1 Introduction

The study of spectral properties of non-self-adjoint Schrödinger operators \(-\,\Delta +V\) in \(L_2({{\mathbb {R}}}^d)\), with a complex-valued potential V, has attracted considerable attention in recent years. In particular, many works have been dedicated to the derivation of non-self-adjoint versions of the famous Lieb–Thirring inequalities (first considered by Lieb and Thirring for real-valued potentials in [37, 38]) and to the problem of finding good upper bounds on individual eigenvalues. Let us mention [4, 10,11,12, 18, 19, 21, 25, 26, 35, 45] as some references for the former topic and [1, 16,17,18, 20, 22, 35, 44] as some references for the latter.

While it is natural to study Schrödinger operators in the Hilbert space \(L_2({{\mathbb {R}}}^d)\), there also exist good reasons (see e.g., [46]) to consider them in \(L_p({{\mathbb {R}}}^d)\), for \(p \ne 2,\) as well. However, from a spectral perspective this is not interesting at all. Indeed, it has been shown in [30] that under weak assumptions on the potential V the \(L_p\)-spectra of self-adjoint Schrödinger operators coincide. Moreover, later results showed that this is the case in the non-self-adjoint setting as well (see [34, 39]). Even more is true: The fact that the underlying manifold is \({{\mathbb {R}}}^d\) does not play a role either. For instance, it was shown in [50] that the \(L_p\)-spectra of the Laplace–Beltrami operator on a complete Riemannian manifold M with Ricci curvature bounded from below are p-independent, provided that the volume of M grows at most sub-exponentially.

One of the simplest manifolds where the \(L_p\)-spectrum of the Laplace–Beltrami operator does depend on p is given by the hyperbolic plane \({{\mathbb {H}}}\). In the half-space model, this manifold is given by

$$\begin{aligned} {{\mathbb {H}}}=\{ (y,t) \in {{\mathbb {R}}}^{2} : y \in {{\mathbb {R}}}, 0< t < \infty \}, \end{aligned}$$

together with the conformal metric \( \hbox {d}s^2 =t^{-2} (\hbox {d}y^2+\hbox {d}t^2)\). It has been shown in [9] that the spectrum of \(-\,\Delta _p\) in \(L_p({{\mathbb {H}}}), 1 \le p < \infty ,\) consists of the parabolic sets

$$\begin{aligned} \Sigma _p:=\left\{ a+ib : a \ge 1/(pp') \text { and } b^2 \le (1-2/p)^2\left( a- {1}/(pp')\right) \right\} ; \end{aligned}$$
(1)

see Fig. 1. Here, \(p'\) denotes the conjugate exponent, i.e.,

$$\begin{aligned} 1/p +1/p'=1. \end{aligned}$$

In particular, the spectrum of the self-adjoint operator \(-\,\Delta _2\) consists of the interval \(\Sigma _2=[1/4,\infty )\) and in case \(p \ne 2\) the spectrum is the set of points on and inside the parabola with vertex \(\lambda =1/(pp')\) and focus \(\lambda =1/4\). We see that \(\Sigma _p=\Sigma _{p'}\), reflecting the fact that (up to a reflection on the real line) the spectra of \(-\,\Delta _p\) and its adjoint \(-\,\Delta _{p'}\) coincide. Moreover, let us remark that, while in case \(p=2\) the spectrum is clearly purely essential, it seems to be unknown whether the same is true for \(p \ne 2\) as well (we conjecture that it is).

Fig. 1
figure 1

Boundaries of \(\Sigma _p=\sigma (-\,\Delta _p)\) drawn for \(p=1\), some general \(p \in (1,2)\) and \(p=2\), respectively (from outer to inner)

Full size image

In the present paper, we will study the Schrödinger operator

$$\begin{aligned} H_p=-\,\Delta _p+V \quad \text {in} \quad L_p({{\mathbb {H}}}), \quad 1< p < \infty , \end{aligned}$$
(2)

given the assumption that

$$\begin{aligned} V \in L_r({{\mathbb {H}}}) \quad \text {for some} \quad r \ge \max (p,p'). \end{aligned}$$
(3)

We will see below that in this case the operator of multiplication by V is \(-\,\Delta _p\)-compact, and hence the essential spectra of \(H_p\) and \(-\,\Delta _p\) coincide. In particular, the topological boundary \(\partial \Sigma _p\), not containing any isolated points, belongs to the essential spectrum of both operators. While the essential spectrum is stable, other parts of the spectrum of \(-\,\Delta _p\) will change with the introduction of the perturbation V. In particular, the spectrum of \(H_p\) in \(\Sigma _p^c\) (the resolvent set of \(-\,\Delta _p\)) will consist of an at most countable number of discrete eigenvalues, which can accumulate at \(\partial \Sigma _p\) only. It is our aim to say more about the speed of this accumulation, and its dependence on p, by deriving suitable Lieb–Thirring-type inequalities. In addition, we will also provide some estimates on individual eigenvalues. Let us mention that in the case of Schrödinger operators with complex potentials in \({{\mathbb {R}}}^d\), it has been shown by Bögli [4] that there exist potentials V such that the discrete eigenvalues of \(-\,\Delta +V\) accumulate at every point of \(\sigma _\mathrm{ess}(-\,\Delta )=[0,\infty )\). We suspect, but do not prove in the present paper, that a similar result is true in the hyperbolic case as well.

As far as we can say, the present paper constitutes the first work on such topics in a non-Hilbert space context. Moreover, we think that our results are even new in the Hilbert space case \(p=2\), where the only existing articles we are aware of are [36, 40], respectively. Here, [36] considers the self-adjoint case only and provides bounds on the number of discrete eigenvalues of \(-\,\Delta _2+V\) in hyperbolic space of dimension \(d \ge 3\), whereas the abstract results of [40] also apply to complex-valued potentials and could, in principle, be used to obtain some estimates on the discrete eigenvalues of \(H_2\) in the half-plane \(\{\lambda \in {{\mathbb {C}}}: {\text {Re}}(\lambda ) < 1/4\}\). In contrast to this, the results we will derive in this paper will provide information on all discrete eigenvalues of \(H_p\) in \(\Sigma _p^c\).

While in the present paper we restrict ourselves to the two-dimensional hyperbolic plane, let us at least mention that in principle we can obtain results for higher-dimensional hyperbolic space as well. Indeed, our results rely on the explicit knowledge of the green kernel of \(-\,\Delta _p\), which is available in all dimensions (though it gets more complicated in case \(d \ge 4\)).

2 Main Results

This section contains the main results of this paper. We use some standard terminology concerning operators and spectra, which is reviewed in ‘Appendix A.1.’

