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We study the effect of non-negative potentials on the spectral gap of one-dimensional Schrödinger operators in the limit of large intervals. We derive upper bounds on the gap for different classes of potentials and show, as a main result, that the spectral gap of a Schrödinger operator with a non-zero and sufficiently fast decaying potential closes strictly faster than the gap of the free Laplacian. We show optimality of this result in some sense and establish a conjecture towards the actual decay rate of the spectral gap.

We classify a class of modules for the Yangian (Y(mathfrak {gl}_2)), where the element (e_{21}in Y(mathfrak {gl}_2)) acts regularly. These modules can be realized by use of the differential operators. Moreover, we compute the central characters of these modules by employing the trick of formal generating series.

Kimoto and Wakayama [Ann. Inst. Henri Poincaré D 10 (2023), 205–275] studied the special values of the spectral zeta function (zeta _Q(s)) associated to the non-commutative harmonic oscillator (Q_{alpha ,beta }). Two kinds of Apéry-like numbers (even case (widetilde{J}_{2s+2}(n)) and odd case (widetilde{J}_{2s+1}(n))) naturally arise in the expressions for the special values of (zeta _Q(s)) at integer points. Supercongruences among these Apéry-like numbers lead one to the modularity of the generating functions of the Apéry-like numbers. Kimoto and Wakayama established a supercongruence among (widetilde{J}_{2s+2}(n)), and conjectured the same type of supercongruence for (widetilde{J}_{2s+1}(n)) as in the even case (widetilde{J}_{2s+2}(n)). In this work, we confirm Kimoto and Wakayama’s supercongruence conjecture in the odd case of Apéry-like numbers (widetilde{J}_{2s+1}(n)).

For a finite dimensional algebra (Lambda ), the problem of whether the unbounded derived category (mathbb {D}(Lambda )) is equal to its localizing subcategory generated by injective (Lambda )-modules was firstly considered by Keller in 2001. If this happens to be true, it is usually said that injectives generate for (Lambda ). Some connections to famous homological conjectures were illuminated by Keller himself. Recently, Rickard presented several classes of rings, including particular types of finite dimensional algebras as well as commutative Noetherian rings, for which injectives generate. He also proved that if injectives generate for (Lambda ), then it satisfies the big finitistic dimension conjecture. The main objective of this paper is to discuss when the reverse statement also holds. We show that, under some mild condition, the injective generation phenomenon and the big finitistic dimension conjecture for (Lambda ) are equivalent.

Suppose that E and F are Banach lattices. It is known that there are several norms on the Fremlin tensor product (E{overline{otimes }} F) that turn it into a normed lattice; in particular, the projective norm (|pi |) (known as the Fremlin projective norm) and the injective norm (|epsilon |) (known as the Wittstock injective norm). Now, assume that E and F are locally convex-solid vector lattices. Although we have a suitable vector lattice structure for the tensor product E and F (known as the Fremlin tensor product and denoted by (E{overline{otimes }}F)), there is a lack of topological structure on (E{overline{otimes }}F), in general. In this note, we consider a linear topology on (E{overline{otimes }}F) that makes it into a locally convex-solid vector lattice, as well; this approach can be taken as a generalization of the projective norm of the Fremlin tensor product between Banach lattices.

Dimitrov and Fioresi introduced an object that they call a generalized root system. This is a finite set of vectors in a Euclidean space satisfying certain compatibilities between angles and sums and differences of elements. They conjecture that every generalized root system is equivalent to one associated to a restriction of a Weyl arrangement. In this note, we prove the conjecture and provide a complete classification of generalized root systems up to equivalence.

It is proved that every Bézout domain admits an embedding into an elementary divisor domain with the same divisibility group. So the prime spectrum of a Bézout domain is homeomorphic to the prime spectrum of an elementary divisor domain. Together with an earlier result, it follows that a topological space is homeomorphic to the maximal spectrum of an elementary divisor domain if and only if it is a serial quasi-compact (T_1)-space.