Abstract
There are several proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a substantial refinement of Cwikel’s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies.
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1 Introduction
We want to find natural bounds, with the right semi-classical behavior, for the number of negative eigenvalues of Schrödinger operators \(P^2+V\), with momentum operator \(P=-i\nabla \), or more general operators like polyharmonic Schrödinger operators \(|P|^{2\alpha }+V\), including the ultra-relativistic operator \(|P|+V\). We also consider operator-valued potentials V.
For the one-particle Schrödinger operator \(P^2+V\) with a real-valued potential V, this type of bound goes back to Cwikel, Lieb, and Rozenblum [9, 31, 32, 43, 44], with very different proofs. They prove
for the number of negative eigenvalues of a Schrödinger operator, where \(L_{0,d}\) is a constant depending only on the dimension. This bound is a semi-classical bound since a simple scaling argument shows that the classical phase-space volume of the region of negative energy is given by
where \(|B_1^d|\) is the volume of the unit ball in \({\mathbb {R}}^d\).
Rozenblum’s paper [43] was an announcement of his result and, typically for the journal, did not contain any proofs. The version with full proofs was published in [44]. Similarly, Lieb’s paper [31] is an announcement of his result and the details of his proof were published later in [32, 50]. The approach of Rozenblum was strongly motivated by the St. Petersburg school of mathematical physics around Birman and Solomyak, whose work had been virtually unnoticed in the west until the mid 1970s, see the “Added notes” on page 378 in [49]. The proofs of Cwikel and Lieb were strongly motivated by Simon [49]. Cwikel’s approach was developed into a more general scheme by Birman and Solomyak, see e.g. [2, 5]. They were able to obtain more general versions of Cwikel’s result in which the \(L^p\) and weak-\(L^p\) spaces appearing in [9] could be replaced by more general spaces. For the most recent developments in this direction, see [28], which builds upon earlier work by Weidl [53, 54].
The intuition behind semi-classical bounds is that the uncertainty principle forces a quantum particle to occupy roughly a classical phase-space volume \((2\pi )^d\). Thus the phase-space volume \({N^\text {cl}(\eta ^2+V)}\) where the classical Hamiltonian energy \(H(\eta ,x) = \eta ^2+V(x)\) is negative, should control \(N(P^2+V)\). The CLR bound (1.1) shows that this is the case up to a factorFootnote 1\(C_{0,d}=L_{0,d} (2\pi )^d/|B_1^d|\). Simon’s profound insights connecting bounds on \(N(P^2+V)\) with known and conjectured interpolation properties of weak operator ideals,Footnote 2 and, in particular, his Conjecture 1 on page 372 in [49], were a major motivation for Cwikel’s work. The discussion in [49] suggested that perhaps some new and more powerful interpolation theorem might yield the weak trace ideal bounds of Conjecture 1 of [49], which would suffice to prove the CLR inequality. As he informed us [10], Cwikel initially tried to see if one of the bilinear interpolation theorems in fundamental papers of Calderón [6, p. 118] and Lions–Peetre [35, p. 14] about interpolation spaces, or some variant of them, might prove Simon’s Conjecture 1. Indeed Proposition 4.2 of [49] can also be obtained from [6, p. 118].
Unfortunately, as shown on page 97 in [9], a proof of Simon’s Conjecture 1 cannot be obtained by any kind of bilinear interpolation. However, as Cwikel strongly emphasized to us [10], some elements of his proof evolved and benefitted greatly from ideas around Lions and Peetre’s Théorème 4.1 of [35, p. 14].
One of our main new contributions is that the CLR bound is intimately related to the fact that certain maximal Fourier multipliers are bounded on \(L^2({\mathbb {R}}^d)\). This leads to a new class of variational problems, see Theorem 1.3, which allows us to improve Lieb’s constants in dimensions \(d\ge 5\). The original bounds on \(C_{0,d}\) in [9] and [31] were explicitly dimension dependent with a considerable growth in the dimension d. The bound due to Lieb grows like \(C_{0,d}= \sqrt{\pi d}(1+O(d^{-1}))\). See [50] or [42, Chapter 3.4] for an excellent discussion of Lieb’s method and Remark 1.2 below for some explicit numbers. However, it is expected that semi-classical arguments work better in high dimensions. In particular, the constant \(C_{0,d}\) should not grow in d. The first dimension independent bound \(C_{0,d}\le 81\) was derived by extending Cwikel’s method to operator-valued potentials in 2002 in [21]. This work extended an induction in the dimension argumentFootnote 3 by Laptev and Weidl [27], who were the first to derive Lieb–Thirring bounds with the sharp classical Lieb–Thirring constant in all dimensions in some cases. Although the upper bound from [21] is dimension independent, it is certainly too large for small dimensions.
For the last 40-plus years it has been believed that any approach based on Cwikel’s method cannot yield any bounds on \(C_{0,d}\) which are comparable to the ones obtained by Lieb in low dimensions. This is wrong, as we will show by drastically simplifying and, at the same time, generalizing the important ideas of Cwikel. A typical result which can be easily achieved with our method is
Theorem 1.1
The number \(N(P^2+V)\) of negative energy bound states of \(P^2+V\) obeys the semiclassical bound
for all \(d\ge 3\), where \(B^d\) is the unit ball in \({\mathbb {R}}^d\), \(|B^d|\) its volume, and the constant \(C_{0,d}\) given in Table 1 below.
Moreover, the same bounds with the same constants also hold in the operator-valued case, see Theorem 1.8.
Remark 1.2
-
(i)
Table 1 below compares the upper bounds on \(C_{0,d}\), obtained with our method, with the best known ones so far for scalar and operator-valued potentials. All bounds on \(C_{0,d}\) in the third column of the table were obtained already in the original work of Lieb more than 40 years ago.Footnote 4 Our bounds on \(C_{0,d}\) also hold in the operator-valued case, see Sect. 6 below. The value in the last column is due to Frank, Lieb and Seiringer [19] and holds for all \(d\ge 3\). Our result also gives the bound \(C_{0,d}\le 5.62080 \) for \(d\ge 9\), see the discussion in Appendix A. For dimensions \(3\le d \le 9\) our upper bounds are compared with the values of the lower bound (1.10) achievable by our method in Table 2 below.
-
(ii)
There have been several previous attempts to improve Lieb’s result, for example, due to Conlon [8], Li and Yau [30], Frank [16], and Weidl [53, 54]. All these very different proofs shed a new light on the Cwikel–Lieb–Rozenblum bound, but failed to give better bounds on the involved constants than already achieved by Lieb.
From the point of view of physics, the other important case is the ultra-relativistic Schrödinger operator \(|P|+V\). For more general so-called polyharmonic Schrödinger operators our method yields the following bound for scalar potentials, which involves an interesting variational problem.
Theorem 1.3
Let \(P=-i\nabla \) be the momentum operator, \(V=V_+-V_-\) be a real-valued potential with positive part \(V_+\in L^1_{{\text {loc}}}\) and negative part \(V_-\in L^{d/\alpha }({\mathbb {R}}^d)\) with \(0<\alpha <d/2\), and \(P^{2\alpha }+V\) the Schrödinger–type operator defined via quadratic form methods on \(L^2({\mathbb {R}}^d)\).
Furthermore, consider the minimization problem
where \(\gamma >2\), the infimum is taken over all \(m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})\), and \(m= m_1*m_2\) denotes the convolution of \(m_1, m_2\) on \({\mathbb {R}}_+\) with measure \(\frac{\mathrm {d}s}{s}\) and let
Then the number \(N(P^{2\alpha }+V)\) of negative energy bound states of \(P^{2\alpha }+V\) is bounded by
with constant \(C_{d/\alpha }\) given by (1.5) for \(\gamma =\frac{d}{\alpha }\).
For \(\alpha =1/2\) and in three dimensions we get the upper bound
which improves the result of Daubechies [12], who gets \(N(|P|+V) \le 6.08 \int _{{\mathbb {R}}^3} V_-(x)^{3}\, \mathrm {d}x\).
A similar result, with the same constants, also holds for operator-valued potentials, see Theorem 1.7.
Remark 1.4
The minimisation problem for \(M_{\gamma }\) in (1.4) is crucial for getting good bounds on the constant in the Cwikel–Lieb–Rozenblum bound. It allows us to obtain the first improvement, in more than 40 years, on the constants derived originally by Lieb [31] in dimensions \(d\ge 5\).
A simple, but not optimal, choice for \(m_{1}\), \(m_2\) is \(m_1(s)= s\mathbf {1}_{\{0<s\le 1\}}\) and \(m_2(s)= 2s^{-1}\mathbf {1}_{\{s>1\}}\), in which case \(\Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}=1\) and \(m(t)= m_1*m_2(t)= \min (t,t^{-1})\), so
This gives
as a possible constant in the CLR inequality and yields \(C_{0,3}\le 10.8\), already an order of a magnitude smaller than Cwikel’s bound. To get the uniform bound claimed in Theorem 1.1 we have to choose better candidates for \(m_1\) and \(m_2\). We can achieve this in small dimensions, see Appendix D. Moreover, combining this with ‘stripping-off-dimensions’ ideas, see Appendix A, with the help of similar bounds for operator-valued potentials presented in Sect. 6, one can get this bound also uniformly in the dimension for the important special case of non-relativistic Schrödinger operators, where \(\alpha =1\).
Choosing \(m_1(s) = s \mathbf {1}_{\{0<s<1\}}\), we can actually solve the minimization problem for \(m_2\), see Proositions C.1 and C.4 in Appendix C. This leads to the upper bound in
Proposition 1.5
For all \(\gamma >2\)
For the proof of the lower bound see Sect. 5.
Remark 1.6
-
(i)
So far the best known bound for polyharmonic Schrödinger operators is due to Frank [16], who proved
$$\begin{aligned} N(P^{2\alpha }+V) \le \left( \frac{d(d +2\alpha )}{(d-2\alpha )^2} \right) ^{(d-2\alpha )/(2\alpha )} \frac{d}{d -2\alpha } \frac{|B_1^d|}{(2\pi )^d} \int _{{\mathbb {R}}^d} V_-(x)^{\frac{d}{2\alpha }}\, \mathrm {d}x, \end{aligned}$$(1.9)based on ideas of Rumin [46, 47]. Even the simple upper bound on \(M_\gamma \) from Remark 1.4 yields better results than (1.9). Computing the ratio of the constants in Frank’s bound and the one from (1.6), using the upper bound in (1.8), one sees that our bound from Theorem 1.3 is better in the whole allowed range of \(0<\alpha <d/2\).
-
(ii)
For the constant \(C_\gamma \) in (1.5), the lower bound from (1.8) yields
$$\begin{aligned} C_\gamma \ge \frac{\gamma ^{\gamma }}{2(\gamma -1)\left( \gamma -2 \right) ^{\gamma -1}} {=:}C^\text {lower}_\gamma , \end{aligned}$$where \(C^\text {lower}_\gamma \) is a probably non-sharp lower bound for the best possible constant achievable by our method.Footnote 5 Thus the upper bound on \(M_\gamma \) from Remark 1.4 gives
$$\begin{aligned} \frac{C_\gamma }{C^\text {lower}_\gamma }\le 4\frac{\gamma -1}{\gamma +2}<4, \end{aligned}$$where \(\gamma =d/\alpha > 2\). This shows that our easy upper bound is less than a factor of 4 off the lower bound.Footnote 6
-
(iii)
The above lower bound also gives the lower bound
$$\begin{aligned} C^\text {lower}_{0,d} = C^\text {lower}_d = \frac{d^{d}}{2(d-1)\left( d-2 \right) ^{d-1}} \end{aligned}$$(1.10)achievable by our method for the constant in Theorem 1.1. In dimensions \(3\le d \le 9\) our results are summarized in Table 2.
In addition,
$$\begin{aligned} C^\text {lower}_{0,d} = \frac{d^2}{2(d-1)(d-2)}\left( 1+\frac{2}{d-2} \right) ^{d-2} \rightarrow \frac{e^2}{2}\ge 3.69452. \end{aligned}$$This comparison shows that there is not too much room to improve on the upper bounds we obtained, even if one finds the sharp value in the minimization problem for \(M_\gamma \) in (1.4).