2.1 Bounds on Eigenvalues

We begin with two results concerning the location of the discrete spectrum \(\sigma _d(H_p)\), starting with the case \(p=2\).

Theorem 2.1

Let \(2 \le r < \infty \) and \(V \in L_r({{\mathbb {H}}})\). If \(\lambda \in \sigma _d(H_2) \), then

$$\begin{aligned} {\text {dist}}(\lambda ,[1/4, \infty ))^{(r-1)} \left( 1+ \frac{1}{2|1/4-\lambda |^{1/2}} \right) \le 2^{3/2} C_0 \Vert V\Vert _r^r, \end{aligned}$$
(4)

where

$$\begin{aligned} C_0=\frac{1 + \frac{\pi }{2} \coth (\frac{\pi }{2} )}{4\pi } \approx 0.216. \end{aligned}$$
(5)

In particular, we see that (4) implies that the distance of the discrete eigenvalues to the essential spectrum is bounded above, i.e., for \(\lambda \in \sigma _d(H_2)\) we have

$$\begin{aligned} {\text {dist}}(\lambda ,[1/ 4, \infty )) \le 2^{3/(2(r-1))} C_0^{1/(r-1)} \Vert V\Vert _r^{r/(r-1)}. \end{aligned}$$

Remark 2.2

Given the same assumptions on V, for the Schrödinger operator \(-\,\Delta +V\) in \(L_2({{\mathbb {R}}}^2)\) it is even known that the imaginary part of a discrete eigenvalue needs to be small if its real part is large, see [18]. Whether a similar statement remains true on the hyperbolic plane is an open question.

In case \(p\ne 2\), the result we obtain is more complicated. For its statement, it is convenient to introduce \(\gamma _p \in [0,1/2]\) by setting

$$\begin{aligned} \gamma _p:= \frac{1}{2} \left| 1-\frac{2}{p}\right| . \end{aligned}$$
(6)

A short computation shows that \(\gamma _p^2=1/4-1/(pp')\), which is the focal length of the parabola \(\Sigma _p=\sigma (-\,\Delta _p)\) (the distance between focus and vertex). In particular, we see that \(\gamma _2=0\) and \(\gamma _p=\gamma _{p'}\).

Theorem 2.3

Let \(2< \max (p,p') \le r < \infty \) and suppose that \(V \in L_r({{\mathbb {H}}})\). If \(\lambda \in \sigma _d(H_p) \cap \Sigma _p^c\), then

$$\begin{aligned} \left( \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|1/4-\lambda |^{1/2}}\right) ^{2r-2}\left( 1 + \frac{|1/4-\lambda |^{1/2}}{8 {\text {dist}}(\lambda ,\Sigma _p)} \right) ^{2r\gamma _p+1} \le 16^{2r-2} C_0 \Vert V\Vert _r^r, \end{aligned}$$
(7)

where \(C_0\) is as defined in (5).

Since \(\gamma _p=\gamma _{p'}\) and \(\Sigma _p=\Sigma _{p'}\) we see that Theorem 2.3 provides the same bounds for the eigenvalues of \(H_p\) and \(H_{p'}\), respectively. This is not a coincidence but follows from the fact that \(H_p^*=H_{p'}\) and hence (up to a reflection on the real line) the spectra and discrete spectra of \(H_p\) and \(H_{p'}\) coincide. This will be proved in Proposition 3.3. The same phenomenon will be observed in other results of this paper.

Remark 2.4

We note that the term \(|\lambda - 1/4|\) in (7) does not play the same role as in (4), since in case \(p\ne 2\) the point 1 / 4 is in the interior of the spectrum.

While (7) puts some restrictions on the location of the discrete eigenvalues, we emphasize that in contrast to the case \(p=2\), in case \(p\ne 2\) the bound (7) does not imply that \({\text {dist}}(\lambda ,\Sigma _p)\) is bounded above for \(\lambda \in \sigma _d(H_p) \cap \Sigma _p^c\). Indeed, in this case (using that for \(p > 2\) we have \(|\lambda - 1/4| \asymp |\lambda |\) for \(\lambda \in \Sigma _p^c\)) we can only conclude that there exists a constant \(C=C(V,p,r)\) such that

$$\begin{aligned} {\text {dist}}(\lambda ,\Sigma _p) \le C |\lambda |^{1/2}, \quad \lambda \in \sigma _d(H_p)\cap \Sigma _p^c. \end{aligned}$$
(8)

We do not know whether this reflects a real difference between the two cases, or whether it is just an artifact of our method of proof.

2.2 Lieb–Thirring Inequalities

We now consider the speed of accumulation of discrete eigenvalues, again starting with the Hilbert space case \(p=2\). In the following estimate, we distinguish between discrete eigenvalues lying in a disk around 1 / 4 (with radius depending on V) and eigenvalues lying outside this disk.

Theorem 2.5

(\(p=2\)). Let \(2 \le r < \infty \) and \(V \in L_r({{\mathbb {H}}})\) . Let \((\lambda _j)\) denote an enumeration of the discrete eigenvalues of \(H_2\), each eigenvalue being counted according to its algebraic multiplicity. Then, for every \(\tau \in (0,1)\) there exist constants C and \(C'\), both depending on \(\tau \) and r, such that the following holds:

  1. (ia)

    If \(2 \le r \le 3-\tau \), then

    $$\begin{aligned} \sum _{|1/4-\lambda _j|^{r-3/2} \le (2\Vert V\Vert _r)^{r}} \left( \frac{{\text {dist}}(\lambda _j,[1/4,\infty ))}{|1/4 -\lambda _j|^{1/2}}\right) ^{r+\tau } \le C\cdot \Vert V\Vert _r^{\frac{r}{2r-3}(r+\tau )}. \end{aligned}$$
  2. (ib)

    If \(r>3-\tau \), then

    $$\begin{aligned} \sum _{|1/4-\lambda _j|^{r-3/2} \le (2\Vert V\Vert _r)^{r}} \frac{{\text {dist}}(\lambda _j,[1/4,\infty ))^{r+\tau }}{|1/4 -\lambda _j|^{3/2}} \le C\cdot \Vert V\Vert _r^{\frac{r}{2r-3}(2r-3+2\tau )}. \end{aligned}$$
  3. (ii)
    $$\begin{aligned} \sum _{|1/4-\lambda _j|^{r-3/2} >(2\Vert V\Vert _r)^{r} } \frac{{\text {dist}}(\lambda _j,[1/4,\infty ))^{r+\tau }}{|1/4 -\lambda _j|^{\frac{3+3\tau }{2}}} \le C' \cdot \Vert V\Vert _r^{\frac{r}{2r-3}(2r-3-\tau )}. \end{aligned}$$

Remark 2.6

The previous theorem has consequences for sequences \((E_j)\) of discrete eigenvalues converging to some \(E \in [1/4,\infty )\). For instance,

  • if \(E > 1/4\), then \(({\text {Im}}E_j) \in l_{r+\tau }\),

  • if \(E=1/4\) and \({\text {Re}}(E_j) \le 1/4\), then \((|E_j-1/4|) \in l_q,\) where

    $$\begin{aligned} q = \left\{ \begin{array}{ll} (r+\tau )/2, &{}\quad \text { if } 2 \le r \le 3-\tau , \\ r-3/2+\tau , &{}\quad \text { if } r > 3-\tau . \end{array}\right. \end{aligned}$$
    (9)

In particular, concerning sequences of eigenvalues converging to the bottom of the essential spectrum we obtain different results for \(r<3\) and \(r > 3\), respectively. Whether this reflects a real difference between the two cases is another interesting open question.

