For an arbitrary nonempty, open set ^<i>n</i>, <i>n</i> of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian (-\Delta)^m|_{{C^\infty_0}(\Omega)}, <i>m</i> , and its Kreinvon Neumann extension <i>A</i>_{<i>K</i>,,<i>m</i>} in <i>L</i>²(). with <i>N</i>(, <i>A</i>_{<i>K</i>,,<i>m</i>)}, (-\Delta)^m|_{{C^\infty_0}(\Omega)}, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of <i>A</i>_{<i>K</i>,,<i>m</i>}, we derive the bound <i>N</i>(, <i>A</i>_{<i>K</i>,,<i>m</i>}) (2<i></i>)^-<i>n</i><i></i>_n||{1 + [2<i>m</i>/(2<i>m</i> + <i>n</i>)]}^{<i>n</i>/(2<i>m</i>)}^{<i>n</i>/(2<i>m</i>)}, &gt; 0, where <i>_n</i> <i></i>^<i>n</i>/2 /((<i>n</i> + 2)/2) denotes the (Euclidean) volume of the unit ball in ^<i>n</i>.

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrdinger operator with an attractive -interaction of a fixed strength, the support of which is a star graph with finitely many edges of an equal length L(0,]. Under the constraint of fixed number of the edges and fixed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle.

An irreducible II_1-subfactor A⊂B is exactly 1-supertransitive if B⊖A is reducible as an A − A bimodule. We classify exactly 1-supertransitive subfactors with index at most 6+1/5, leaving aside the composite subfactors at index exactly 6 where there are severe difficulties. Previously, such subfactors were only known up to index 3+\sqrt{5} ≈ 5.23. Our work is a significant extension, and also shows that index 6 is not an insurmountable barrier.There are exactly three such subfactors with index in (3+\sqrt{5},6+1/5], all with index 3+2\sqrt{2}. One of these comes from SO(3)_q at a root of unity, while the other two appear to be closely related, and are ‘braided up to a sign’.

Let <i>X, Y</i> be compact Hausdorff spaces and <i>E, F</i> be Banach spaces. A linear map <i>T: C(X, E) C(Y, F)</i> is separating if <i>Tf, Tg</i> have disjoint cozeroes whenever <i>f, g</i> have disjoint cozeroes. We prove that a biseparating linear bijection <i>T</i> (that is, <i>T</i> and <i>T</i>^-1 are separating) is a weighted composition operator <i>Tf = h f</i> o . Here, <i>h</i> is a function from <i>Y</i> into the set of invertible linear operators from <i>E</i> onto <i>F</i>, and , is a homeomorphism from <i>Y</i> onto <i>X</i>. We also show that <i>T</i> is bounded if and only if <i>h(y)</i> is a bounded operator from <i>E</i> onto <i>F</i> for all <i>y</i> in <i>Y</i>. In this case, <i>h</i> is continuous with respect to the strong operator topology.

We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically in the subcritical case. Furthermore, we show that the model then has a rich resonance structure.