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Peter Kuchment

Publications and source records attributed to Peter Kuchment.

At least 19 recordsLinked to original sources
arXiv2026

Directional Ballistic Transport in Quantum Waveguides

We study the transport properties of Schr\"odinger operators on $\mathbb{R}^d$ with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce surface states that are weakly confined near the support of the potential. We show that a natural set of surface states exhibits directional ballistic transport, characterized by ballistic transport in the periodic directions and its absence in the others. To prove this, we develop a Floquet theory that captures the analytic variation of surface states. The main idea consists of reformulating the eigenvalue problem for surface states as a Fredholm problem via the Dirichlet-to-Neumann map.

math.SP↗
arXiv2024

Stability of spectral partitions with corners

A spectral minimal partition of a manifold is a decomposition into disjoint open sets that minimizes a spectral energy functional. While it is known that bipartite minimal partitions correspond to nodal partitions of Courant-sharp Laplacian eigenfunctions, the non-bipartite case is much more challenging. In this paper, we unify the bipartite and non-bipartite settings by defining a modified Laplacian operator and proving that the nodal partitions of its eigenfunctions are exactly the critical points of the spectral energy functional. Moreover, we prove that the Morse index of a critical point equals the nodal deficiency of the corresponding eigenfunction. Some striking consequences of our main result are: 1) in the bipartite case, every local minimum of the energy functional is in fact a global minimum; 2) in the non-bipartite case, every local minimum of the energy functional minimizes within a certain topological class of partitions. Our results are valid for partitions with non-smooth boundaries; this introduces considerable technical challenges, which are overcome using delicate approximation arguments in the Sobolev space $H^{1/2}$.

math.AP↗
arXiv2024

Half-time Range description for the free space wave operator and the spherical means transform

The forward problem arising in several hybrid imaging modalities can be modeled by the Cauchy problem for the free space wave equation. Solution to this problems describes propagation of a pressure wave, generated by a source supported inside unit sphere $S$. The data $g$ represent the time-dependent values of the pressure on the observation surface $S$. Finding initial pressure $f$ from the known values of $g$ consitutes the inverse problem. The latter is also frequently formulated in terms of the spherical means of $f$ with centers on~$S$. Here we consider a problem of range description of the wave operator mapping $f$ into $g$. Such a problem was considered before, with data $g$ known on time interval at least $[0,2]$ (assuming the unit speed of sound). Range conditions were also found in terms of spherical means, with radii of integration spheres lying in the range $[0,2]$. However, such data are redundant. We present necessary and sufficient conditions for function $g$ to be in the range of the wave operator, for $g$ given on a half-time interval $[0,1]$. This also implies range conditions on spherical means measured for the radii in the range $[0,1]$.

math.AP↗
arXiv2024

On open book analogs of quantum graphs

Quantum graphs have become in this century a favorite playground for mathematicians, mathematical physicists, and chemists, due to their manifold applications as models of thin structures, as well as presenting sometimes simpler playground for hard higher dimensional problems. It was clear from some applications that thin surface structures (looking as stratified varieties) also arise, for instance in photonic crystals theory and dynamical systems. However, both justification and studying of these models is much harder and very little progress has been made by now. The goal of this note is to set down some basic notions and results for such structures. The name ``open book'' has been used for such geometric structures in topology and comes from an image of several smooth $n$-dimensional ``pages'' bound to an $(n-1)$- dimensional ``binding.''

math-ph↗
arXiv2020

Deep learning for 2D passive source detection in presence of complex cargo

Methods for source detection in high noise environments are important for single-photon emission computed tomography (SPECT) medical imaging and especially crucial for homeland security applications, which is our main interest. In the latter case, one deals with passively detecting the presence of low emission nuclear sources with significant background noise (with Signal To Noise Ratio ($SNR$) $1\%$ or less). In passive emission problems, direction sensitive detectors are needed, to match the dimensionalities of the image and the data. Collimation, used for that purpose in standard Anger $γ$-cameras, is not an option. Instead, Compton $γ$-cameras (and their analogs for other types of radiation) can be utilized. Backprojection methods suggested before by two of the authors and their collaborators enable detection in the presence of a random uniform background. In most practical applications, however, cargo packing in shipping containers and trucks creates regions of strong absorption and scattering, while leaving some streaming gaps open. In such cases backprojection methods prove ineffective and lose their detection ability. Nonetheless, visual perception of the backprojection pictures suggested that some indications of presence of a source might still be in the data. To learn such features (if they do exist), a deep neural network approach is implemented in 2D, which indeed exhibits higher sensitivity and specificity than the backprojection techniques in a low scattering case and works well when presence of complex cargo makes backprojection fail completely.