For the next result in case \(p \ne 2\) we again recall that \(1/(pp')\) is the vertex of \(\Sigma _p\).

Theorem 2.7

(\(p \ne 2\)). Let \(2< \max (p,p') \le r < \infty \) and \(V \in L_r({{\mathbb {H}}})\) . Let \((\lambda _j)\) denote an enumeration of the discrete eigenvalues of \(H_p\) in \(\Sigma _p^c\), each eigenvalue being counted according to its algebraic multiplicity. Moreover, set

$$\begin{aligned} k:=k(r,p):=r(2-2\gamma _p)-2 \quad (\in (2,\infty )), \end{aligned}$$

where \(\gamma _p\) is as defined in (6). Then, for every \(\tau \in (0,1)\) there exist \(0< \varepsilon _1,\varepsilon _2,\varepsilon _3< 4\tau \) and constants \(C,C'\), depending on \(\tau , r\) and p, such that the following holds:

  1. (i)
    $$\begin{aligned} \sum _{|\frac{1}{pp'}-\lambda _j|^{(k-1)/2} \le (2\Vert V\Vert _r)^{{r}}} {\text {dist}}(\lambda _j,\Sigma _p)^{k+\varepsilon _1} \le C \Vert V\Vert _r^r \left( \Vert V\Vert _r^{\frac{r}{k-1}}+\gamma _p\right) ^{k+1+\varepsilon _2}. \end{aligned}$$
  2. (ii)
    $$\begin{aligned} \sum _{|\frac{1}{pp'}-\lambda _j|^{(k-1)/2} > (2\Vert V\Vert _r)^{r}} \frac{{\text {dist}}(\lambda ,\Sigma _p)^{k+\varepsilon _1} }{(|\lambda -\frac{1}{pp'}|^{1/2}+2\gamma _p)^{k+1+\varepsilon _3}} \le C' \Vert V\Vert _r^r \left( \Vert V\Vert _r^{\frac{r}{k-1}}+\gamma _p\right) ^{-\tau }. \end{aligned}$$

Remark 2.8

We see that in contrast to the case \(p=2\) (where the parabola \(\Sigma _p\) degenerates to an interval) here we obtain the same information on all sequences of eigenvalues, independent of the fact whether they converge to the vertex \(1/(pp')\) or to a generic point of the boundary of \(\Sigma _p\). Still, also here differentiating between ‘small’ and ‘large’ eigenvalues has its value, since the estimate in (ii) also provides information on sequences \((E_j)\) of eigenvalues diverging to \(\infty \).

To see how the above estimates depend on p, let us assume that \(V \in L_r( {{\mathbb {H}}})\) for some fixed \(r > 2\) and let (without restriction) \(2<p\le r\). Suppose that \((E_j)\) is a sequence of discrete eigenvalues of \(H_p=-\,\Delta _p+V\) converging to some \(E \in \partial \Sigma _p\). Then, \(({\text {dist}}(E_j,\Sigma _p)) \in l_{k+\varepsilon _1}\), where

$$\begin{aligned} k = r(2-2\gamma _p)-2 = r(1+ 2/p)-2. \end{aligned}$$

In particular, we see that k decreases for increasing p. This can be interpreted as saying that the constraints on sequences of eigenvalues of \(H_p, 2 < p \le r,\) are getting more severe with increasing p and are maximal for \(p=r\), in which case \(({\text {dist}}(E_j,\Sigma _p)) \in l_{r+\varepsilon _1}\) (just like in the Hilbert space case).

Finally, let us emphasize that the results of Theorems 2.5 and 2.7 are not ‘continuous’ in p, but have a ‘discontinuity’ at \(p=2\). To see this, let \((E_j)\) again denote a sequence of eigenvalues of \(H_p\), converging to some \(E \in \partial \Sigma _p {\setminus } \{1/(pp')\}\). Then, in case \(p=2\) the sequence \(({\text {dist}}(E_j,\Sigma _p))\) is ‘almost’ in \(l_r\), while in case \(p=2+\varepsilon \) (with \(\varepsilon \) sufficiently small) it is only ‘almost’ in \(l_{2r-2}\). Since \(2r-2 > r\) for \(r>2\), the latter result is weaker than the former. Whether this discontinuity corresponds to a real phenomenon seems like a further interesting question for future research.

2.3 On Proofs and How the Paper Proceeds

The results in Sect. 2.1 will be proved using the Birman–Schwinger principle, which requires us to obtain good upper bounds on the norm of the Birman–Schwinger operator \(V(-\,\Delta _p-\lambda )^{-1}\). We will obtain such bounds via corresponding Schatten–von Neumann norm estimates (in case \(p=2\)) and summing norm estimates (in case \(p \ne 2\)), respectively.

The Lieb–Thirring estimates in Sect. 2.2 will be obtained using a method first introduced in [5, 13]: We will construct suitable holomorphic functions (perturbation determinants) whose zeros coincide with the eigenvalues of \(H_p\) and we will then use a complex analysis result of Borichev et al. [5] to study these zeros. While this method has been applied in many different cases for operators in Hilbert spaces (see the citations at the beginning of the introduction), we seem to be the first to apply it in a general Banach space context. In order to make this work, we will rely on a general theory of perturbation determinants in Banach spaces recently obtained in [27].

The paper will proceed as follows: In the next section, we will provide the precise definitions of \(-\,\Delta _p\) and \(H_p\), respectively, and we will derive and recall some of their properties. In Sect. 4, we will derive various norm estimates on the resolvent of \(-\,\Delta _p\) and on the Birman–Schwinger operator \(V(-\,\Delta _p-\lambda )^{-1}\). These results will be used in Sect. 5 to prove the results of Sect. 2.1. In Sect. 6, we will derive an abstract Lieb–Thirring estimate, which will be applied in Sect. 7 to prove the results of Sect. 2.2. The paper is concluded by an appendix with three parts: In the first part, we recall some standard results concerning operators and their spectra; in the second part, we review the theory of perturbation determinants in Banach spaces and we introduce the Schatten–von Neumann and (p, q)-summing ideals; finally, in the third part we recall some results from complex interpolation theory which are required in Sect. 4.

