, and are of standard form, ∑i≤qxi∂∂xi−∑i>qxi∂∂xi, near the zeros (critical points). As an application, the authors discuss the integration homomorphism relating the de Rham complex and the Thom–Smale complex with coefficients in a representation of the fundamental group.

We consider the Dirichlet Laplacian in a family of narrow unbounded domains. As the width of these domains goes to 0, we study the asymptotic behavior of the eigenvalues that lie below the essential spectrum and the asymptotic behavior of the corresponding eigenfunctions.

We show that the Szego regularized determinant of a zeroth order operator differs from the zeta regularized determinant of the same operator by a local term. In addition, we compute multiplicative anomalies for the zeta regularized determinant.

In the paper, we derive a formula for computing the determinant of a Schrodinger operator on a compact metric graph. This formula becomes very explicit in the case of the Laplacian with the Neumann boundary conditions.

We prove that, for a metric graph different from a polygon, the spectrum of the Laplacian is generically simple.