physics.ins-det↗
arXiv2020

Three-representation problem

We provide the proof of a previously announced result that resolves the following problem posed by A.~A.~Kirillov. Let $T$ be a presentation of a group $\mathcal{G}$ by bounded linear operators in a Banach space $G$ and $E\subset G$ be a closed invariant subspace. Then $T$ generates in the natural way presentations $T_1$ in $E$ and $T_2$ in $F:=G/E$. What additional information is required besides $T_1, T_2$ to recover the presentation $T$? In finite-dimensional (and even in infinite dimensional Hilbert) case the solution is well known: one needs to supply a group cohomology class $h\in H^1(\mathcal{G},Hom(F,E))$. The same holds in the Banach case, if the subspace $E$ is complemented in $G$. However, every Banach space that is not isomorphic to a Hilbert one has non-complemented subspaces, which aggravates the problem significantly and makes it non-trivial even in the case of a trivial group action, where it boils down to what is known as the three-space problem. This explains the title we have chosen. A solution of the problem stated above has been announced by the author in 1976, but the complete proof, for non-mathematical reasons, has not been made available. This article contains the proof, as well as some related considerations of the functor $Ext^1$ in the category \textbf{Ban} of Banach spaces.

math.FA↗
arXiv2020

Generic properties of dispersion relations for discrete periodic operators

An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrödinger operator $-Δ+V(x)$ in $R^n$ with periodic potential near the edges of the spectrum. A well known conjecture says that generically (with respect to perturbations of the periodic potential) the extrema are attained by a single branch of the dispersion relation, are isolated, and have non-degenerate Hessian (i.e., dispersion relations are graphs of Morse functions). The important notion of effective masses in solid state physics, as well as Liouville property, Green's function asymptotics, etc. hinge upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples exist in some discrete situations. We start establishing the following dichotomy: the non-degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters. The known counterexample has only two free parameters. This might be too tight for genericity to hold. We consider the maximal $Z^2$-periodic two-atomic nearest-cell interaction graph, with nine edges per unit cell and the discrete "Laplace-Beltrami" operator on it. We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph. We show three different approaches to the genericity, which might be suitable in various situations. It is also proven in this case that adding more parameters does not destroy the genericity result. We list all "bad" periodic subgraphs of the one we consider and discover that in all these cases genericity fails for "trivial" reasons only.

math-ph↗
arXiv2019

Liouville-Riemann-Roch theorems on abelian coverings

The classical Riemann-Roch theorem has been extended by N. Nadirashvili and then M. Gromov and M. Shubin to computing indices of elliptic operators on compact (as well as non-compact) manifolds, when a divisor mandates a finite number of zeros and allows a finite number of poles of solutions. On the other hand, Liouville type theorems count the number of solutions that are allowed to have a "pole at infinity." Usually these theorems do not provide the exact dimensions of the spaces of such solutions (only finite-dimensionality, possibly with estimates or asymptotics of the dimension. An important case has been discovered by M. Avellaneda and F. H. Lin and advanced further by J. Moser and M. Struwe. It pertains periodic elliptic operators of divergent type, where, surprisingly, exact dimensions can be computed. This study has been extended by P. Li and Wang and brought to its natural limit for the case of periodic elliptic operators on co-compact abelian coverings by P. Kuchment and Pinchover. Comparison of the results and techniques of Nadirashvili and Gromov and Shubin with those of Kuchment and Pinchover shows significant similarities, as well as appearance of the same combinatorial expressions in the answers. Thus a natural idea was considered that possibly the results could be combined somehow in the case of co-compact abelian coverings, if the infinity is "added to the divisor." This work shows that such results indeed can be obtained, although they come out more intricate than a simple-minded expectation would suggest. Namely, the interaction between the finite divisor and the point at infinity is non-trivial.

math.SP↗
arXiv2019

Spectra of "fattened" open book structures

We establish convergence of spectra of Neumann Laplacian in a thin neighborhood of a branching 2D structure in 3D to the spectrum of an appropriately defined operator on the structure itself. This operator is a 2D analog of the well known by now quantum graphs. As in the latter case, such considerations are triggered by various physics and engineering applications.