3 The Operators

Some standard results (and terminology) for operators and spectra used throughout this section are compiled in ‘Appendix A.1.’

3.1 The Hyperbolic Plane, Its Laplace–Beltrami Operator and Green’s Function

Most of the material in this section is taken from [8, Section 5.7] (see also [6]).

(a) As noted in the Introduction, in the half-space model the hyperbolic plane is described by

$$\begin{aligned} {{\mathbb {H}}}=\{ x=(y,t) \in {{\mathbb {R}}}^{2} : y \in {{\mathbb {R}}}, 0< t < \infty \}. \end{aligned}$$

Equipped with the conformal metric \( \hbox {d}s^2 =t^{-2} (\hbox {d}y^2+\hbox {d}t^2)\), it is a complete Riemannian manifold with volume element

$$\begin{aligned} d\mu (x):= t^{-2} \hbox {d}y\, \hbox {d}t. \end{aligned}$$

The Riemannian distance \(d(x,x')\) between two points \(x=(y,t),x'=(y',t')\) can be computed via the identity

$$\begin{aligned} {\cosh (d(x,x'))+1}= \frac{|y-y'|^2+(t+t')^2}{2tt'}. \end{aligned}$$

It is sometimes convenient to use the so-called geodesic polar coordinates, see, e.g., [52, Section 3.1]: We fix \(x_0 \in {{\mathbb {H}}}\) and identify \(x\in {{\mathbb {H}}}{\setminus } \{x_0\}\) with the pair

$$\begin{aligned} (r, \xi ) \in (0,\infty ) \times {{\mathbb {S}}}^1, \end{aligned}$$

where \(r:=d(x,x_0)\) and \(\xi \in {{\mathbb {S}}}^1\) denotes the unit vector at \(x_0\) which is tangent to the geodesic ray that starts at \(x_0\) and contains x. (Here, we identify the unit tangent space at \(x_0\) with the sphere \({{\mathbb {S}}}^1\).) The volume element in geodesic polar coordinates is given by

$$\begin{aligned} \sinh (r) \hbox {d}r \,\hbox {d}\xi , \end{aligned}$$

with \(d\xi \) denoting the surface measure on \({{\mathbb {S}}}^1\).

(b) The Laplace–Beltrami operator on \({{\mathbb {H}}}\) is given by

$$\begin{aligned} -\,\Delta = -\, t^2(\partial ^2/\partial y^2 +\partial ^2/\partial t^2). \end{aligned}$$

It is essentially self-adjoint on \(C_c^\infty ({{\mathbb {H}}})\) and so its closure (also denoted by \(-\,\Delta \)) is self-adjoint in \(L_2({{\mathbb {H}}})\). Since \(-\,\Delta \) is positive, \(\Delta \) generates a contraction semigroup \(e^{t\Delta }\) on \(L^2({{\mathbb {H}}}),\) which can be shown to be submarkovian (i.e., it is positivity preserving and a contraction on \(L_\infty ({{\mathbb {H}}})\)). In particular, this implies that \(e^{t\Delta }\) maps \(L_1({{\mathbb {H}}}) \cap L_\infty ({{\mathbb {H}}})\) into itself and \(e^{t\Delta }|_{L_1\cap L_\infty }\) can be extended to a submarkovian semigroup \(T_p(t)\) on \(L_p({{\mathbb {H}}})\) for every \(p \in [1,\infty ]\). Moreover, these semigroups are consistent, i.e., \(T_p(t)|_{L_p \cap L_q}=T_q(t)|_{L_p\cap L_q}\) for \(p\ne q\). In case that \(p \in [1,\infty )\) they are also strongly continuous. In the following, we denote the generator of \(T_p(t), 1 \le p < \infty ,\) by \(\Delta _p\) (in particular, \(-\,\Delta = -\,\Delta _2\)). Note that the domain of \(-\,\Delta _p\) coincides with the Sobolev space \(W_2^p({{\mathbb {H}}})\), see, e.g., [49] and [53, Section 7.4.5]. Identifying the adjoint space of \(L_p({{\mathbb {H}}})\) with \(L_{p'}({{\mathbb {H}}})\), the adjoint of \(-\,\Delta _p, 1<p<\infty ,\) is equal to \(-\,\Delta _{p'}\). The spectrum of \(-\,\Delta _p\) is equal to the set \(\Sigma _p\) defined in (1). Concerning the structure of the spectrum let us mention that for \(p>2\) each point in the interior of \(\Sigma _p\) is an eigenvalue, see [51].

Remark 3.1

In general, it seems to be unknown whether \(\sigma (-\,\Delta _p)\) is purely essential.

(c) For \(\lambda \in \varrho (-\,\Delta _p)=\Sigma _p^c\) the resolvent \((-\,\Delta _p-\lambda )^{-1}\) is an integral operator on \(L_p({{\mathbb {H}}})\) whose kernel (green function) \(G_\lambda (x,x')\) depends on the Riemannian distance \(d(x,x')\) only. In order to present an explicit formula for this kernel, it is convenient to first map \({{\mathbb {C}}}{\setminus } [1/4,\infty )\) conformally onto the half-plane \(\{ \lambda : {\text {Re}}(\lambda )> 1/2\}\) by setting

$$\begin{aligned} s:= s(\lambda )= {1}/2 + \sqrt{ {1}/4 - \lambda }, \qquad \lambda \in {{\mathbb {C}}}{\setminus } \left[ {1}/ 4,\infty \right) , \end{aligned}$$
(10)

i.e., \(\lambda =-\,s(s-1)\).

Remark 3.2

We note that throughout this article we use the branch of the square root on \({{\mathbb {C}}}{\setminus } (-\,\infty ,0]\) which has positive real part.

With \(d=d(x,x')\), we have

$$\begin{aligned} G_\lambda (x,x') = \frac{1}{2^{3/2} \pi } \int _d^\infty \frac{e^{-r(s-\frac{1}{2})}}{(\cosh (r)-\cosh (d))^{1/2}} \hbox {d}r, \end{aligned}$$
(11)

see, e.g., [9, Formula (2.13)].