-
(iv)
It is known that if \(\alpha \ge d/2\), the operator \(P^{2\alpha }-U\) always has bound states for nontrivial \(U\ge 0\), so a quantitative bound of the form \(N(P^{2\alpha }-U)\lesssim \int _{{\mathbb {R}}^d}U(x)^{d/\alpha }\) cannot hold if \(\alpha \ge d\). For \(\alpha =1\) see [48] or [24, Problem 2 in §45]. For more general cases, see [26, 37, 39], and [20] for a simple proof of how the existence/ non-existence of a CLR type bound for operators of the form \(T(P)+V\) for a large class of functions \(T:{\mathbb {R}}^d\rightarrow [0,\infty )\) is related to the behavior of the symbol T close to its zero set.
As mentioned before, our method can be generalized to operator-valued potentials. To formulate this, we need some additional notation. An operator-valued potential V is a map \(V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})\) with \(V(x):\mathcal {G}\rightarrow \mathcal {G}\) a bounded self-adjoint operator on an auxiliary Hilbert spaceFootnote 7\(\mathcal {G}\) for almost all \(x\in {\mathbb {R}}^d\). We denote by \(\mathcal {B}(\mathcal {G})\) the set of bounded operators on \(\mathcal {G}\) and by \({\mathcal {S}}_{p}(\mathcal {G})\) the von Neumann–Schatten ideal of compact operators on \(\mathcal {G}\) with p-summable singular values, see for example [51] for a background on von Neumann–Schatten ideals.
Theorem 1.7
(Operator-valued version of Theorem 1.3) Let \(\mathcal {G}\) be a Hilbert space and \(V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})\) an operator valued potential with positive part \(V_+\in L^1_{\text {loc}}({\mathbb {R}}^d,\mathcal {B}(\mathcal {G}))\) and negative part \(V_-\in L^{d/(2\alpha )}({\mathbb {R}}^d, {\mathcal {S}}_{d/(2\alpha )}(\mathcal {G}))\). Then the number of negative energy bound states of \(P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}+V\) is bounded by
with the same constant \(C_{d/\alpha }\) as in Theorem 1.3.
For the physically most interesting case \(\alpha =1\) this enables us to get considerable improvements on the constants in the Cwikel–Lieb–Rozenblum bound.
Theorem 1.8
(Operator-valued version of Theorem 1.1) Let \(\mathcal {G}\) be a Hilbert space and \(V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})\) an operator valued potential with positive part \(V_+\in L^1_{\text {loc}}({\mathbb {R}}^d,\mathcal {B}(\mathcal {G}))\) and negative part \(V_-\in L^{d/2}({\mathbb {R}}^d, {\mathcal {S}}_{d/2}(\mathcal {G}))\). Then the number of negative energy bound states of \(P^{2}\otimes \mathbf {1}_{\mathcal {G}}+V\) is bounded by
with
where \(C_n\) is given by (1.5) for \(\gamma =n\).
Remark 1.9
Table 1 lists upper bounds on \(C_{0,d}^{\mathrm {op}}\) for dimensions \(3\le d \le 9\), see also Appendix D. The constant for \(d=9\) is also an upper bound on \(C_{0,d}^{\mathrm {op}}\) in any dimension \(d\ge 10\) by (1.13).
The structure of the paper is as follows. In Sect. 2 we present the main ideas of our method in the case of a standard non-relativistic Schrödinger operator. The extension to more general kinetic energies is done in Sect. 3.
In Sect. 4 we explain the surprising connection of semiclassical bounds and maximal Fourier multiplier estimates, which is probably the most important new part of our method.
Although we cannot explicitly find minimizers of the variational problem from Theorem 1.3, there is a natural lower bound, which is discussed in Sect. 5. The numerical study to find reasonable upper bounds for this variational problem is presented in Appendix D.
The extension to the operator-valued setting is done in Sects. 6 and 7. In particular, in Sect. 7 we prove a fully operator-valued version of Cwikel’s original weak trace ideal bound.
2 The splitting trick
Let \(U{:=}V_-\ge 0\). As quadratic forms \(P^2+V\ge P^2-U\). This and the Birman–Schwinger principle shows
where \(n(A;\kappa )\) is the number of singular values \((s_j(A))_{j\in {\mathbb {N}}}\) greater than \(\kappa >0\) of a compact operator A.
We denote by \(\mathcal {F}\) the Fourier transform and by \(\mathcal {F}^{-1}\) its inverse, by \(M_h\) the operator of multiplication with a function h, and \(A=A_{f,g} = M_f\mathcal {F}^{-1}M_g\) for f, g non-negative (measurable) functions on \({\mathbb {R}}^d\). When \(f(x)=U(x)^{1/2}\) and \(g(\eta )= |\eta |^{-1}\), then \(A A^* = U^{1/2}|P|^{-2}U^{1/2}\), which has the same non-zero eigenvalues as \(A^*A\). Thus
In particular, the Chebyshev–Markov inequality gives
for any \(0<\mu <1\). The first main idea, going already back to Cwikel [9], is to split \(A_{f,g}= B_{f,g}+ H_{f,g}\), where \(B_{f,g}\) is bounded and \(H_{f,g}\) is a Hilbert–Schmidt operator, and note that Ky Fan’s inequality for the singular values [51, Theorem 1.7] yields
for all \(j\in {\mathbb {N}}\). So if \(\Vert B_{f,g}\Vert \le \mu <1\) we get
where \(\Vert H\Vert _{HS}\) denotes the Hilbert–Schmidt norm of the operator H.
In order to make the above argument work, one has to be able to split \(A_{f,g}= B_{f,g}+H_{f,g}\) in such a way that the Hilbert–Schmidt norm of \(H_{f,g}\) is easy to calculate and one has a good bound on the operator norm of \(B_{f,g}\). Writing out the inverse Fourier transform, one sees that \(A_{f,g}\) has a kernel
that is,
at least for nice enough \(\varphi \). In order to write \(A_{f,g}\) as a sum of a bounded and a Hilbert–Schmidt operator, set \(t=f(x) g(\eta )\), split \(t= m(t) +t-m(t)\) for some bounded, measurable function \(m:[0,\infty )\rightarrow {\mathbb {R}}\), and define \(B_{f,g,m}\) and \(H_{f,g,m}\) via their kernels
It is then clear that \(A_{f,g}= B_{f,g,m}+H_{f,g,m}\). Our starting point is that the Hilbert–Schmidt norm of \(H_{f,g,m}\) is straightforward to calculate; the main difficulty is to get an explicit bound on the operator norm of \(B_{f,g,m}\) on \(L^2\) under suitable assumptions on m. For the special choice \(g(\eta ) = |\eta |^{-1}\) one has \(\Vert H_{f,g}\Vert _{HS}^2= c \int _{R^d}f(x)^d\, \mathrm {d}x\), see (2.9), so the right hand side of (2.1) has exactly the right (semi-classical) scaling in f. But, in order to use this in (2.1), it also enforces that the upper bound \(\mu \) on the operator norm of \(B_{f,g}\) has to be independent of f. This has an important consequence:
Since for a given \(\varphi \in L^2\) one can freely choose \(f\ge 0\) as to make \(|B_{f,g,m}\varphi |\) as big as possible, this leads naturally to the associated maximal operator \(\mathcal {B}_{g,m}(\varphi ){:=}\sup _{f\ge 0}|B_{f,g,m}\varphi |\). Although this is not explicitly written in the paper by Cwikel, getting a useful bound on such a type of maximal operator is exactly what he achieved in [9], using a dyadic decomposition in the ranges of f and g and collecting suitable terms. We will do this in a much simpler and more efficient way. This enables us to get a constant which is more than 10 times smaller than the original constant by Cwikel.
It turns out that one can always calculate the Hilbert–Schmidt norm of \(H_{f,g,m}\). The maximal operator \(\mathcal {B}_{g,m}\) corresponding to \(B_{f,g,m}\) can be bounded in operator norm under an additional structural assumption on m, which we present first.
Theorem 2.1
Let g be a measurable non-negative function on \({\mathbb {R}}^d\) for \(d\ge 1\) and assume that m is given by a convolution,
with \(m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})\). Then the maximal operator given by \(\mathcal {B}_{g,m}(\varphi ) {:=} \sup _{f\ge 0}| B_{f,g,m}\varphi | \) extends to a bounded operator on \(L^2({\mathbb {R}}^d)\) with
for its operator norm.
We emphasize that this maximal operator bound provides an upper bound for the operator norm of \(B_{f,g,m}\) independently of the choice of f, as it has to be. It also turns out to be independent of g. The maximal operator bound is a natural consequence of the convolution structure of m, see Sect. 4, where we show that it is equivalent to maximal Fourier multiplier bounds. Concerning the Hilbert–Schmidt norm of \(H_{f,g,m}\) we have
Theorem 2.2
Let f, g be non-negative measurable functions on \({\mathbb {R}}^d\), \(d\ge 1\), and m be a measurable function on \({\mathbb {R}}_+\). The Hilbert–Schmidt norm of \(H_{f,g,m}\) is given by
where the function \(G_{g,m}\) is given by
Remark 2.3
In its applications to nonrelativistic Schrödinger operators \(P^2 + V\), the function g is given by \(g(\eta )=|\eta |^{-1}\). We would like to emphasize that g is never in \(L^2({\mathbb {R}}^d)\), due to its slow decay at infinity, which is an ultraviolet problem. Choosing m with \(m(t)\sim t\) for small \(t>0\) makes the integrand in (2.7) vanish for large frequencies. This can be thought of as an ultraviolet regularization: the right hand side of (2.7) is finite if and only if g is locally square integrable (near its singularity), which is an infrared problem. Clearly, \(g(\eta )=|\eta |^{-1}\) is locally square integrable only in dimension \(d\ge 3\). This explains the well-known fact that the CLR bound for non–relativistic Schrödinger operators holds only in dimensions \(d\ge 3\).
For a generalized Schrödinger operator \(T(P)+V\), where the kinetic energy (frequency–energy relation of the free particle) is given by a measurable function \(T\ge 0\), we have \(g=T^{-1/2}\). In this case a CLR–type bound holds if \(T^{-1}\) is locally integrable near the zero set of T. This is sharp, since we know from [20] that weakly coupled negative energy bound states of \(T(P)+V\) exist for arbitrary weak attractive potentials V when \(T^{-1}\) is not locally integrable near the zero set of T.
Proof of Theorem 2.2
Since the operator \(H_{f,g,m}\) has a kernel given by the right-hand side of (2.5), we compute its Hilbert–Schmidt norm as
using the Fubini–Tonelli Theorem and the definition of \(G_{g,m}\). \(\square \)
In the rest of this section we will discuss how Theorems 2.1, 2.2, and the bound (2.1) lead to the Cwikel–Lieb–Rozenblum bound for a non-relativistic single-particle Schrödinger operator. In this case \(g(\eta )=|\eta |^{-1}\), and a simple scaling in the \(\eta \) integral gives
Going to spherical coordinates shows
where \(|S^{d-1}|\) is the surface area of the unit sphere in \({\mathbb {R}}^d\) and \(|B_1^d|=|S^{d-1}|/d\) is the volume of the unit ball in \({\mathbb {R}}^d\).
Now we repeat the derivation of (2.1), except that we also scale f by \(\kappa >0\), using \(\kappa A_{f,g} = A_{\kappa f,g}= B_{\kappa f,g,m}+ H_{\kappa f,g,m}\). The argument leading to (2.1) then gives
as long as \(\kappa > \mu \ge \Vert B_{\kappa f,g,m}\Vert \). Clearly, the last factor in (2.11) has the correct dependence on the potential U. Thanks to Theorem 2.1, we can use \(\mu = \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}\) as an upper bound for \(\Vert B_{f,g,m}\Vert \), which is independent of f, so the same bound holds for \(\Vert B_{\kappa f,g,m}\Vert \) for any \(\kappa >0\). Using this, we can now freely optimize (2.11) in \(\kappa >\mu \) to get
with the constant
This gives most of the main ideas of our proof of Theorem 1.1. The last new idea, which is crucially important for the proof of Theorem 2.1, is the connection between the bound on the norm of the operator \(B_{f,g,m}\), more precisely, the bound (2.6) on the operator norm of the associated maximal operator \(\mathcal {B}_{g,m}(\varphi ){:=}\sup _{f\ge 0}|B_{f,g,m}\varphi |\), and bounds for maximal Fourier multipliers on \(L^2\). This is explained in Sect. 4.
