Number of Bound States of Schrödinger Operators with Matrix-Valued Potentials

Abstract

Abstract. We give a CLR type bound on the number of bound states of Schrö-dinger operators with matrix-valued potentials using the functional integral method of Lieb. This significantly improves the constant in this inequality obtained earlier by Hundertmark.

Cwikel's theorem and the CLR inequality

“…In this paper we shall derive the CLR inequality (1.1) from (1.2) and we shall extend (1.2) to L 2 (R d ) ⊗ G with constants independent of the dimension of the auxiliary Hilbert space G. Both results are new and go beyond [Ru1,Ru2]. Our results in the operator-valued case improve upon previous results of [Hu1] (who follows [Cw] and has larger constants) and [FrLiSe1] (who can only deal with (−∆) s for 0 < s ≤ 1). Moreover, we show that a modification of Rumin's proof of (1.2) yields an easy proof of Cwikel's theorem mentioned at the beginning.…”

“…(3.5) gives 0.228 for s = 1 (to be compared with 0.174 from [FrLiSe1]). We emphasize again that the methods of [Li1,Da,FrLiSe1] are restricted to s ≤ 1. The above constants are the best ones available for 1 < s < d/2; see the comparison with the constant from [Cw,Hu1] after Theorem 3.1.…”

“…In the case of Cwikel's theorem this strategy was implemented in [Hu1]. The constant in the CLR inequality for Schrödinger operators with matrix-valued potentials was improved in [FrLiSe1]. In this subsection we show that Rumin's proof can also be modified to yield an operator-valued version of Cwikel's theorem.…”

Spectral inequalities for Schrödinger operators with surface potentials

“…Indeed, according to [HLW] and [DLL] the operator-valued inequality (2.16) holds with an additional factor of π/ √ 3 on the right hand side if γ ≥ 1, with an additional factor of 2 if 1/2 ≤ γ < 1 and d = 1 and with an additional factor of 2π/ √ 3 if γ ≥ 1/2 and d ≥ 2. Similarly, one can use the operator-valued inequality from [FLS2] (see also [H1]) to prove (2.4) for γ ≥ 0 and d ≥ 3 and to obtain (2.12). Extending the original proof of Lieb and Thirring [LT] to the operator-valued case would yield (2.4) for γ > 0 and d = 2.…”

High energy bounds on wave operators

“…Under certain conditions our results transfer to this situation (Corollary 4.14). In the case where H A is the negative Laplacian, in particular in applications to quantum physics, these H 0 are known as matrix-valued Schrödinger operators or Schrödinger operators with matrix-valued potentials (see, e.g., [GKM02,FLS07,KR08,CJLS16]), although we allow the matrices to become infinite-dimensional (Example 4.15). A particular problem occurs here for low-dimensional Laplacians if H B is of infinite rank; we discuss this in Example 4.16.…”