3.2 The Schrödinger Operator

Let \(p > 1\) and \(V \in L_p({{\mathbb {H}}}) +L_\infty ({{\mathbb {H}}})\). We use the same symbol V to denote the maximal operator of multiplication by V in \(L_p({{\mathbb {H}}})\). The Sobolev embedding theorems, see e.g., [29], imply that \({\text {Dom}}(-\,\Delta _p)=W_2^p({{\mathbb {H}}}) \subset L_p(\mathcal {H}) \cap L_\infty (\mathcal {H})\) and hence the Schrödinger operator

$$\begin{aligned} H_p= -\,\Delta _p + V \end{aligned}$$

is well defined on \({\text {Dom}}(H_p) := {\text {Dom}}(-\,\Delta _p)\). We now assume that V satisfies the stronger assumption (3), i.e., \(V \in L_r({{\mathbb {H}}})\) for some \(r \ge \max (p,p')\). Then, in case \(p \ge 2\) we will prove in Theorem 4.11 that V is \(-\,\Delta _p\)-compact and hence \(H_p\) is closed and \(\sigma _\mathrm{ess}(H_p)=\sigma _\mathrm{ess}(-\,\Delta _p)\), see ‘Appendix A.1 (b).’

The case \(1<p<2\) can be reduced to the case \(p > 2\) with the help of the following proposition.

Proposition 3.3

Let \(1<p<2\) and suppose that \(V \in L_r({{\mathbb {H}}})\) for some \(r\ge p'=p/(p-1)\). Then, \( H_p = H_{p'}^*.\) In particular, up to a reflection on the real line the essential and discrete spectra of \(H_p\) and \(H_{p'}\) coincide.

Note that some standard properties of the adjoint operator are reviewed in ‘Appendix A.1 (c).’

Proof of the proposition

Just for this proof let us write \(V_p\) for the maximal operator of multiplication by V in \(L_p({{\mathbb {H}}})\), so \(V_p^*=V_{p'}\). Since \(0 \in \varrho (-\,\Delta _{p'})\), we then have

$$\begin{aligned} H_{p'} = (-\,\Delta _{p'}+ V_{p'})=(I+V_{p'}(-\,\Delta _{p'})^{-1})(-\,\Delta _{p'}). \end{aligned}$$

Since \(p'>2\) by the previous discussion (or Theorem 4.11) the operator \((I+V_{p'}(-\,\Delta _{p'})^{-1})\) is bounded on \(L_{p'}({{\mathbb {H}}})\), so we obtain

$$\begin{aligned} H_{p'}^*= & {} (-\,\Delta _{p'})^*(I+V_{p'}(-\,\Delta _{p'})^{-1})^* = (-\,\Delta _{p})(I+[V_{p'}(-\,\Delta _{p'})^{-1}]^*). \end{aligned}$$

Now, general theory only allows us to conclude that \([V_{p'}(-\,\Delta _{p'})^{-1}]^* \supset (-\,\Delta _p)^{-1}V_p\). However, since the operator on the left-hand side of this inclusion is bounded on \(L_{p}\) (even compact), it coincides with the closure of the operator on the right-hand side and hence \(H_{p'}^* = (-\,\Delta _{p})(I+\overline{(-\,\Delta _{p})^{-1}V_p})\). But here the domain of the product on the right is equal to \({\text {Dom}}(-\,\Delta _p)\) and on this set the operators \(\overline{(-\,\Delta _{p})^{-1}V_p}\) and \((-\,\Delta _{p})^{-1}V_p\) coincide. So finally we see that

$$\begin{aligned} H_{p'}^* = (-\,\Delta _{p})(I+(-\,\Delta _{p})^{-1}V_p) = -\,\Delta _p + V_p = H_p. \end{aligned}$$

\(\square \)

4 A Variety of Estimates

In this section, we derive various estimates on the resolvent and the resolvent kernel of \(-\,\Delta _p\) and on the Birman–Schwinger operator \(V(-\,\Delta _p-\lambda )^{-1}\). To this end, it will be necessary to first map the resolvent set \(\Sigma _p^c = \varrho (-\,\Delta _p)\) conformally onto the right half-plane

$$\begin{aligned} {{\mathbb {C}}}_+ := \{ \lambda \in {{\mathbb {C}}}: {\text {Re}}(\lambda ) > 0\}. \end{aligned}$$

Since (1) shows that \(\Sigma _p^c\) is just the set outside the parabola parameterized by \({{\mathbb {R}}}\ni t \mapsto 1/(pp') + t^2+it(1-2/p)\), such a conformal map (or rather its inverse) is given by

$$\begin{aligned} \Psi _p: {{\mathbb {C}}}_+\rightarrow & {} \Sigma _p^c \nonumber \\ z\mapsto & {} 1/(pp')-z^2-z|1-2/p|=:\lambda . \end{aligned}$$
(12)

Using \(\gamma _p=1/2|1-2/p|\) as defined in (6), a short calculation shows that

$$\begin{aligned} \lambda =\Psi _p(z) = 1/4-(z+\gamma _p)^2 \end{aligned}$$
(13)

and

$$\begin{aligned} z = \Psi _p^{-1}(\lambda ) = - \gamma _p+\sqrt{1/4-\lambda }. \end{aligned}$$
(14)

We note that with \(s=s(\lambda )\) as defined in (10) we have

$$\begin{aligned} s=\Psi _p^{-1}(\lambda )+ 1/2 +\gamma _p, \quad \lambda \in \Sigma _p^c. \end{aligned}$$
(15)

The following lemma will allow us to freely switch between estimates in terms of \(\lambda \) and z, respectively.

Lemma 4.1

Let \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _p(z), 1 \le p < \infty \).

  1. (i)

    If \(p=2\), then

    $$\begin{aligned} |z| {\text {Re}}(z) \le {\text {dist}}(\lambda ,[1/4,\infty )) \le 2 {\text {Re}}(z) |z|. \end{aligned}$$
    (16)
  2. (ii)

    If \(p \ne 2\), then

    $$\begin{aligned} \frac{|z+\gamma _p| {\text {Re}}(z)}{4} \le {\text {dist}}(\lambda ,\Sigma _p) \le 16 |z+\gamma _p| {\text {Re}}(z). \end{aligned}$$
    (17)

Proof

(i) In case \(p=2\), a short computation shows that with \(\lambda = 1/4-z^2\):

$$\begin{aligned} {\text {dist}}(\lambda ,[1/4,\infty )) = \left\{ \begin{array}{ll} |z|^2, &{}\quad |{\text {Im}}(z)| \le {\text {Re}}(z), \\ 2 {\text {Re}}(z) |{\text {Im}}(z)|, &{}\quad |{\text {Im}}(z)| > {\text {Re}}(z). \\ \end{array}\right. \end{aligned}$$

Since \(|z| \ge {\text {Re}}(z)\), and \(\sqrt{2}|{\text {Im}}(z)| > |z|\) if \(|{\text {Im}}(z)| > {\text {Re}}(z)\), we obtain the lower bound in (16). Similarly, since \(|z| \le \sqrt{2} |{\text {Re}}(z)|\) if \(|{\text {Im}}(z)| \le {\text {Re}}(z)\), and since \(|{\text {Im}}(z)| \le |z|\), we obtain the upper bound as well.