Before we do this let us point out that our approach leads to new results also for more general kinetic energies.
3 General kinetic energies
First we consider the case where \(P^2\) is replaced by \(P^{2\alpha }\) and give the
Proof of Theorem 1.3
Replacing \(g(\eta )=|\eta |^{-1}\) by \(g(\eta )=|\eta |^{-\alpha }\) one simply reruns the argument from the previous section. Calculating, again by scaling,
and
one sees that the argument leading to (2.11) remains virtually unchanged, only d gets replaced by by \(d/\alpha \). Thus
with constant
For \(m_1\) and \(m_2\) we make the simple choice from Remark 1.4. Then \(m(t)=m_1*m_2(t)= \min (t,t^{-1})\) and \(\mu =\Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}=1\). Hence,
and collecting terms finishes the proof of Theorem 1.3. \(\square \)
Remark 3.1
For the number of negative energy bound states of \(P^{2\alpha }+U\) the so-far best bounds are due to Frank [16, 17]. Using ideas from Rumin [46, 47], he got the bound
Even with the non-optimal choice of \(m_1\) and \(m_2\) above, one sees that the bound from Theorem 1.3 is better as long as \(2< \left( 1+2\alpha /d\right) ^{d/(2\alpha )}\). Since \(0<\delta \mapsto \left( 1+1/\delta \right) ^{\delta } \) is strictly increasing, this is the case as soon as \(d>2\alpha \), that is, the whole range of allowed values of \(\alpha \).
For more general kinetic energies of the form T(P) with T a non-negative measurable function which is locally bounded we have
Theorem 3.2
The number of negative energy bound states of a Schrödinger–type operator \(T(P)+V\), defined suitably with the help of quadratic form methods on \(L^2\), obeys the bound
for any \(\lambda >0\), with \( V_-=\max (-V,0)\), the negative part of V and
where \(\alpha _+= \max (\alpha ,0)\) is the positive part.
Proof
In this case we use \(g(\eta )= T(\eta )^{-1/2}\), \(f(x)= V_-(x)\), and again make the choice \(m_1(s)=s\mathbf {1}_{\{0<s\le 1\}}\) and \(m_2(s)= 2s^{-1}\mathbf {1}_{\{s\ge 1\}}\). So \(\mu =\Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}=1\). With \(\lambda =\kappa -\mu =\kappa -1\), the same argument leading to (2.10) now gives
for any \(\lambda >0\). Using Theorem 2.2 to calculate the Hilbert–Schmidt norm shows
since \(m(t)=m_1*m_2(t)=\min (t,t^{-1})\). \(\square \)
Remark 3.3
-
(i)
The bound given in Theorem 3.2 improves the bound from [20], which was based on Cwikel’s original method. Clearly, \(G_T\) given by (3.2) is increasing in \(u>0\). Moreover, since T is assumed to be locally bounded it is easy to see that \(G_T(u)\) is finite if and only if \(\eta \mapsto T(\eta )^{-1}\) is integrable over the set \(\{T<u\}\). The result proven in [20] shows that under some rather mild general conditions on the kinetic energy symbol T the operator \(T(P)+V\) has weakly coupled bound states for any non-trivial potential \(V\le 0\), no matter how small |V| is, if \(T^{-1}\) is not integrable over the set \(\{T<u\}\) for all small \(u>0\), which is equivalent to \(G_T(u)=\infty \) for all small \(u>0\) and, by monotonicity, equivalent to \(G_T(u)=\infty \) for all \(u>0\). This shows that the bound given by Theorem 1.1 is quite natural.
-
(ii)
Let \(g(u)=(u^{1/2}- u^{-1/2})_+^2\). Then \(g'(t)=0\) for \(0<t<1\) and \(g'(t)= 1-t^{-2}\) for \(t>1\). The layer cake principle yields
$$\begin{aligned} \int G_T(V_-(x))\, \mathrm {d}x&= \int _0^\infty g'(t) \iint \mathbf {1}_{\{T(\eta )< V_-(x)/t\}} \, \frac{\mathrm {d}x\,\mathrm {d}\eta }{(2\pi )^d} \, \mathrm {d}t \\&=\int _0^\infty g'(t) N^{cl}(T+ t^{-1}V) \, \mathrm {d}t \end{aligned}$$with the classical phase–space volume
$$\begin{aligned} N^{cl}(T+ V) {:=}\iint \mathbf {1}_{\{T(\eta ) + V(x) <0\}} \, \frac{\mathrm {d}x\,\mathrm {d}\eta }{(2\pi )^d} . \end{aligned}$$(3.3)Hence, in terms of the classical phase–space volume Theorem 3.2 gives an upper bound of the form
$$\begin{aligned} N(T(P)+V) \le \lambda ^{-2} \int _1^\infty N^{cl}(T+ t^{-1}(\lambda +1)^2 V) \, (1-t^{-2})\, \mathrm {d}t\, \end{aligned}$$(3.4)for any \(\lambda >0\). One can interpret (3.4) as a quantum correction to the classical phase-space guess (3.3). The integral on the right hand side is finite if and only if the classical phase-space volume is small enough for small potentials. A bound of the form (3.4), with \((1-t^{-2})\) replaced by 1, was also derived in [20]. In most cases where one can explicitly calculate or find explicit upper bounds for \(G_T\), one shows, in fact, that
$$\begin{aligned} \int _1^\infty N^{cl}(T+ t^{-1}V) (1-t^{-2})\, \mathrm {d}t \lesssim N^{cl}(T+ V), \end{aligned}$$(3.5)see the discussion in Section 6 of [20]. In these cases, Theorem 3.2 gives an upper bound for the number of negative bound states of \(T(P)+V\), under very weak conditions on the dispersion relation T, solely in terms of the classical phase-space volume,
$$\begin{aligned} N(T(P)+V) \le C \lambda ^{-2} N^{cl}(T+(1+\lambda )^2V), \end{aligned}$$(3.6)for some constant C and all \(\lambda >0\). However, the bound (3.5), hence also the bound (3.6), does not hold in critical cases, where it is known that logarithmic corrections to the classical phase space guess appear [3, 4, 52].
4 The connection with maximal Fourier multipliers
In this section we give the proof of Theorem 2.1. The important observation is the connection to maximal Fourier multipliers, as we discuss now. Recall that given functions \(f,g:{\mathbb {R}}^d\rightarrow [0,\infty )\) and a bounded, measurable function \(m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}_+ \), the operator \(B_{f,g,m}\) is given by
at least for nice enough \(\varphi \), e.g., Schwartz functions. We would like to conclude that \(B_{f,g,m}\) is a bounded operator on \(L^2({\mathbb {R}}^d)\), which might suggest to look for results which show that a pseudo-differential operator with symbol \(a(x,\eta )= m(f(x)g(\eta ))\) is bounded. A classical example of such a result is the Calderón–Vaillancourt theorem, see for instance [36, Proposition 9.4]. However, typical in the study of pseudo-differential operators, this needs high enough differentiability of the symbol a, which we do not have. More importantly, we need an estimate independent of f, which one cannot get without looking more closely into the structure of the problem. To see how the product structure \(f(x) g(\eta )\) helps in the operator bound, we rewrite \(B_{f,g,m}\) as
This suggest to look at the Fourier multiplier \(B_{t,g,m}\) defined by
and the associated maximal operator
It is clear that one has \(|B_{f,g,m}(\varphi )|\le B^*_{g,m}(\varphi )\), hence also \( \mathcal {B}(\varphi )=\sup _{f\ge 0}|B_{f,g,m}(\varphi )| \le |B^*_{g,m}(\varphi )|\), for any Schwarz function \(\varphi \). On the other hand, choosing f(x) in such a way as to make \(|B_{f,g,m}\varphi (x)|\) arbitrarily close to \(B^*_{g,m}\varphi (x)\), shows the ‘reverse bound’ \(\mathcal {B}_{g,m}(\varphi )=\sup _{f\ge 0}|B_{f,g,m}\varphi | \ge B^*_{g,m}(\varphi )\) for a given fixed Schwartz function \(\varphi \). Thus \(\mathcal {B}_{g,m}(\varphi )= B^*_{g,m}(\varphi )\).
In particular, \(\Vert \mathcal {B}_{g,m}\Vert = \Vert B^*_{g,m}\Vert \) for the corresponding operator norms on \(L^2\). So a bound for the maximal operator \(\mathcal {B}_{g,m}(\varphi )=\sup _{f\ge 0}|B_{f,g,m} (\varphi )|\)– which yields a bound for the operator norm of \(B_{f,g,m}\) which is uniform in the choice of the function f– is equivalent to having a bound for the maximal Fourier multiplier \(B^*_{g,m}\). This is our starting point for the proof of Theorem 2.1.
Remark 4.1
One should be a little bit careful in the definition (4.4) of the maximal operator \(B_{g,m}^*\). If \(\varphi \) is a Schwartz function and \(m:[0, \infty )\rightarrow {\mathbb {R}}\) is bounded and measurable, then both \(B_{f, g, m}\varphi (x)\) and \(B_{t, g, m}\varphi (x)\) are well-defined for all \(x\in {\mathbb {R}}^d\), \(t\ge 0\), and \(f,g\ge 0\) measurable. To ensure measurability of \(x\mapsto B^*_{g,m} \varphi (x)\) one has to impose stronger conditions on m, for example \(m:[0,\infty ) \rightarrow {\mathbb {R}}\) bounded and continuous is enough. In this case, \(t\mapsto B_{t,g,m}\varphi (x)\) is continuous for each \(x\in {\mathbb {R}}^d\) and the supremum in t can be taken over any dense subset. For example, \(B^*_{g,m}\varphi (x)= \sup _{t\in {\mathbb {Q}}_+} | B_{t,g,m} \varphi (x)|\), with \({\mathbb {Q}}_+\) the positive rationals. Note that for the choice of m in Theorem 2.1 the function m is continuous. Indeed, if m is given by a convolution of \(m_1, m_2 \in L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})\), then it is easy to see that it has a canonical continuous representative with \(\lim _{t\rightarrow 0} m(t) = 0 = \lim _{t\rightarrow \infty } m(t)\).
Theorem 4.2
Let g be a measurable non-negative function on \({\mathbb {R}}^d\) and assume that m is given by a convolution,
with \(m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})\). Then the maximal Fourier multiplier \(B^*_{g,m}\), defined in (4.4), extends to a bounded operator on \(L^2({\mathbb {R}}^d)\) with
for its operator norm.
Remark 4.3
There are several different but related proofs of boundedness of maximal Fourier multipliers available in the literature, see, e.g., [7, 11, 45]. These works concentrate on getting \(L^p\) bounds and do not care much about the involved constants. For us the \(L^2\) boundedness is important, with good bounds on the operator norm.
Proof
When m is given by a convolution and \(\varphi \) is a Schwartz function, we have
Interchanging the integrals, applying the triangle, and then the Cauchy–Schwarz inequality for the \(\mathrm {d}s/s\) integration yields
Since the measure \(\mathrm {d}s/s\) is invariant under scaling, we can scale s by a fixed factor t to see that
that is, the right hand side of (4.5) is independent of \(t>0\). So
In particular,
Using Fubini–Tonelli to interchange the integrals and Plancherel’s theorem for the \(L^2\) norm of the Fourier transform, one sees that
Assume for the moment that \(0<g<\infty \) everywhere. Then interchanging the integration and using the same scaling argument as before to scale out \(g(\eta )\) yields
Hence
so \(B^*_{g,m}\) is continuous at zero in \(L^2({\mathbb {R}}^d)\). Since this maximal operator is the supremum of linear operators, it is sublinear and continuity at zero implies that it is locally uniformly continuous. Thus \(B^*_{g,m}\) can be extended to a bounded operator on \(L^2({\mathbb {R}}^d)\).
