David Krejčiřík

Bound States in Curved Quantum Layers

Abstract: We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in ℝ3. We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hard-wall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of geometrically induced bound states.

Bound States in Weakly Deformed Strips and Layers

Abstract. We consider Dirichlet Laplacians on straight strips in

\(\mathcal {PT}\) -Symmetric Waveguides

{\Bbb R}^2

Topologically nontrivial quantum layers

or layers in

Pseudospectra in non-Hermitian quantum mechanics

{\Bbb R}^3

Calculation of the metric in the Hilbert space of a {\cal P}{\cal T} -symmetric model via the spectral theorem

with a weak local deformation. First we generalize a result of Bulla et al. to the three-dimensional situation showing that weakly coupled bound states exist if the volume change induced by the deformation is positive;we also derive the leading order of the weak-coupling asymptotics. With the knowledge of the eigenvalue analytic properties, we demonstrate then an alternative method which makes it possible to evaluate the next term in the asymptotic expansion for both the strips and layers. It gives,in particular, a criterion for the bound-state existence in the critical case when the added volume is zero.

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

Abstract.We introduce a planar waveguide of constant width with non-Hermitian

A sharp upper bound for the first Dirichlet eigenvalue and the growth of the isoperimetric constant of convex domains

\mathcal {PT}