(ii) In case \(p \ne 2\), we proceed more indirectly. Let \(\phi : {{\mathbb {D}}}\rightarrow {{\mathbb {C}}}_+\) denote an arbitrary conformal mapping between the unit disk \({{\mathbb {D}}}=\{w \in {{\mathbb {C}}}: |w| \le 1\}\) and the right half-plane. Then, the Koebe distortion theorem (see [43], page 9) implies that with \(z = \phi (w):\)

$$\begin{aligned} \frac{1}{4} |\phi '(w)| (1-|w|) \le {\text {Re}}(z)={\text {dist}}(z,\partial {{\mathbb {C}}}_+) \le 2 |\phi '(w)| (1-|w|). \end{aligned}$$
(18)

The function \(f:= \Psi _p \circ \phi : {{\mathbb {D}}}\rightarrow \Sigma _p^c\) is conformal as well, so applying the distortion theorem a second time we obtain with \(\lambda =\Psi _p(\phi (w))=\Psi _p(z)\):

$$\begin{aligned} \frac{1}{4} |f'(w)| (1-|w|) \le {\text {dist}}(\lambda ,\Sigma _p) \le 2 |f'(w)| (1-|w|). \end{aligned}$$
(19)

But \(f'(w)= \Psi _p'(z)\cdot \phi '(w)\), so (18) and (19) together imply that

$$\begin{aligned} {\text {dist}}(\lambda ,\Sigma _p) \le 2 |\Psi _p'(z)| |\phi '(w)| (1-|w|) \le 8 |\Psi _p'(z)| {\text {Re}}(z). \end{aligned}$$

Since \(\Psi _p'(z)=-\,2(z+\gamma _p)\), this proves the upper bound in (17). The lower bound is proved similarly. \(\square \)

4.1 Kernel Estimates

In the following, we present a series of estimates on the green function \(G_\lambda (.,.)\) defined in (11), starting with the following one due to Elstrodt. As above, we write \(\Vert .\Vert _p\) for the norm in \(L_p({{\mathbb {H}}})\).

Proposition 4.2

([15], Corollary 7.3. (see also [14])). For \(\lambda \in {{\mathbb {C}}}{\setminus } [\frac{1}{4}, \infty )\) and \(s=s(\lambda )\) as defined in (10), the following holds:

$$\begin{aligned} \sup _{x \in {{\mathbb {H}}}} \Vert G_\lambda (x,.)\Vert _{2}^2 \le \left\{ \begin{array}{ll} \frac{|{\text {Im}}(\psi (s))|}{2 \pi |{\text {Im}}(\lambda )|}, &{}\quad \lambda \in {{\mathbb {C}}}{\setminus } {{\mathbb {R}}}, \\ \frac{\psi '(s)}{4 \pi (s-\frac{1}{2})}, &{}\quad \lambda \in {{\mathbb {R}}}. \end{array}\right. \end{aligned}$$
(20)

Here, \(\psi (s)=\frac{d}{ds} \ln (\Gamma (s))\) denotes the Digamma function.

It is convenient to rewrite this estimate as follows.

Corollary 4.3

For all \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _2(z)=\frac{1}{4} -z^2\), we have

$$\begin{aligned} \sup _{x \in {{\mathbb {H}}}} \Vert G_\lambda (x,.)\Vert _{2}^2 \le C_0 \frac{1}{|z+\frac{1}{2}| ({\text {Re}}(z))}, \end{aligned}$$
(21)

where \(C_0\) is as defined in (5).

Proof of the corollary

Let us first consider the case \(\lambda \in {{\mathbb {C}}}{\setminus } {{\mathbb {R}}}, \lambda =-\,s(s-1)\). Since

$$\begin{aligned} \psi (s) = -\,\gamma + \sum _{k=1}^\infty \left( \frac{1}{k} - \frac{1}{k+s-1}\right) , \end{aligned}$$

where \(\gamma \) is the Euler–Mascheroni constant (see [2, Formula 6.3.16]), we can use the fact that \({\text {Re}}(s) > 1/2\) to obtain that

$$\begin{aligned} |{\text {Im}}(\psi (s))|= & {} |{\text {Im}}(s)| \sum _{k=0}^\infty \frac{1}{(k+{\text {Re}}(s))^2 + {\text {Im}}(s)^2} \le |{\text {Im}}(s)| \sum _{k=0}^\infty \frac{1}{k^2+|s|^2} \\= & {} \frac{|{\text {Im}}(s)|}{2} \left( \frac{1}{|s|^2} + \frac{\pi \coth (\pi |s|)}{|s|}\right) \le \frac{|{\text {Im}}(s)|}{2|s|} \left( 2 + \pi \coth \left( \frac{\pi }{2} \right) \right) . \end{aligned}$$

Moreover, a short computation shows that

$$\begin{aligned} {\text {Im}}(\lambda )= 2 {\text {Im}}(s) \left( \frac{1}{2}-{\text {Re}}(s)\right) . \end{aligned}$$

Hence, for \(\lambda \in {{\mathbb {C}}}{\setminus } {{\mathbb {R}}}\) we obtain that

$$\begin{aligned} \frac{|{\text {Im}}(\psi (s))|}{2 \pi |{\text {Im}}(\lambda )|} \le \frac{1 + \frac{\pi }{2} \coth (\frac{\pi }{2} )}{4\pi } \frac{1}{|s| ({\text {Re}}(s)-\frac{1}{2})}. \end{aligned}$$
(22)

Similarly, for \(\lambda < 1/ 4\) (and hence \(s>1/2\)) we use that

$$\begin{aligned} \psi '(s) = \sum _{k=0}^\infty \frac{1}{(s+k)^2} \le \sum _{k=0}^\infty \frac{1}{s^2+k^2} \le \frac{1}{s} \left( 1 + \frac{\pi }{2} \coth \left( \frac{\pi }{2}\right) \right) \end{aligned}$$

to obtain

$$\begin{aligned} \frac{\psi '(s)}{4 \pi (s-\frac{1}{2})} \le \frac{1 + \frac{\pi }{2} \coth (\frac{\pi }{2} )}{4\pi } \frac{1}{s(s-\frac{1}{2})}, \qquad \lambda < 1/4. \end{aligned}$$
(23)

Taking into account that by (15) we have \(s=z+\frac{1}{2}\), the estimates (22), (23) and (20) conclude the proof. \(\square \)

Now, we estimate the \(L_1\)-norm of the Green function.

