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Denis Borisov

Publications and source records attributed to Denis Borisov.

33 records · Page 2Linked to original sources
arXiv2012

On the spectrum of deformations of compact double-sided flat hypersurfaces

We study the asymptotic behaviour of the eigenvalues of the Laplace-Beltrami operator on a compact hypersurface in \mathds{R}^{n+1} as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem corresponds to the Dirichlet and Neumann problems on one side of this flat (Euclidean) limit, and derive an explicit three-term asymptotic expansion for the eigenvalues where the remaining two terms are of orders \varepsilon^2\log\varepsilon and \varepsilon^2.

math.AP↗
arXiv2012

Spectral gaps for self-adjoint second order operators

We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that the potential part of this operator is non-negative. We add a localized perturbation assuming that it produces two negative isolated eigenvalues being the two lowest spectral values of the resulting perturbed operator. The main result is a lower bound on the gap between these two eigenvalues. It is given explicitly in terms of the geometric properties of the domain and the coefficients of the perturbed operator. We apply this estimate to several asymptotic regimes studying its dependence on various parameters. We discuss specific examples of operators to which the bounds can be applied.

math.SP↗
arXiv2012

Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

We consider a magnetic Schroedinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum.

math.AP↗
arXiv2011

Planar waveguide with "twisted" boundary conditions: discrete spectrum

We consider a planar waveguide with combined Dirichlet and Neumann conditions imposed in a "twisted" way. We study the discrete spectrum and describe it dependence on the configuration of the boundary conditions. In particular, we show that in certain cases the model can have discrete eigenvalues emerging from the threshold of the essential spectrum. We give a criterium for their existence and construct them as convergent holomorphic series.

math.SP↗
arXiv2011

Asymptotics for the expected lifetime of Brownian motion on thin domains in R^n

We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in R^n, and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we also see that the symmetry of the perturbation plays a role in the expansion. As in the case of eigenvalues, these expansions may also be used to approximate the exit time for domains where the scaling parameter is not necessarilly close to zero.

math.AP↗
arXiv2011

The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions

The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schroedinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schroedinger operators.

math.SP↗
arXiv2010

Low lying spectrum of weak-disorder quantum waveguides

We study the low-lying spectrum of the Dirichlet Laplace operator on a randomly wiggled strip. More precisely, our results are formulated in terms of the eigenvalues of finite segment approximations of the infinite waveguide. Under appropriate weak-disorder assumptions we obtain deterministic and probabilistic bounds on the position of the lowest eigenvalue. A Combes-Thomas argument allows us to obtain so-called 'initial length scale decay estimates' at they are used in the proof of spectral localization using the multiscale analysis.

math.SP↗
arXiv2010

On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet-Bloch decomposition, the two-terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.

math.SP↗
arXiv2009

Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in R^d

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues for a large class of domains, under very mild assumptions. As an application, we derive a three--term asymptotic expansion for the first eigenvalue of d-dimensional ellipsoids.

math.AP↗
arXiv2008

PT-symmetric waveguides

We introduce a planar waveguide of constant width with non-Hermitian PT-symmetric Robin boundary conditions. We study the spectrum of this system in the regime when the boundary coupling function is a compactly supported perturbation of a homogeneous coupling. We prove that the essential spectrum is positive and independent of such perturbation, and that the residual spectrum is empty. Assuming that the perturbation is small in the supremum norm, we show that it gives rise to real weakly-coupled eigenvalues converging to the threshold of the essential spectrum. We derive sufficient conditions for these eigenvalues to exist or to be absent. Moreover, we construct the leading terms of the asymptotic expansions of these eigenvalues and the associated eigenfunctions.

math-ph↗
arXiv2007

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.

math.SP↗
arXiv2007

Asymptotics for the solutions of elliptic systems with fast oscillating coefficients

We consider a singularly perturbed second order elliptic system in the whole space. The coefficients of the systems fast oscillate and depend both of slow and fast variables. We obtain the homogenized operator and in the uniform norm sense we construct the leading terms of the asymptotics expansion for the resolvent of the operator described by the system. The convergence of the spectrum is established. The convergence of the spectrum is established. The examples are given.

math-ph↗
arXiv2007

On the spectrum of two quantum layers coupled by a window

We consider the Dirichlet Laplacian in a domain two three-dimensional parallel layers having common boundary and coupled by a window. The window produces the bound states below the essential spectrum; we obtain two-sided estimates for them. It is also shown that the eigenvalues emerge from the threshold of essential spectrum as the window passes through certain critical shapes. We prove the necessary condition for the window to be of critical shape. Under an additional assumption we show that this condition is sufficient and obtain the asymptotic expansion for the emerging eigenvalue as well as for the associated eigenfunction.

math-ph↗
arXiv2004

Spectrum of the Magnetic Schrodinger Operator in a Waveguide with Combined Boundary Conditions

We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann boundary condition (see Section 2, Eq. (2.2) for the definition of this boundary condition}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.

math-ph↗