If g attains the values 0 or \(\infty \), we set \({\widetilde{\varphi }} = \mathbf {1}_{\{0<g<\infty \}} \varphi \). Since \(m(0) = m(\infty ) = 0\), we have \(B_{t, g, m}\varphi = B_{t, g, m} {\widetilde{\varphi }}\), hence also \(B^*_{g, m} {\varphi }= B^*_{g, m} {\widetilde{\varphi }}\) and with \(\Vert {\widetilde{\varphi }}\Vert _{L^2} \le \Vert \varphi \Vert _{L^2}\) the above argument proves the claim in the case of general g. \(\square \)
The next result, which also yields the proof of Theorem 2.1, is a direct consequence of Theorem 4.2.
Corollary 4.4
Let f, g be measurable non-negative functions on \({\mathbb {R}}^d\) and assume that \(m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) is given by a convolution \(m = m_1*m_2\), with \(m_1,m_2\in L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})\). Then the operator \(B_{f,g,m}\), defined by (2.4), i.e., given by the kernel
is bounded on \(L^2({\mathbb {R}}^d)\) with
Proof
By definition of the maximal Fourier multiplier we have \(|B_{f,g,m}\varphi (x)|\le B^*_{g,m}\varphi (x)\) and thus also \(\sup _{f\ge 0}|B_{f,g,m}\varphi (x)|\le B^*_{g,m}\varphi (x)\) for almost every \(x\in {\mathbb {R}}^d\).
Since the \(L^2\)–bound from Theorem 4.2 is independent of \(g\ge 0\), we can also take the supremum in \(g\ge 0\), after taking the \(L^2\)–norm. \(\square \)
5 A lower bound for the variational problem \(M_{\gamma }\)
Recall that the variational problem, which comes up in a natural way in our bound on the number of bound states is
where the convolution \(m_1*m_2\) is on \({\mathbb {R}}_+\) with its scaling invariant measure \(\frac{\mathrm {d}s}{s}\), and the infimum is taken over all functions \(m_1,m_2:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) .
Theorem 5.1
For all \(\gamma >2\) we have the lower bound
Proof
Notice that \(\Vert m\Vert _\infty \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\) for \(m =m_1*m_2\). Thus
In order to minimize the integral \(\int _0^\infty (t- m(t))^2 t^{-\gamma -1}\,\mathrm {d}t\) under the pointwise constraint \(\ell = \Vert m\Vert _{L^{\infty }} \ge |m|\) for \(\ell >0\), one has to choose m in such a way that \((t-m(t))^2\) is as small as possible for each \(t>0\). Thus, for fixed \(\ell >0\), the minimizer is given by \(m_{\ell }(t)=\min (t,\ell )\). Since
this yields the lower bound for \(M_{\gamma }\). \(\square \)
6 Extension to operator–valued potentials
In this section we extend our method to operator–valued potentials and give the proof of Theorem 1.7, i.e. we prove that the number of negative bound states of \(P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}+ V\) is bounded by
where \(V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})\) is an operator valued potential with positive part \(V_+\in L^1_{\text {loc}}({\mathbb {R}}^d,\mathcal {B}(\mathcal {G}))\) and negative part \(V_-\in L^{d/(2\alpha )}({\mathbb {R}}^d, {\mathcal {S}}_{d/(2\alpha )}(\mathcal {G}))\).
Let \(U(x)= V(x)_-\) be the negative part of V(x) defined by spectral calculus. The Birman–Schwinger operator corresponding to \(|P|^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}-U\) is given by
and we again have
Now we factor K as \(K= \widetilde{A}_{f,g}^* \widetilde{A}_{f,g}\) where \(\widetilde{A}_{f,g}\) has kernel
\(g(\eta )= |\eta |^{-\alpha }\) is real–valued (even positive), and \(f(x)=\sqrt{U(x)}\) takes values in the self-adjoint positive operators on \(\mathcal {G}\). We split this as
with a function \(m:[0,\infty )\rightarrow {\mathbb {R}}\), so that
and
where \(\varphi \) is a function from a nice dense subset of \(L^2({\mathbb {R}}^d,\mathcal {G})\), so that the integrals converge and m(tf(x)) is an operator on \(\mathcal {G}\) defined via functional calculus.
Remark 6.1
With a slight abuse of notation, we write \(\mathcal {F}\) in the definition of \(\widetilde{B}_{f,g,m}\), which strictly speaking denotes the Fourier transform on \(L^2({\mathbb {R}}^d)\), instead of \(\mathcal {F}\otimes \mathbf {1}_{\mathcal {G}}\), the Fourier transform on \(L^2({\mathbb {R}}^d,\mathcal {G}) = L^2({\mathbb {R}}^d)\otimes \mathcal {G}\). In addition, in the definition of \(\widetilde{B}_{f,g,m}\) and \(\widetilde{H} _{f,g,m}\) above we swapped the role of f and g compared to the discussion in Sect. 4. This is convenient, since by assumption \(g(\eta )\) is a multiplication operator on \(\mathcal {G}\), and this makes a maximal Fourier multiplier estimate, now with g instead of f, easier. The general case can be reduced to this setting, see Sect. 7 below.
The following theorem is the replacement of Theorems 2.1 and 2.2 in the operator-valued setting.
Theorem 6.2
\(\widetilde{H}_{f,g,m}\) is a Hilbert–Schmidt operator on \(\mathcal {H}=L^2({\mathbb {R}}^d,\mathcal {G})\) with Hilbert–Schmidt norm given by
where \(G_{g,m}\) is again given by
If, moreover, \(m=m_1*m_2\) then for all measurable non-negative functions g and non-negative operator-valued functions f the operator \(\widetilde{B}_{f,g,m}\) is bounded on \(\mathcal {H}\) with
for all \(\varphi \in \mathcal {H}\).
Proof
To prove (6.3), we note that the Hilbert–Schmidt operators on \(\mathcal {H}= L^2({\mathbb {R}}^d,\mathcal {G})\) are isomorphic to operators with kernels in \(L^2({\mathbb {R}}^d\times {\mathbb {R}}^d, {\mathcal {S}}_2(\mathcal {G}))\) and
see Lemma B.3.
Using the explicit form of the ‘kernel’ of \(\widetilde{H}_{f,g,m}\) given in (6.2) this shows
by the definition of \(G_{g,m}\) and the spectral theorem.
Concerning the boundedness of \(\widetilde{B}_{f,g,m}\) we recall (6.1) and, if \(m=m_1*m_2\),
Thus,
due to the scaling invariance of ds/s. We therefore have a maximal operator bound
In particular,
and
where we again used that, by scaling \(\int _0^\infty m_1(r/s)^2\, \frac{\mathrm {d}s}{s}= \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})}^2\) for all \(r>0\), so by functional calculus
Altogether, we get the operator-valued version of our previous maximal Fourier multiplier bound in the form
and it is easy to see that
which completes the proof of Theorem 6.2. \(\square \)
The proof of Theorem 1.7 is straightforward: one simply does the same steps as in the scalar case with (2.10) replaced by
where now \(\mu \ge \Vert \widetilde{B}_{\kappa f,g,m}\varphi \Vert _{\mathcal {H}}\). As before, Theorem 6.2 gives a bound for \(\Vert \widetilde{B}_{\kappa f,g,m}\varphi \Vert _{\mathcal {H}}\) independent of \(\kappa \), in particular, we can take any \(\mu \ge \Vert m_1\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+, \frac{\mathrm {d}s}{s})} \). It also allows us to calculate the Hilbert–Schmidt norm. For \(g(\eta )=|\eta |^{-\alpha }\) we get
so
Using this in the above bound for \(N(P^{2\alpha }\otimes \mathbf {1}_{\mathcal {G}}-U)\) and minimizing over \(\kappa \), as in the scalar case, finishes the proof of Theorem 1.7.
7 Trace ideal bounds
In this section we show how the ideas developed so far can be used to prove a fully operator-valued version of Cwikel’s theorem. Such an inequality was first proved in [16].
In this setting let \((X,\mathrm {d}x)\) and \((Y,\mathrm {d}y)\) be sigma-finite measure spaces and \(\mathcal {H}, \mathcal {G}\) (separable) Hilbert spaces. We denote by \(L^p(X,{\mathcal {S}}_p(\mathcal {H}))\) the set of measurable functions \(f: X\rightarrow {\mathcal {S}}_p(\mathcal {H})\), where \({\mathcal {S}}_p(\mathcal {H})\) is the space of p-summable compact operators, i.e. the von Neumann–Schatten class, on \(\mathcal {H}\), such that
Similarly, we denote by \(L^p_\text {w}(Y, \mathcal {B}(\mathcal {G}))\) the set of of all measurable functions \(g:Y\rightarrow \mathcal {B}(\mathcal {G})\), with values in the bounded operators on \(\mathcal {G}\), such that
A map \(A: L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})\) is in the weak trace–ideal \({\mathcal {S}}_{p,\text {w}}= {\mathcal {S}}_{p,\text {w}}(L^2(X,\mathcal {H}), L^2(Y,\mathcal {G}))\) if
where \(s_n(A)\) are the singular values of A, i.e. the eigenvalues of \(A^* A:L^2(X,\mathcal {H})\rightarrow L^2(X,\mathcal {H})\).
Theorem 7.1
(Fully operator valued version of Cwikel’s theorem) Let \(\Phi : L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})\) be a unitary operator, which is also bounded from \(L^1(X,\mathcal {H})\) into \(L^\infty (Y,\mathcal {G})\).
If \(p>2\) and \(f\in L^p(X,{\mathcal {S}}_p(\mathcal {H}))\) and \(g\in L^p_\text {w}(Y, \mathcal {B}(\mathcal {G}))\), then \(f\Phi ^* g\) is in the weak trace ideal \({\mathcal {S}}_{p,w}(L^2(X,\mathcal {H}),L^2(Y,\mathcal {G}))\) and
where \(Q_{p}\) is given in (C.2).
Remark 7.2
Theorem 7.1 improves the result of Frank in [16],
The value of \(Q_p\) comes from choosing \(m_1(s)=s\mathbf {1}_{\{0<s\le 1\}}\) and then finding an optimal \(m_2\), see Appendix C. Making the simple choice of Remark 1.4 for \(m_2\) leads to an upper bound for the weak–trace ideal norm with \(Q_p\) replaced by \(8(p(p-2)(p+2))^{-1}\) in (7.2). It is easy to see that this simple choice of \(m_1\) and \(m_2\) yields a bound which is already a factor of \((p+2)/4\) smaller than the one in [16]. In addition, the bound in [16] in the scalar case, when \(\Phi \) is the usual Fourier transform, is worse than the one in Theorem 7.1, with the above easy choice for \(m_1\) and \(m_2\), by a factor of \(\frac{1}{2}(1+2/p)^{p/2}>1\) in the allowed range \(p>2\).
Proof
First we note that one can reduce the result to the case when g is pointwise a positive multiple of the identity operator on \(\mathcal {G}\). As operators on \(\mathcal {G}\) one has \(g(y) g(y)^* \le \Vert g(y)\Vert _{\mathcal {B}(\mathcal {G})}^2\mathbf {1}_{\mathcal {G}}\). Thus with \(A_1= f\Phi ^* g\) we have
with \(A_2= f\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})}\mathbf {1}_{\mathcal {G}}= f\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})} \) where, for simplicity, we wrote \(\Vert g\Vert _{\mathcal {B}(\mathcal {G})}\) for \(\Vert g\Vert _{\mathcal {B}(\mathcal {G})}\mathbf {1}_\mathcal {G}\). Since the singular values of \(A_1\) are the square roots of the eigenvalues of \(A_1^* A_1\), which has the same non-zero-eigenvalues as \( A_1 A_1^*\) we see that the nonzero singular values of \(A_1\) obey the bound \(s_n(A_1) \le s_n(A_2)\).