Lemma 4.4

For all \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _1(z)=\frac{1}{4} -(z+\frac{1}{2})^2\), we have

$$\begin{aligned} \sup _{x \in {{\mathbb {H}}}} \Vert G_\lambda (x,.)\Vert _{1} \le \frac{1}{{\text {Re}}(z)({\text {Re}}(z)+1)}. \end{aligned}$$
(24)

Proof

We note that with \(d=d(x,x')\) we obtain from (11) and (15) that

$$\begin{aligned} G_\lambda (x,x') = \frac{1}{2^{\frac{3}{2}} \pi } \int _d^\infty \frac{e^{-a(z+\frac{1}{2})}}{(\cosh (a)-\cosh (d))^{1/2}} \hbox {d}a. \end{aligned}$$

Switching to geodesic polar coordinates, centered at x, we can thus compute

$$\begin{aligned} \Vert G_\lambda (x,.)\Vert _{1}= & {} \int _{{\mathbb {H}}}\mu (dx') |G_\lambda (x,x')| \\\le & {} \frac{1}{2^{\frac{3}{2}} \pi } \int _{{{\mathbb {S}}}^1} \hbox {d}\xi \int _0^\infty \hbox {d}r \sinh (r) \int _r^\infty \hbox {d}a \frac{e^{-a({\text {Re}}(z)+\frac{1}{2})}}{(\cosh (a)-\cosh (r))^{1/2}} \\= & {} \frac{1}{2^{\frac{1}{2}}} \int _0^\infty \hbox {d}a\, e^{-a({\text {Re}}(z)+\frac{1}{2})}\int _0^a \hbox {d}r \frac{\sinh (r) }{(\cosh (a)-\cosh (r))^{1/2}} \\= & {} \, 2^{\frac{1}{2}} \int _0^\infty \hbox {d}a\, e^{-a({\text {Re}}(z){+}\frac{1}{2})} (\cosh (a)-1)^{1/2}\\= & {} \int _0^\infty \hbox {d}a\, e^{-a({\text {Re}}(z){+}\frac{1}{2})} (e^{a/2}-e^{-a/2}) \\= & {} \frac{1}{{\text {Re}}(z)} - \frac{1}{{\text {Re}}(z)+1} = \frac{1}{{\text {Re}}(z)({\text {Re}}(z)+1)}. \end{aligned}$$

Here, in the last equality between the integrals, we used that \(2(\cosh (a)-1)=(e^{a/2}-a^{-a/2})^2\). \(\square \)

Finally, we generalize the previous two lemmas using complex interpolation; see ‘Appendix A.3.’

Lemma 4.5

Let \(1 \le p < 2\). Then, for \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _p(z)=\frac{1}{4} -(z+\gamma _p)^2\) we have

$$\begin{aligned} \sup _{x \in {{\mathbb {H}}}} \Vert G_\lambda (x,.)\Vert _{p} \le C_0^{1-1/p} \left( \frac{1}{{\text {Re}}(z)({\text {Re}}(z)+\frac{1}{2})}\right) ^{1/p}, \end{aligned}$$
(25)

where \(C_0\) was defined in (5).

Proof

We use the terminology of Appendix A.3. Let \(S:=\{ w : 0 \le {\text {Re}}(w) \le 1\}\). For fixed \(x \in {{\mathbb {H}}}\) and \(z \in {{\mathbb {C}}}_+\), we consider the function

$$\begin{aligned} f : S \rightarrow L_2({{\mathbb {H}}})+L_1({{\mathbb {H}}}), \quad w \mapsto G_{\frac{1}{4} - (z + \frac{1}{2} w)^2}(x,.) \end{aligned}$$

The explicit expression (11) for the kernel and our above estimates show that this function is in \(\mathcal G(L_2({{\mathbb {H}}}),L_1({{\mathbb {H}}}))\), i.e., it is continuous and bounded on S and analytic in the interior of S. Moreover, by (21)

$$\begin{aligned} A_0^2 := \sup _{y \in {{\mathbb {R}}}} \Vert f(iy)\Vert _{2}^2 \le C_0 \sup _{y \in {{\mathbb {R}}}}\frac{1}{|z+i\frac{y}{2} +\frac{1}{2}| ({\text {Re}}(z+i \frac{y}{2}))} {=} C_0 \frac{1}{({\text {Re}}(z){+}\frac{1}{2})({\text {Re}}(z))} \end{aligned}$$

and by (24)

$$\begin{aligned} A_1 := \sup _{y \in {{\mathbb {R}}}} \Vert f(1+iy)\Vert _{1} \le \sup _{y \in {{\mathbb {R}}}} \frac{1}{{\text {Re}}(z+i \frac{y}{2})({\text {Re}}(z+ i \frac{y}{2})+1)} {=} \frac{1}{{\text {Re}}(z)({\text {Re}}(z){+}1)}. \end{aligned}$$

Hence, from Proposition A.4 we obtain that for \(\theta \in (0,1)\) and \(1/p= (1-\theta )/2+\theta \) we have \(f(\theta ) \in L_p({{\mathbb {H}}})=[L_2({{\mathbb {H}}}),L_1({{\mathbb {H}}})]_\theta \) and

$$\begin{aligned} \Vert f(\theta )\Vert _{p} =\Vert f(\theta )\Vert _{[L_2,L_1]_\theta } \le A_0^{1-\theta }A_1^\theta . \end{aligned}$$

But using that \(\theta = 2/p-1, \gamma _p=1/p-1/2\) and \(\lambda =\frac{1}{4} - (z+\gamma _p)^2=\frac{1}{4} - (z+\frac{1}{2} \theta )^2\) the last bound translates into

$$\begin{aligned} \Vert G_\lambda (x,.)\Vert _{p}\le & {} C_0^{(1-1/p)} \left( \frac{1}{({\text {Re}}(z){+}\frac{1}{2})({\text {Re}}(z))}\right) ^{(1-1/p)} \left( \frac{1}{{\text {Re}}(z)({\text {Re}}(z){+}1)}\right) ^{2/p-1} \\\le & {} C_0^{(1-1/p)} \left( \frac{1}{{\text {Re}}(z)({\text {Re}}(z)+\frac{1}{2})}\right) ^{1/p}, \end{aligned}$$

where in the last step we used that \({\text {Re}}(z)+1 > {\text {Re}}(z)+1/2\). \(\square \)

4.2 A Resolvent Norm Estimate

We continue with an estimate on the operator norm of the resolvents of \(-\,\Delta _p\). Here and in the following, we write \(\Vert T\Vert _{p,q}\) for the operator norm of \(T:L_p({{\mathbb {H}}}) \rightarrow L_q({{\mathbb {H}}})\).