math-ph↗
arXiv2019

Analyticity of the spectrum and Dirichlet-to-Neumann operator technique for quantum graphs

In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the Grassmanian of all possible vertex conditions was used, thus creating an erroneous impression that something ``wrong'' can happen at the boundaries of the chart. Here we show that the analyticity of the corresponding ``dispersion relation'' holds over the whole Grassmannian, as well as over other parameter spaces. We also address the Dirichlet-to-Neumann (DtN) technique of relating quantum and discrete graph operators, which allows one to transfer some results from the discrete to the quantum graph case, but which has issues at the Dirichlet spectrum. We conclude that this difficulty, as in the first part of the paper, stems from the use of specific coordinates in a Grassmannian and show how to avoid it to extend some of the consequent results to the general situation.

math-ph↗
arXiv2018

Boris R. Vainberg (on his 80th birthday)

Boris R. Vainberg was born on March 17, 1938, in Moscow. His father was a Lead Engineer in an aviation design institute. His mother was a homemaker. From early age, Boris was attracted to mathematics and spent much of his time at home and in school working through collections of practice problems for the Moscow Mathematical Olympiad. His first mathematical library consisted of the books he received as one of the prize-winners of these olympiads.

math.HO↗
arXiv2018

Some binomial formulas for non-commuting operators

Let $D$ and $U$ be linear operators in a vector space (or more generally, elements of an associative algebra with a unit). We establish binomial-type identities for $D$ and $U$ assuming that either their commutator $[D,U]$ or the second commutator $[D,[D,U]]$ is proportional to $U$. Operators $D=d/dx$ (differentiation) and $U$- multiplication by $e^{λx}$ or by $\sin λx$ are basic examples, for which some of these relations appeared unexpectedly as byproducts of an authors' previous medical imaging research.

math.CA↗
arXiv2017

On Parseval frames of exponentially decaying composite Wannier functions

Let $L$ be a periodic self-adjoint linear elliptic operator in $\R^n$ with coefficients periodic with respect to a lattice $\G$, e.g. Schrödinger operator $(i^{-1}\partial/\partial_x-A(x))^2+V(x)$ with periodic magnetic and electric potentials $A,V$, or a Maxwell operator $\nabla\times\varepsilon (x)^{-1}\nabla\times$ in a periodic medium. Let also $S$ be a finite part of its spectrum separated by gaps from the rest of the spectrum. We address here the question of existence of a finite set of exponentially decaying Wannier functions $w_j(x)$ such that their $\G$-shifts $w_{j,\g}(x)=w_j(x-\g)$ for $\g\in\G$ span the whole spectral subspace corresponding to $S$. It was shown by D.~Thouless in 1984 that a topological obstruction sometimes exists to finding exponentially decaying $w_{j,\g}$ that form an orthonormal (or any) basis of the spectral subspace. This obstruction has the form of non-triviality of certain finite dimensional (with the dimension equal to the number of spectral bands in $S$) analytic vector bundle (Bloch bundle). It was shown in 2009 by one of the authors that it is always possible to find a finite number $l$ of exponentially decaying Wannier functions $w_j$ such that their $\G$-shifts form a tight (Parseval) frame in the spectral subspace. This appears to be the best one can do when the topological obstruction is present. Here we significantly improve the estimate on the number of extra Wannier functions needed, showing that in physical dimensions the number $l$ can be chosen equal to $m+1$, i.e. only one extra family of Wannier functions is required. This is the lowest number possible in the presence of the topological obstacle. The result for dimension four is also stated (without a proof), in which case $m+2$ functions are needed. The main result of the paper was announced without a proof in Bull. AMS, July 2016.

math-ph↗
arXiv2016

Inversion of Weighted Divergent Beam and Cone Transforms

In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in any dimension. The results of numerical implementation of some of the formulas in dimensions two and three are also provided.

math.NA↗
arXiv2016

3D Image Reconstruction from Compton camera data

In this paper, we address analytically and numerically the inversion of the integral transform (\emph{cone} or \emph{Compton} transform) that maps a function on $\mathbb{R}^3$ to its integrals over conical surfaces. It arises in a variety of imaging techniques, e.g. in astronomy, optical imaging, and homeland security imaging, especially when the so called Compton cameras are involved. Several inversion formulas are developed and implemented numerically in $3D$ (the much simpler $2D$ case was considered in a previous publication). An admissibility condition on detectors geometry is formulated, under which all these inversion techniques will work.

physics.data-an↗