Similarly, \(|f(x)|{:=}\sqrt{f(x)^*f(x)}\) is a non negative operator on \(\mathcal {H}\) and
with \( A_3= |f|\Phi ^* \Vert g\Vert _{\mathcal {B}(\mathcal {G})}\). So the singular values of \(A_2\) are the same as the singular values of \(A_3\) and without loss of generality, we can assume that g is a non-negative function and f takes values in the non-negative operators on \(\mathcal {H}\). By scaling, we can also assume that \(\Vert f\Vert _{L^p(X,{\mathcal {S}}_p(\mathcal {H}))}= \Vert g\Vert _{L^p_\text {w}(Y)}^p=1\).
Since \(\Phi : L^1(X,\mathcal {H})\rightarrow L^\infty (Y,\mathcal {G})\) is bounded, Lemma B.4 shows that it has a kernel \(\Phi (\cdot ,\cdot )\) such that for all \(f\in L^2(X,\mathcal {H})\) and almost all y\(\in \)Y
Moreover, \(\sup _{(y,x)\in Y\times X}\Vert \Phi (y,x)\Vert _{\mathcal {B}(\mathcal {H},\mathcal {G})}= \Vert \Phi \Vert _{L^1\rightarrow L^\infty }\) Having reduced the estimate to scalar non-negative functions g and non-negative operator-valued functions f we can rewrite \(\widetilde{A}_{f,g}= g\Phi f\) as
using that g(y) is now a non-negative scalar. Thus, we can take again an arbitrary function \(m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) with \(m(0)=0\) and split
The above expressions are well-defined by the spectral theorem, since g is a non-negative function and f takes values in the non-negative operators on \(\mathcal {H}\), so m(g(y)f(x)) is a bounded operator on \(\mathcal {H}\) for almost all y and x, when m is bounded. Thus the integrals in (7.4) and (7.4) converge for all \(\varphi \) from a dense subset of \(L^2(X,\mathcal {H})\), for example the piecewise constant functions.
Scaling in f by \(\kappa >0\), we get from Ky Fan’s inequality
where we take \(\mu = \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}\), the upper bound on the norm of \(\widetilde{B}_{\kappa f,g,m}\) from Lemma 7.3 below and we used \(s_n(H)\le n^{-1}\sum _{j=1}^n s_j(H)^2\le n^{-1}\Vert H\Vert _{HS}^2\), for any Hilbert–Schmidt operator, due to the monotonicity of its singular values. Thus using the bound (7.7) one gets
with \(D= \int _0^\infty (1-t^{-1}m(t))^2 t^{1-p}\, \mathrm {d}t\), and minimizing this over \(\kappa >0\) we have
for the singular values for all \(n\in {\mathbb {N}}\).
Now we make the choice \(m_1(s)= s\mathbf {1}_{\{0<s\le 1\}}\) and minimize over all admissible \(m_2\). Proposition C.4 shows that this leads to \(\mu ^{p-2}D= Q_p\), with \(Q_p\) defined in (C.2). In view of Remark 7.4 (ii), the minimizer for \(Q_p\) is admissible in Lemma 7.3. \(\square \)
Lemma 7.3
Let \(p>2\), \(\mathcal {H}\) and \(\mathcal {G}\) auxiliary Hilbert spaces, \((X,\mathrm {d}x)\) and \((Y,\mathrm {d}y)\) \(\sigma \)–finite measure spaces, \(0\le g\in L^p_{\text {w}}(Y)\), \(0\le f\in L^p(X,{\mathcal {S}}_p(\mathcal {H}))\), \(\Phi : L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})\) unitary and also bounded from \(L^1(X,\mathcal {H})\rightarrow L^\infty (Y,\mathcal {G})\). Then for all continuous and piecewise differentiable bounded functions \(m:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) with \(m(0)=0\) and \(\partial _t(t-m(t))^2\ge 0\) for all \(t>0\), the operator \(\widetilde{H}_{f,g,m}\) defined in (7.5) is a Hilbert–Schmidt operator and
Moreover, if \(m=m_1*m_2\), then the operator \(\widetilde{B}_{f,g,m}\) defined in (7.4) is bounded from \(L^2(X,\mathcal {H})\) to \( L^2(Y,\mathcal {G})\) and
Remark 7.4
-
(i)
As the proof of Lemma 7.3 shows one even has a bound on \(\widetilde{B}_{f,g,m}\) of the form
$$\begin{aligned} \sup _{f\ge 0} \big \Vert \sup _{g\ge 0}\Vert \widetilde{B}_{f,g,m}\varphi \Vert _\mathcal {G}\big \Vert _{L^2(Y)} \le \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert \varphi \Vert _{L^2(X,\mathcal {H})} \end{aligned}$$where the first supremum is taken over all functions \(g:Y\rightarrow [0,\infty )\) and the second supremum is taken over all non-negative operator-valued functions \(f:X\rightarrow \mathcal {B}(\mathcal {H})\).
-
(ii)
The condition \(\partial _t(t-m(t))^2\ge 0\) might look weird at first, but there is a large class of functions m for which it holds: A simple choice is \(m_1(s)=s\mathbf {1}_{\{0<s\le 1\}}\) and \(m_2(s)=2s^{-1}\mathbf {1}_{\{s\ge 1\}}\). In this case \(m(t)=m_1*m_2(t)=\min (t,t^{-1})\), so this simple choice of \(m_1\) and \(m_2\) is admissible in Lemma 7.3. More generally, setting \(m_2(t)= -h'(t^{-1})\) for some absolutely continuous function h with \(h(0)=1\) and \(\lim _{t\rightarrow \infty }h(t)=0\), the proof of Proposition C.4 shows that \( t-m(t) = th(t^{-1}) \) for all \(t>0\),
$$\begin{aligned} \int _0^\infty (t-m(t))^2 t^{1-p}\mathrm {d}t&= \int _0^\infty h(t)^2 t^{p-2} \frac{\mathrm {d}t}{t}, \end{aligned}$$(7.9)and
$$\begin{aligned} \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})} \Vert m_2\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}&= \left( \frac{1}{2}\int _0^{\infty } h'(s)^2\,\frac{\mathrm {d}s}{s}\right) ^{\frac{p-2}{2}} . \end{aligned}$$(7.10)Such a choice for \(m_1\) and \(m_2\) then leads to the variational problem (C.1), which we solve in Proposition C.1. Moreover, \(\partial _t (t-m(t))^2 = \partial _t (th(t^{-1}))^2= 2th(t^{-1})(h(t^{-1})-t^{-1}h'(t^{-1}))\ge 0\) for any decreasing function \(h\ge 0\). Fortunately, the minimizers for the variational problem (C.1) have this property and thus can be used in Lemma 7.3 which leads to the constant in Theorem 7.1.
Proof
We freely use results for the operator-valued setting given in Appendix B. For notational simplicity we set
and note
Because g is real-valued, even positive, and f takes values in the non-negative, hence self-adjoint, operators
so, setting \( G(u){:=}\int _{Y}\left[ u g(y) -m(ug(y)) \right] ^2\, \mathrm {d}y\), we have
With \(k(t)= (t-m(t))^2 \), the layer-cake principle shows
By definition \(|\{y\in Y: g(y)>t\}|\le t^{-p}\Vert g\Vert _{L^p_\text {w}(Y)}^p\) for all \(t>0\). By assumption, \(k' \ge 0\), thus
An integration by parts argument would show that \(\int _0^\infty k'(t) t^{-p}\, \mathrm {d}t = p\int _0^\infty k(t) t^{1-p}\, \mathrm {d}t\), but due to the singularity of the integrand this requires that k vanishes at zero fast enough and that k does not grow too fast at infinity. Instead, we prefer to use non-negativity of \(k'\). Note that
Since the integrand in the double integral is non–negative, we can use the Fubini–Tonelli Theorem to freely interchange the order of integration. Hence
Thus the formal integration by parts argument is justified. Moreover, this argument shows that if one side is infinite, so is the other. With (7.11) we get
Integrating this over X finishes the proof of (7.7).
To prove (7.8) we introduce
for \(t\ge 0\) (note that \(\widetilde{B}_{f,0,m}\varphi =0\) since \(m(0)=0\)). If \(m=m_1*m_2\), then a by now familiar calculation yields
and therefore the Cauchy–Schwarz inequality gives
By scaling, the right hand side above does not depend on \(t>0\) anymore. Hence we get the bound
for the associated maximal operator \(\widetilde{B}_{f,m}^*\varphi (y){:=}\sup _{t\ge 0}\Vert \widetilde{B}_{f,t,m}\varphi (y)\Vert _\mathcal {G}\). In particular,
Interchanging the integrals, the last factor on the right hand side of (7.13) is given by
As functions of the real variable \(r\ge 0\) the scaling invariance of the measure ds/s on \({\mathbb {R}}_+ \) and \(m_1(0)=0\) give \( \int _0^\infty m_1(sr)^2\, \frac{\mathrm {d}s}{s} = \Vert m_1\Vert _{L^2({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})}^2\mathbf {1}_{\{r>0\}}\), so the spectral theorem implies
Using this in (7.13) shows
which proves (7.8), since \(\Vert \widetilde{B}_{f,g,m}\varphi (y)\Vert _\mathcal {G}\le \widetilde{B}_{f,m}^*\varphi (y)\) for all \(y\in Y\). \(\square \)
Notes
Parts of this connection were already known to the St. Petersburg school of mathematical physics around Birman and Solomyak, see the above mentioned “Added notes” in [49].
See also [25] for some indication of the induction in dimension trick.
The numbers are taken from Roepstorff’s book [42, Table 3.1].
Which is of course not necessarily the best possible constant.
Using the upper bound on \(M_\gamma \) given in Proposition 1.5, one can actually derive the better estimate
$$\begin{aligned} \frac{C_\gamma }{C^\text {lower}_\gamma } \le \frac{2(\gamma -1)}{\gamma } \frac{1}{\Gamma (\frac{2}{\gamma })^{\gamma }} \left( \frac{\gamma -2}{2} \frac{\pi }{\sin (\frac{2\pi }{\gamma })} \right) ^{\frac{\gamma }{2}} = \frac{2(\gamma -1)}{\gamma } \left( \frac{\Gamma (2-\frac{2}{\gamma })}{\Gamma (1+\frac{2}{\gamma })} \right) ^{\frac{\gamma }{2}}. \end{aligned}$$The right-hand side can be shown to be increasing in \(\gamma \) with limit \(\lim _{\gamma \rightarrow \infty }\frac{2(\gamma -1)}{\gamma } \left( \frac{\Gamma (2-\frac{2}{\gamma })}{\Gamma (1+\frac{2}{\gamma })} \right) ^{\frac{\gamma }{2}} = 2 \mathrm {e}^{2\gamma ^*-1}\le 2.34\), where \(\gamma ^*\) is the Euler-Mascheroni constant. We will however not elaborate this further.
We follow the convention that all Hilbert spaces are considered to be separable, unless stated otherwise ;-). Physically, this auxiliary Hilbert space corresponds to other degrees of freedom, for example spin.
For the equality \(L^1(Y) \otimes L^1(X) = L^1(Y\times X)\) one should be a wee bit more precise about the involved topologies in the tensor products: For a Banach space E, the algebraic tensor product \(L^1(Y) \otimes _{\mathrm {alg}} E\) is the vector space of finite linear combinations \(\sum _{l=1}^N g_l\otimes f_l\), where \(g_l\in L^1(Y)\) and \(f_l\in E\). One equips this vector space with the norm \(\Vert z\Vert _\pi {:=}\inf \{ \sum _l \Vert g_l\Vert _{L^1(Y)} \Vert f_l\Vert _E :\, z= \sum _{l} g_l\otimes f_l \} \). Then for the closure \(L^1(Y){\widehat{\otimes }} E{:=}\overline{L^1(Y) \otimes _{\mathrm {alg}}E}^{\Vert \cdot \Vert _\pi }\), called the projective tensor product, one has \(L^1(Y){\widehat{\otimes }} E = L^1(Y,E)\), see [55, Proposition III.B.28] or [14, Example VIII.10]. In particular, one has \(L^1(Y){\widehat{\otimes }}L^1(X)= L^1(Y, L^1(X))= L^1(Y\times X)\). We will not dwell on this fine point any further ;-).
