We develop an abstract method to identify spectral points of definite type in the spectrum of the operator $T_1\otimes I_2 + I_1\otimes T_2$. The method is applicable in particular for non-self-adjoint waveguide type operators with symmetries. Using the remarkable properties of the spectral points of definite type, we obtain new results on realness of weakly coupled bound states and of low lying essential spectrum in the $\mathcal{P}\mathcal{T}$-symmetric waveguide. Moreover, we show that the pseudospectrum has a normal tame behavior near the low lying essential spectrum and exclude the accumulation of non-real eigenvalues to this part of the essential spectrum. The advantage of our approach is particularly visible when the resolvent of the unperturbed operator cannot be explicitly expressed and most of the mentioned spectral conclusions are extremely hard to prove using direct methods.
Spatial cruciform quantum waveguides (the Dirichlet problem for Laplace operator) are constructed such that the total multiplicity of the discrete spectrum exceeds any preassigned number.
We study the essential spectra of formally self-adjoint elliptic systems on doubly periodic planar domains perturbed by a semi-infinite periodic row of foreign inclusions. We show that the essential spectrum of the problem consists of the essential spectrum of the purely periodic problem and another component, which is the union of the discrete spectra of model problems in the infinite perturbation strip; these model problems arise by an application of the partial Floquet-Bloch-Gelfand transform.
We consider a soft quantum waveguide described by a two-dimensional Schrödinger operators with an attractive potential in the form of a channel of a fixed profile built along an infinite smooth curve which is not straight but it is asymptotically straight in a suitable sense. Using Birman-Schwinger principle we show that the discrete spectrum of such an operator is nonempty if the potential well defining the channel profile is deep and narrow enough. Some related problems are also mentioned.
We investigate properties of a particle confined to a hard-wall spiral-shaped region. As a case study we analyze in detail the Archimedean spiral for which the spectrum above the continuum threshold is absolutely continuous away from the thresholds. The subtle difference between the radial and perpendicular width implies, however, that in contrast to `less curved' waveguides, the discrete spectrum is empty in this case. We also discuss modifications such a multi-arm Archimedean spirals and spiral waveguides with a central cavity; in the latter case bound state already exist if the cavity excee…
We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schrödinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolutely continuous non-negative spectrum.
We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $κ$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schrödinger operator on $L^2(\mathbb R)$ \[ \mathcal{L}_e := -\frac{d^2}{ds^2} - \frac{κ^2}{π^2} \] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $\varepsilon$ from the essential spectrum, where $2\varepsilon$ is the width of the waveguide. This is in contr…
Let $Ω\subset \mathbb R^3$ be a broken sheared waveguide, i.e., it is built by translating a cross-section in a constant direction along a broken line in $\mathbb R^3$. We prove that the discrete spectrum of the Dirichlet Laplacian operator in $Ω$ is non-empty and finite. Furthermore, we show a particular geometry for $Ω$ which implies that the total multiplicity of the discrete spectrum is equals 1.
We consider a family $\{Ω^\varepsilon\}_{\varepsilon>0}$ of periodic domains in $\mathbb{R}^2$ with waveguide geometry and analyse spectral properties of the Neumann Laplacian $-Δ_{Ω^\varepsilon}$ on $Ω^\varepsilon$. The waveguide $Ω^\varepsilon$ is a union of a thin straight strip of the width $\varepsilon$ and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period $\varepsilon$, along the strip upper boundary. For $\varepsilon\to 0$ we prove a (kind of) resolvent convergence of $-Δ_{Ω^\varepsilon}$ to a certa…
We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order $ h >0$. The essential spectrum of the problem is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as $h \to 0$.
We examine the band-gap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter $\varepsilon>0$. The periodicity cell itself contains a string of holes at a distance $O(\varepsilon)$ between them. Under assumptions on the symmetry of the holes, we derive and justify asymptotic formulas for the endpoints of the spectral bands in the low-frequency range of the spectrum as $\varepsilon \to 0$. We demonstrate that, for $\varepsilon$ small enough, some spectral gaps are open. The position…
We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.