For \(\alpha \in (0,1)\), one has \(K_{\alpha }(t) \sim \sqrt{\frac{\pi }{2t}} \mathrm {e}^{-t}\) as \(t\rightarrow \infty \) [38, Eq. 10.25.3] and \(K_{\alpha }(t) \sim \frac{1}{2} \Gamma (\alpha ) (\frac{t}{2})^{-\alpha }\) as \(t\rightarrow 0\) [38, Eq. 10.30.2], while \(I_{\alpha }(t)\) grows exponentially fast as \(t\rightarrow \infty \) [38, Eq. 10.30.4], so it can never satisfy the boundary condition at infinity.
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Acknowledgements
Special thanks go to Michael Cwikel and Barry Simon for many helpful comments and remarks on an earlier version. We would also like to thank the Mathematisches Forschungsinstitut Oberwolfach (MFO) and the Centre International de Rencontres Mathématiques (CIRM Luminy) for their research in pairs programs, where part of this work was conceived. Many thanks also go to Rafael Benguria for comments and discussions at an early stage of this work.
Funding
Work funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173. Dirk Hundertmark also thanks the Alfried Krupp von Bohlen und Halbach Foundation for financial support. Open Access funding enabled and organized by Projekt DEAL.
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Appendices
Appendix A. Induction in dimension
In this section we prove Theorem 1.8, that is, we prove that the number of negative bound states of \(P^2 \otimes \mathbf {1}_{\mathcal {G}}+ V\) is bounded by
and, moreover,
where \(C_n\) is given by (1.5) for \(\gamma =n\). Here, \(V:{\mathbb {R}}^d\rightarrow \mathcal {B}(\mathcal {G})\) is an operator valued potential with positive part \(V_+\in L^1_{\text {loc}}({\mathbb {R}}^d,\mathcal {B}(\mathcal {G}))\) and negative part \(V_-\in L^{d/2}({\mathbb {R}}^d, {\mathcal {S}}_{d/2}(\mathcal {G}))\).
In order to do this, we need the following operator-valued extension of the well-known Lieb–Thirring bounds for suitable moments \(\theta \):
where \(L_{\theta , d}^{\mathrm {op}} = C_{\theta , d}^{\mathrm {op}} \, L_{\theta , d}^{\mathrm {cl}}\) with the classical Lieb–Thirring constant
It is important that the constant \(L_{\theta , d}^{\mathrm {op}}\), respectively, \(C_{\theta , d}^{\mathrm {op}}\) does not depend on the auxiliary Hilbert space \(\mathcal {G}\).
The bound (A.1) was first proven in the seminal work of Laptev and Weidl [27] for all dimensions \(d\in {\mathbb {N}}\) and moments \(\theta \ge \frac{3}{2}\), moreover, they showed \(C_{\theta , d}^{\mathrm {op}}=1\) in this case. This was later simplified in [1]. For moments \(\theta \ge \frac{1}{2}\) and again all dimensions \(d\in {\mathbb {N}}\) the bound (A.1) was shown to hold in [23], moreover, \(C_{\theta , d}^{\mathrm {op}}\le 2\) for \(\frac{1}{2}\le \theta <\frac{3}{2}\), see also [15] and, recently, [18] for improvements when \(\theta =1\). The limiting case \(\theta =0\), that is, the operator–valued version of the CLR bound was then proven in [21], with improvements on the constant later in [19].
The possibility that a bound of the form A.1 allows to strip off one dimension in the Lieb–Thirring bounds was crucially used in Laptev–Weidl [27], see also [25]. The possibility of stripping off more than one dimension was realized in [21].
In the short proof below, which we give for the convenience of the reader, we follow the discussion in [21].
Lemma A.1
For \(n\le d\) we have
In particular, for \(d\ge 3\),
Proof
For \(n\le d\) we factor \({\mathbb {R}}^d = {\mathbb {R}}^n \times {\mathbb {R}}^{d-n}\), that is, \(x = (x_<, x_>) \in {\mathbb {R}}^n \times {\mathbb {R}}^{d-n}\), and split the the kinetic energy as \(P^2 = P^2_< + P^2_>\), more precisely,
Moreover, observe that
As quadratic forms on \(L^2({\mathbb {R}}^d, \mathcal {G})\), we then have
with the operator-valued potential \(W(x_<)= \big (P_>^2 \otimes \mathbf {1}_{\mathcal {G}}+ V(x_<, \cdot ) \big )_- : L^2({\mathbb {R}}^{d-n},\mathcal {G}) \rightarrow L^2({\mathbb {R}}^{d-n},\mathcal {G})\). Note that \(W(x_<)\) is the negative part of a Schrödinger operator in \(d-n\) dimensions where one freezes the \(x_<\) coordinate in the potential. Inequality (A.1) can therefore be applied and yields
Since by assumption \(\int _{{\mathbb {R}}^{d}} \mathop {\mathrm {tr}} \nolimits _{\mathcal {G}} V_- (x)^{\theta +\frac{d}{2}} \, \mathrm {d}x <\infty \), the Fubini–Tonelli theorem shows that \(W(x_<)\) is compact (even in the von Neumann–Schatten ideal \({\mathcal {S}}_{\theta +\frac{n}{2}}(L^2({\mathbb {R}}^{d-n},\mathcal {G}))\)) for almost all \(x_<\in {\mathbb {R}}^n\). Taking traces in inequality (A.3) gives the estimate
where we also used the operator-valued Lieb–Thirring inequality (A.1) and combined the integrals using the Fubini–Tonelli theorem. It follows that
A short calculation, see below, shows
so (A.4) and the definition of \(C_{\theta , d}^{\mathrm {op}}\) imply the sub-multiplicativity
which proves is the first claim of Lemma A.1. In particular, for \(\theta =0\) and \(3\le n \le d-1\), we get
since Laptev–Weidl [27] showed \(C_{\theta ,m}^{\mathrm {op}} = 1\) if \(m\in {\mathbb {N}}\) and \(\theta \ge \frac{3}{2}\). This proves the second claim in Lemma A.1.
It remains to show (A.5), which follows from the definition of the classical Lieb–Thirring constant and the Fubini–Tonelli Theorem:
The third equality follows from scaling, setting \(\eta _<= (1-\eta _>^2)_+^{1/2}\xi \) with \(\xi \in {\mathbb {R}}^n\). \(\square \)
Proof of Theorem 1.8
Lemma A.1 shows that
and the reverse inequality clearly holds. Moreover, the case \(\alpha =1\) in Theorem 1.7 shows the bound
with the constant \(C_{\gamma =n}\) from (1.5). \(\square \)
Appendix B. Auxiliary bounds for the operator-valued case
In this appendix we gather three results, which we needed for extending our method from the scalar case to the operator-valued case. These results are probably well-known to specialists; we give short proofs for the convenience of the reader.
First we consider operators of the form \(A^* A\) and \(A A^*\) for some bounded operator \(A:\mathcal {H}\rightarrow \mathcal {G}\), where \(\mathcal {H}, \mathcal {G}\) are two auxiliary (separable) Hilbert spaces. Let \(N(A)= \{ f\in \mathcal {H}:\, Af=0\}\subset \mathcal {H}\) be the null space of A, \(N(A^*)= \{ g\in \mathcal {G}:\, A^*g=0\}\subset \mathcal {G}\) the null space of the adjoint \(A^*:\mathcal {G}\rightarrow \mathcal {H}\), and \(N(A)^\perp {:=}\{h\in \mathcal {H}:\, \langle h, f\rangle _\mathcal {G}=0 \text { for all } f\in N(A)\}\subset \mathcal {H}\), respectively \(N(A^*)^\perp {:=}\{g\in \mathcal {G}:\, \langle g, f\rangle _\mathcal {H}=0 \text { for all } f\in N(A^*)\}\subset \mathcal {G}\), the orthogonal complement of N(A) in \(\mathcal {H}\), respectively \(N(A^*)\) in \(\mathcal {G}\).
Lemma B.1
Let \(\mathcal {H},\mathcal {G}\) be Hilbert spaces and \(A:\mathcal {H}\rightarrow \mathcal {G}\) be a bounded operator. Then \(A^*A\big |_{N(A)^\perp }\) is unitarily equivalent to \(A A^*\big |_{N(A^*)^\perp }\). In particular, if \(A:\mathcal {H}\rightarrow \mathcal {G}\) is compact, then its non-zero singular values, including multiplicities, are the same as the non-zero singular values of \(A^*:\mathcal {G}\rightarrow \mathcal {H}\).
Remark B.2
In Theorem 3 in [13] a stronger result, which allows for unbounded operators is proven, we need it only for bounded operators \(A:\mathcal {H}\rightarrow \mathcal {G}\).
Proof
The polar decomposition, e.g., Theorem VI.10 in [41], of a bounded operator easily extends to a two Hilbert space situation: For a bounded operator \(A:\mathcal {H}\rightarrow \mathcal {G}\) there exists a partial isometry \(U:\mathcal {H}\rightarrow \mathcal {G}\) with \(N(U)= N(A)\) and range \(\mathrm {Ran}(U)=\overline{\mathrm {Ran}(A)}\), and a symmetric operator |A| with \(|A|^2= A^* A\) such that \(A=U|A|\).
Moreover, \(U:\overline{\mathrm {Ran}(A^*)}=N(A)^\perp \rightarrow \overline{\mathrm {Ran}(A)}= N(A^*)^\perp \) is an isometry, and
so \(A A^*\vert _{N(A^*)^\perp }\) is unitarily equivalent to \(A^* A\vert _{N(A)^\perp }\).
Since the singular values of A are the square roots of the eigenvalues of \(A^*A\) and the singular values of \(A^*\) the square roots of the eigenvalues of \(AA^*\), the last claim in Lemma B.1 is evident from the unitary equivalence above. \(\square \)
Given a Hilbert space \(\mathcal {H}\) and a \(\sigma \)-finite measure space \((X,\mathrm {d}x)\) we denote by \(L^p(X,\mathcal {H})\) the space of measurable functions \(f:X\rightarrow \mathcal {H}\) for which
when \(1\le p<\infty \), respectively,
when \(p=\infty \). Since \(\mathcal {H}\) is assumed to be separable, Pettis’ measurability theorem [40], see also [14], shows that the weak and strong notions of measurability for functions \(X\ni x\mapsto f(x)\) coincide. If \(\mathcal {H}={\mathbb {C}}\), we simply write \(L^p(X,{\mathbb {C}})= L^p(X)\). Moreover, we denote by \({\mathcal {S}}_2\big (L^2(X,\mathcal {H}), L^2(Y,\mathcal {G})\big )\), the space of Hilbert–Schmidt operators \(H:L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})\) with scalar-product
and associated norm \(\Vert H\Vert _{{\mathcal {S}}_2}{:=}\langle H, H \rangle _{{\mathcal {S}}_2}^{1/2} \) and by \(L^2(Y\times X,{\mathcal {S}}_2\big (\mathcal {H},\mathcal {G})\big )\), the \(L^2\)–space of operator-valued kernels \(K: Y\times X\rightarrow {\mathcal {S}}_2(\mathcal {H},\mathcal {G})\) with scalar product
The next result extends the well-known one-to-one correspondence of Hilbert–Schmidt operators from \(L^2(X)\) to \(L^2(Y)\) with kernels in \( L^2(Y\times X)\) to the operator-valued setting.
Lemma B.3
Let \((X,\mathrm {d}x)\) and \((Y,\mathrm {d}y)\) be \(\sigma \)-finite measure spaces and \(\mathcal {H},\mathcal {G}\) two auxiliary Hilbert spaces. Then \({\mathcal {S}}_2\big (L^2(X,\mathcal {H}), L^2(Y,\mathcal {G})\big )\) is isomorphic to \(L^2\big (Y\times X,{\mathcal {S}}_2(\mathcal {H},\mathcal {G})\big )\), that is, for any \(H\in {\mathcal {S}}_2\big (L^2(X,\mathcal {H}), L^2(Y,\mathcal {G})\big )\) there exists a unique \(K_H\in L^2(Y\times X,{\mathcal {S}}_2(\mathcal {H},\mathcal {G})) \) such that for any \(f\in L^2(X,\mathcal {H})\) and almost all \(y\in Y\)
and vice versa. Moreover, the Hilbert–Schmidt norm of the operator \(H\in {\mathcal {S}}_2\big (L^2(X,\mathcal {H}), L^2(Y,\mathcal {G})\big )\) can be calculated as
Proof
The proof is a modification of the proof in the scalar-valued case. We sketch it for the convenience of the reader. Any kernel \(K\in L^2\big (Y\times X,{\mathcal {S}}_2(\mathcal {H},\mathcal {G})\big )\) yields a bounded operator \(H_K: L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})\) by defining
Indeed, since
by Cauchy–Schwarz, we get
since the Hilbert–Schmidt norm bounds the operator norm. So the map \(K\mapsto H_K\) from kernels to operators \(L^2(X,\mathcal {H}) \rightarrow L^2(Y,\mathcal {G})\) is bounded with norm \(\le \Vert K\Vert _{L^2}\), and it is clearly injective.
Given two orthonormal bases \((\alpha _m)_{m\in {\mathbb {N}}}\) of \(\mathcal {H}\) and \((\beta _m)_{m\in {\mathbb {N}}}\) of \(\mathcal {G}\), the space \(S_2(\mathcal {H},\mathcal {G})\) has a basis for almost all given by the rank-one operators \(|\beta _m \rangle \langle \alpha _n|:\mathcal {H}\rightarrow \mathcal {G}\), \(f\mapsto \beta _m \langle \alpha _n,f\rangle _\mathcal {H}\). Furthermore, let \((\varphi _j)_{j\in {\mathbb {N}}}\) and \((\psi _l)_{l\in {\mathbb {N}}}\) be bases for \(L^2(Y)\) and \(L^2(X)\). Then \((\Psi _{l,n})_{l,n\in {\mathbb {N}}}\), given by the \(\mathcal {H}\)-valued functions \( X\ni x\mapsto \Psi _{l,n}(x) = \psi _l(x) |\alpha _n\rangle \), is a basis for \(L^2(X,\mathcal {H})= L^2(X)\otimes \mathcal {H}\) and \((\Phi _{l,m})_{k,m\in {\mathbb {N}}}\), given by the \(\mathcal {G}\)-valued functions \( Y\ni y\mapsto \Phi _{l,n}(y) = \varphi _k(y) |\beta _m\rangle \), is a basis for \(L^2(Y,\mathcal {G})\). Thus any kernel \(K\in L^2\big (Y\times X, {\mathcal {S}}_2(\mathcal {H},\mathcal {G})\big ) = L^2(Y)\otimes L^2(X)\otimes {\mathcal {S}}_2(\mathcal {H},\mathcal {G})\) can be written in the form
and a short calculation shows
Let \(R\in {\mathbb {N}}\) and
which is the kernel of the finite rank operator
Since \(\Vert K-K_R\Vert _{L^2}\rightarrow 0\) the bound (B.4) shows \(\Vert H_{K}- H_{K_R}\Vert \rightarrow 0\) as \(R\rightarrow \infty \), so any \(H_K\) is the limit in the operator norm of finite-rank operators, hence a compact operator. Using the basis \((\Psi _{l,n})_{l,n\in {\mathbb {N}}}\) to calculate the trace, a straightforward calculation shows
so \(H_K\in {\mathcal {S}}_2(L^2(X,\mathcal {H}), L^2(Y,\mathcal {G}))\) and \(\Vert H_K\Vert _{{\mathcal {S}}_2}= \Vert K\Vert _{L^2}\).
So far we have shown that the map \(K\mapsto H_K\) is an isometry from \(L^2\big (Y\times X, {\mathcal {S}}_2(\mathcal {H},\mathcal {G})\big )\) into \({\mathcal {S}}\big (L^2(X,\mathcal {H}), L^2(Y,\mathcal {G})\big )\) so its range is closed. The finite rank operators \(F:L^2(X,\mathcal {H})\rightarrow L^2(Y,\mathcal {G})\) are of the form
with \(c_{r,s}\not = 0\) for finitely many \(r,s\in {\mathbb {N}}\) and \(\widetilde{\Phi }_r\in L^2(Y,\mathcal {G})\), \(\widetilde{\Psi }_s\in L^2(X,\mathcal {H}) \). Expanding \(\widetilde{\Psi }_s\) in the basis \((\Phi _{l,n})_{l,n\in {\mathbb {N}}}\) and similarly for \(\widetilde{\Phi }_r\), one sees that finite rank operators of the above form can be arbitrarily well approximated, in operator norm, by finite rank operators of the form (B.7). Since the finite rank operators are dense in the Hilbert–Schmidt operators, the operators of the form (B.7) are also dense and hence the range of \(K\mapsto H_K\) is all of \({\mathcal {S}}\big (L^2(X,\mathcal {H}), L^2(Y,\mathcal {G})\big )\). \(\square \)
The last result concerns an operator-valued version of Dunford’s theorem. For this we need some more notation. For background on integration in Banach spaces, we refer to [14].
We denote by \(\mathcal {B}(\mathcal {H},\mathcal {G})\) the Banach space of bounded operators from \(\mathcal {H}\) to \(\mathcal {G}\) equipped with the operator norm.
We write \(L^\infty _s(Y\times X,\mathcal {B}(\mathcal {H},\mathcal {G}))\) for the space of functions \(K:Y\times X \rightarrow \mathcal {B}(\mathcal {H},\mathcal {G})\) such that
and for all \(h\in \mathcal {H}\) the map
is strongly measurable (with respect to the topology on \(\mathcal {G}\)). Since \(\mathcal {G}\) is a separable Hilbert space, Pettis’ measurability theorem implies that this the case if and only if it is weakly measurable, i.e., for any \(\psi \in \mathcal {G}\),
is measurable. In this case, for \(f\in L^1(X,\mathcal {H})\), integrals of the form
are well-defined elements in \(\mathcal {G}\) for almost all \(y\in Y\), with
Thus, for \(K\in L^\infty _s(Y\times X,\mathcal {B}(\mathcal {H},\mathcal {G}))\), the map \(\Phi _K: L^1(X, \mathcal {H}) \rightarrow L^{\infty }(Y,\mathcal {G})\) is bounded with
The next Lemma shows that the map \(K\mapsto \Phi _K\) is even an isometry.
Lemma B.4
For any bounded operator \(\Phi :L^1(X,\mathcal {H})\rightarrow L^\infty (Y,\mathcal {G})\) there exists a kernel \(K_\Phi \in L^\infty _s\big (Y\times X,\mathcal {B}(\mathcal {H},\mathcal {G})\big )\) such that
for any \(f\in L^1(X,\mathcal {H})\) and almost all \(y\in Y\). Moreover,
Proof
If \(K\in L^\infty _s\big (Y\times X,\mathcal {B}(\mathcal {H},\mathcal {G})\big )\), the discussion above shows that the map \(\Phi _K\) defined in (B.8) is bounded from \(L^1(X,\mathcal {H})\) to \(L^{\infty }(Y,\mathcal {G})\) and
Conversely, assume that \(\Phi \) is a bounded map from \(L^1(X,\mathcal {H})\) into \(L^\infty (Y,\mathcal {G})\) and choose orthonormal bases \((\alpha _n)_{n\in {\mathbb {N}}}\) in \(\mathcal {H}\) and \((\beta _m)_{m\in {\mathbb {N}}}\) in \(\mathcal {G}\). Then any function \(f\in L^1(X,\mathcal {H})\) can be identified with a sequence of functions \(f=(f_1, f_2,\ldots )\), where \(f_l\in L^1(X)\) and \(\Vert f\Vert _{L^1(X,\mathcal {H})}= \Vert (\sum _{l\in {\mathbb {N}}}|f_l|^2)^{1/2}\Vert _{L^1(X)}\), and similarly for \(L^1(Y,\mathcal {G})\). So without loss of generality, we can assume that \(\mathcal {H}=\mathcal {G}=l^2({\mathbb {N}})\), i.e., the bounded operators from \(\mathcal {H}\rightarrow \mathcal {G}\) correspond to infinite matrices which map \(l^2({\mathbb {N}})\) boundedly into itself. Finally, let \((e_j)_{j\in {\mathbb {N}}}\) be the canonical basis of \(l^2({\mathbb {N}})\).
For \(N\in {\mathbb {N}}\) and \(g_l\in L^1(Y)\), \(f_l\in L^1(X)\), \(l=1,\ldots , N\), the finite linear combinationsFootnote 8 of the form
are dense in \(L^1(Y\times X)\). Now assume that \(\Phi :L^1(X,l^2({\mathbb {N}}))\rightarrow L^\infty (Y,l^2({\mathbb {N}}))\) is bounded. For \(m,n\in {\mathbb {N}}\) let
which defines a linear functional on the finite linear combinations and is bounded by \(\Vert S_{m,n}\Vert \le \Vert \Phi \Vert \). Thus it has a continuous extension to all of \(L^1(Y\times X)\) and since the dual \(L^1(Y\times X)^* = L^\infty (Y\times X)\), there exist measurable functions \(K^{m,n}_{\Phi } \in L^\infty (Y\times X)\), \(m,n\in {\mathbb {N}}\), such that
Taking unions of countably many zero sets, we can assume that the kernels \(K^{m,n}_\Phi (\cdot ,\cdot )\) are well–defined for any \(m,n\in {\mathbb {N}}\), up to a common zero set in \(Y\times X\).
Let \(l^2_{\text {fin}}({\mathbb {N}})\) be the set of sequences \(\alpha =(\alpha _1,\alpha _2,\ldots )\) with only finitely many \(\alpha _j\) non–zero, which is dense in \(l^2({\mathbb {N}})\). For \(\alpha \in l^2_{\text {fin}}({\mathbb {N}})\) and \((y,x)\in Y\times X\) we define the sequence \(K_\Phi (y,x)\alpha \in {\mathbb {C}}^{\mathbb {N}}\) as
From the construction it is clear that for \(\alpha ,\beta \in l^2_{\text {fin}}({\mathbb {N}})\), the map \((y,x)\mapsto \langle \beta ,K_\Phi (y,x)\alpha \rangle _{l^2}\) is measurable. The next step is to show that for almost all \((y,x)\in Y\times X\) one has \(K_\Phi (y,x)\in \mathcal {B}(l^2({\mathbb {N}}),l^2({\mathbb {N}}))\). Since \(l^2_{\text {fin}}({\mathbb {N}})\) is dense in \(l^2({\mathbb {N}})\) one has
Moreover, let \(L^1_{\text {fin}}(X,l^2({\mathbb {N}}))\) be the set of functions \(f=(f_1,f_2,\ldots )\in L^1(X,l^2({\mathbb {N}}))\) with only finitely many nonzero \(f_j\), which is dense in \(L^1(X,l^2({\mathbb {N}}))\), and similarly for \(L^1_{\text {fin}}(Y,l^2({\mathbb {N}}))\). For any \(g\in L^1_{\text {fin}}(Y,l^2({\mathbb {N}}))\), \(f\in L^1_{\text {fin}}(X,l^2({\mathbb {N}}))\), we clearly have from the above
and with \(A= \big \{(g,f)\in L^1_{\text {fin}}(Y, l^2({\mathbb {N}}))\times L^1_{\text {fin}}(X,l^2({\mathbb {N}}))\big |\, \Vert g\Vert _{L^1(Y,l^2({\mathbb {N}}))}= \Vert f\Vert _{L^1(X,l^2({\mathbb {N}}))}=1 \big \}\), which is dense in \(L^1(Y, l^2({\mathbb {N}}))\times L^1(X,l^2({\mathbb {N}}))\), one sees
Thus the kernel \(K_\Phi (y,x)\) maps \(l^2({\mathbb {N}})\) boundedly into itself uniformly in \((y,x)\in Y\times X\). Taking limits, measurability of \((y,x)\mapsto \langle \beta ,K_\Phi (y,x)\alpha \rangle _{l^2}\) extends from \(\alpha ,\beta \in l^2_{\text {fin}}({\mathbb {N}})\) to all of \(l^2({\mathbb {N}})\). Thus \(K_\Phi \) is weakly, hence strongly measurable. From (B.10) one also gets \(\Phi = \Phi _{K_\Phi }\). In addition, the last bound together with (B.9) shows
so the map \(L^\infty _s\big (Y\times X, \mathcal {B}(l^2({\mathbb {N}}),l^2({\mathbb {N}}))\big )\ni K\mapsto \Phi _K\in \mathcal {B}\big (L^1(X,l^2({\mathbb {N}})), L^\infty (Y,l^2({\mathbb {N}}))\big )\) is an isometry. \(\square \)
Appendix C. Solution of an auxiliary minimization problem
In this section we introduce an auxiliary minimization problem \(Q_{\gamma }\) which on one hand can be solved explicitly, and on the other hand provides an upper bound on the minimization problem \(M_{\gamma }\) defined in (1.4).
Proposition C.1
For any \(\gamma >2\) the minimization problem
has the solution
Moreover, h is a minimizer if and only if \(h(s) = h_*(\lambda s)\) for arbitrary \(\lambda >0\), where
and \(K_{\alpha }\) denotes the modified Bessel function of the second kind with parameter \(\alpha \in (0,1)\).
Remark C.2
As the form of the minimization problem suggests, any minimizer should be decreasing and, using known properties of Bessel functions, one can see that the above minimiser \(h_*\) is strictly monotone decreasing.
The point is that the above minimization problem is quadratic, hence it can be solved by completing the square. First we make the change of coordinates \(s = t^{\frac{2}{\gamma }}\), which gives
This is immediate upon setting \(g(t) = h(t^{\frac{2}{\gamma }})\) in the above integrals. Defining the variational problem
we obtain
Hence, Proposition C.1 is a direct consequence of
Lemma C.3
For any \(\gamma >2\) and \(u>0\), the variational problem (C.3) has the solution
The unique minimizer is given by
Proof
Given a real Hilbert space \(\mathcal {H}\) with scalar product \(\langle \cdot ,\cdot \rangle \) and linear operators A, B on \(\mathcal {H}\) consider the functionals
and the associated constrained minimization problem
for \(u>0\). Note that directional derivatives of F and G are given by
when \(h,\varphi \) are in the domains of A and B, but we are, intentionally, a bit vague at this point concerning domain questions.
Assume that \(\psi \) is a weak solution of the Euler–Lagrange equation
for some \(\lambda \ge 0\) and all h, more precisely, all h in the intersection of the domains of A and B and also assume that \(\psi \) fulfils the constraint: \(G(\psi )=u\). Given an arbitrary \(\varphi \in \mathcal {H}\) with \(G(\varphi )=u\), we write it as \(\varphi =\psi +h\). Then
so, since \(\langle B\psi , B\psi \rangle =u\), we have \(2\langle Bh, B\psi \rangle = -\langle Bh, Bh\rangle \) and from (C.9) we get
Thus
so \(\psi \) is a mininimizer. Moreover, if equality holds, i.e., if \(F(\varphi )=F(\psi )\), then \( \langle Ah, Ah\rangle =0\), i.e., h is in the kernel of A, and if in addition \(\lambda >0\), the h is also in the kernel of B.
We apply the above with the choice
of real-valued functions on \(R_+\), which are square integrable w.r.t. the weighted Lebesgue measure \( t^{1-\frac{4}{\gamma }}\mathrm {d} t\). The operator B is the identity on \(L^2({\mathbb {R}}_+, t^{1-\frac{4}{\gamma }}\mathrm {d} t)\) and A is the (weak) derivative,
with domain \(\mathcal {D}(A) = \{\varphi \in L^2({\mathbb {R}}_+, t^{1-\frac{4}{\gamma }}\mathrm {d} t):\, \varphi '\in L^2({\mathbb {R}}_+, t^{1-\frac{4}{\gamma }}\mathrm {d} t)\} \).
In this setting, we have \(q_{u,\gamma }= \mathcal {Q}_u\). Integration by parts shows that the Euler–Lagrange equation is given by
which can be transformed into a modified Bessel differential equation upon setting \(g(t) = (t\sqrt{\lambda })^{\frac{2}{\gamma }} {\tilde{g}}(t \sqrt{\lambda })\). Then \({\tilde{g}}\) satisfies the modified Bessel equation
with solution space spanned by the modified Bessel functions \(I_{\frac{2}{\gamma }}, K_{\frac{2}{\gamma }}\). Using the well-known asymptotics of modified Bessel functions,Footnote 9 it is easy to see that the function
is the unique solution of (C.12) satisfying \(g_{\lambda }(0) = 1\) and \(\lim _{t\rightarrow \infty } g_{\lambda }(t) = 0\). We now use that for \(\alpha \in (0,1)\),
which together with the asymptotics of Bessel functions (see Footnote 9) implies that
Identity (C.13) follows from integration by parts and \(\frac{\mathrm {d}}{\mathrm {d}t}[\frac{t^2}{2} K_{1-\alpha }(t)K_{1+\alpha }(t)] = t^2 K_{\alpha }(t) K_{\alpha }'(t)\), which is a consequence of the relations \(K_{1-\alpha }(t) + K_{\alpha +1}(t) = -2 K_{\alpha }'(t)\) and \(K_{\alpha }'(t) = \frac{\alpha }{t} K_{\alpha }(t) - K_{\alpha +1}(t) = - \frac{\alpha }{t} K_{\alpha }(t) - K_{1-\alpha }(t)\) [38, Eqs. 10.29.1, 10.29.2] for modified Bessel functions. Hence, using that \(\Gamma (1+\alpha ) = \alpha \Gamma (\alpha )\) and \(\Gamma (\alpha )\Gamma (1-\alpha ) = \frac{\pi }{\sin (\pi \alpha )}\) for \(\alpha \in (0,1)\), we obtain
Note that (C.15) determines the relation between the Lagrange multiplier \(\lambda >0\) and the constraint \(u>0\) in (C.6). Similarly, again using the above relation for the derivative of the modified Bessel function \(K_{\alpha }\), we have
Therefore, (C.14) and the above functional equations for the \(\Gamma \)-function give
This proves (C.5). To show uniqueness of the minimizer \(g_\lambda \), note that since \(\lambda >0\), we have \(0=Bh=h\) if \(F(\varphi ) = F(g_\lambda )\) and \(G(\varphi )=u=G(g_\lambda )\) by (C.11). \(\square \)
Proposition C.4
For any \(\gamma >2\),
Proof
The choice \(m_1(s) = s \mathbf {1}_{\{0<s\le 1\}}\) in the minimization problem for \(M_{\gamma }\) gives
so
where we used that \(\int _0^{\infty } m_1(s)^2\,\frac{\mathrm {d}s}{s} = \frac{1}{2}\). Setting
it follows that
Moreover, h is absolutely continuous with \(h'(t) = -m_2(t^{-1})\), and
so h cannot oscillate too fast at infinity. Finiteness of \( \int _0^{\infty } t^{\gamma -2} h(t)^2 \, \frac{\mathrm {d}t}{t} \) then implies that \(h(t) \rightarrow 0\) as \(t\rightarrow \infty \). Indeed, let \(t_2\ge t_1\ge 1\). Then, using Cauchy–Schwarz,
where we also used \(\gamma >2\). Since \(0<t\mapsto t^{\gamma -2}h(t)^2\) is integrable on \((0,\infty )\) w.r.t. \(\frac{\mathrm {d}t}{t}\), the above bound shows that \(h(t)^2\) is Cauchy in the limit \(t\rightarrow \infty \). Hence \(\lim _{t\rightarrow \infty }h(t)^2\) exists. Moreover, using again that \(t\mapsto t^{\gamma -2}h(t)^2 \in L^1((0,\infty ), \frac{\mathrm {d}t}{t})\) and \(\gamma >2\), this forces \(\lim _{t\rightarrow \infty }h(t)^2=0\), i.e., \(\lim _{t\rightarrow \infty }h(t)=0\).
In addition, using again the Cauchy–Schwarz inequality, we have
so \(\lim _{t\rightarrow 0}h(t) = 1\). Hence,
for any absolutely continuous function \(h:\mathbb {R}_+ \rightarrow \mathbb {R}\) with \(h'\in L^2(\mathbb {R}_+;\frac{\mathrm {d}s}{s})\) satisfying the boundary conditions \(h(0) = 1\) and \(\lim _{s\rightarrow \infty } h(s) = 0\). The bound \(M_{\gamma }\le Q_{\gamma } \) follows by taking the infimum over these functions. \(\square \)
Appendix D. Numerical results
In this section we derive upper bounds on the the constants in Theorem 1.3 and 1.7, in particular, the constant \(C_{0,d}\) in the bound for the number of bound states of a non-relativistic one-particle Schrödinger operator from Corollary 1.1, given in Table 1.
Recall that the best constant in our approach is related to the minimization problem for
The choice of \(m_1,m_2\) is quite arbitrary. It is important, however, to have \(m_1*m_2(s)\sim s\) for small s, in order to make the integral \(\int _{0}^{\infty } \left( 1-\frac{(m_1 * m_2)(s)}{s}\right) ^2 \, s^{2-\gamma }\,\frac{\mathrm {d}s}{s}\) finite.
We reformulate the above problem by making the ansatz
where \(\xi ,\psi : {\mathbb {R}}_+ \rightarrow {\mathbb {R}}\) are such that \(\int _0^{\infty } \xi (r)\,\frac{\mathrm {d}r}{r} = \int _{0}^{\infty } \psi (r) \,\frac{\mathrm {d}r}{r} =1\).
Then the convolution of \(m_1\) and \(m_2\) is given by
and a short calculation, taking into account the above normalization of \(\xi \) and \(\psi \), shows
The \(L^2\)-norms of \(m_1, m_2\) can be expressed in terms of \(\xi \) and \(\psi \) by
and
Thus, an upper bound on \(M_{\gamma }\) can be obtained by minimizing the functional
over all functions \(\psi , \xi \in L^1({\mathbb {R}}_+,\frac{\mathrm {d}s}{s})\) satisfying the constraint
Finding the minimizer, even finding that a minimizer exists for the new minimization problem given by (D.2) and (D.3), is a very challenging problem, as challenging as for the original minimization problem. However, to get a reasonable upper bound on the minimal value, it suffices to take suitable trial functions. To get the constants given in Table 1, in our calculations, which where done with Mathematica, we used the following family of trial functions
with parameters \(\alpha , p, \beta , q >0\), i.e., Gamma distributions on \({\mathbb {R}}_+\).
The normalization condition is easily verified. For integer \(p,q \ge 1\), the calculation of \(I[\xi , \psi ]\) can be reduced to calculating the integral
as from J we can get \(I[\xi , \psi ]\) by taking derivatives,
Similarly, the “\(L^2\)-norm integrals” are given by
for \(q\in {\mathbb {N}}\) and \(\beta >1\), as well as
where
for \(\alpha >1\), \(p\in {\mathbb {N}}\).
In our numerical calculations with Mathematica, we made the choice \(p=2, q=3\), for dimensions \(d=3,4\), and optimized in the parameters \(\alpha , \beta >1\), while for dimensions \(d \ge 5\) the values were obtained with \(p=3, q=2\), and minimization in \(\alpha , \beta >1\). More specifically, we got the values in Table 1 by the choice of parameters listed in Table .
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Hundertmark, D., Kunstmann, P., Ried, T. et al. Cwikel’s bound reloaded. Invent. math. 231, 111–167 (2023). https://doi.org/10.1007/s00222-022-01144-7
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DOI: https://doi.org/10.1007/s00222-022-01144-7