Lemma 4.6

Let \(1 \le p < \infty \) and let \(z \in {{\mathbb {C}}}_+\) and \(\lambda = \Psi _p(z)= \frac{1}{4} - (z+\gamma _p)^2\).

  1. (i)

    If \(p=2\), then

    $$\begin{aligned} \Vert (-\,\Delta -\lambda )^{-1}\Vert _{2,2} = \frac{1}{{\text {dist}}(\lambda ,[1/4,\infty ))} \le \frac{1}{|z| {\text {Re}}(z)}. \end{aligned}$$
  2. (ii)

    If \(p \ne 2\), then

    $$\begin{aligned} \Vert (-\,\Delta _p-\lambda )^{-1}\Vert _{p,p} \le \left( \frac{1}{({\text {Re}}(z))^{2-2\gamma _p}(1+{\text {Re}}(z))^{2\gamma _p}} \right) . \end{aligned}$$

Proof

(i) The identity follows from the fact that \(-\,\Delta =-\,\Delta _2\) is self-adjoint with \(\sigma (-\,\Delta _2)=[1/4,\infty )\). The inequality follows from Lemma 4.1 (i).

(ii) In case \(p=1\), we can use Lemma 4.4 to compute for \(\lambda = \frac{1}{4} - (z+\frac{1}{2})^2\)

$$\begin{aligned} \Vert (-\,\Delta _1-\lambda )^{-1}\Vert _{1,1} \le \sup _{x \in {{\mathbb {H}}}} \Vert G_\lambda (.,x)\Vert _{1} = \sup _{x \in {{\mathbb {H}}}} \Vert G_\lambda (x,.)\Vert _{1} \le \frac{1}{{\text {Re}}(z)({\text {Re}}(z)+1)}.\nonumber \\ \end{aligned}$$
(26)

Now, we treat the case \(1<p<2\) by interpolation (see again Appendix A.3): Let \(S=\{ w \in {{\mathbb {C}}}: 0 \le {\text {Re}}(w) \le 1\}\) and fix \(z \in {{\mathbb {C}}}_+\). Define

$$\begin{aligned} T_w:= \left( -\,\Delta -\frac{1}{4} + \left( z+\frac{1}{2} w\right) ^2\right) ^{-1}. \end{aligned}$$

Then, for all simple functions \(f,g: {{\mathbb {H}}}\rightarrow {{\mathbb {C}}}\) the function

$$\begin{aligned} S \ni w\mapsto & {} \int _{{\mathbb {H}}}T_wf(x) g(x) \mu (dx) \end{aligned}$$

is continuous and bounded on S and analytic in the interior of S. Moreover, for every simple function f we have

$$\begin{aligned} \sup _{y \in {{\mathbb {R}}}} \Vert T_{iy}f\Vert _{2}&\le \Vert f\Vert _2 \sup _{y \in {{\mathbb {R}}}} \Vert \left( -\,\Delta -\frac{1}{4} + (z+\frac{1}{2} (iy))^2\right) ^{-1}\Vert _{2,2} \\&\le \Vert f\Vert _2 \sup _{y \in {{\mathbb {R}}}} \frac{1}{|z+i\frac{y}{2}|{\text {Re}}(z+i \frac{y}{2})} = \Vert f\Vert _2 \frac{1}{({\text {Re}}(z))^2} \end{aligned}$$

and

$$\begin{aligned} \sup _{y \in {{\mathbb {R}}}} \Vert T_{1+iy}f\Vert _{1}&\le \Vert f\Vert _1 \sup _{y \in {{\mathbb {R}}}} \Vert \left( -\,\Delta _1-\frac{1}{4} + (z+\frac{1}{2}(1+iy))^2\right) ^{-1}\Vert _{1,1} \\&\le \Vert f\Vert _1 \sup _{y \in {{\mathbb {R}}}} \frac{1}{{\text {Re}}(z+i\frac{y}{2})({\text {Re}}(z+i\frac{y}{2})+1)}\\&= \Vert f\Vert _1 \frac{1}{{\text {Re}}(z)(1+{\text {Re}}(z))}. \end{aligned}$$

Hence, the Stein interpolation theorem (Theorem A.6) implies that for \(\theta \in (0,1)\) and \(\frac{1}{p} = (1-\theta ) \frac{1}{2} + \theta \), the operator \(T_\theta \) extends to a bounded operator on \(L_p({{\mathbb {H}}})\) satisfying

$$\begin{aligned} \Vert T_\theta \Vert _{p,p} \le \left( \frac{1}{({\text {Re}}(z))^2}\right) ^{1-\theta }\left( \frac{1}{{\text {Re}}(z)(1+{\text {Re}}(z))}\right) ^\theta . \end{aligned}$$

Since \(1<p<2\), we have \(\theta = (2/p-1)=2\gamma _p\) (see Definition (6)) and \(\lambda =\frac{1}{4} -(z+\gamma _p)^2=\frac{1}{4} -(z+\frac{1}{2} \theta )^2\). Hence, the previous estimate implies

$$\begin{aligned} \Vert (-\,\Delta _p-\lambda )^{-1} \Vert _{p,p} \le \frac{1}{({\text {Re}}(z))^{2-2\gamma _p}(1+{\text {Re}}(z))^{2\gamma _p}}. \end{aligned}$$

Finally, the case \(p>2\) follows by duality using the fact that \(\gamma _p= \gamma _{p'}\). \(\square \)

4.3 Summing Norm Estimates

In this section, \((\Pi _r, \Vert .\Vert _{\Pi _r})\) and \((\Pi _{r,q},\Vert .\Vert _{\Pi _{r,q}})\) denote the r-summing and (r, q)-summing operators on \(L_p({{\mathbb {H}}})\), respectively. Some properties of these operator ideals are reviewed in ‘Appendix A.2’ (see Examples A.1 and A.2, in particular).

Lemma 4.7

Let \(p \ge 2\), \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _p(z)\). If \(V \in L_\infty ({{\mathbb {H}}})\), then

$$\begin{aligned} \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{p,p} \le \Vert V\Vert _\infty \cdot \left\{ \begin{array}{ll} \frac{1}{{\text {dist}}(\lambda ,[1/4,\infty ))}, &{}\quad p=2, \\ \frac{1}{({\text {Re}}(z))^{1+\frac{2}{p}}(1+{\text {Re}}(z))^{1-\frac{2}{p}}}, &{}\quad p > 2. \end{array}\right. \end{aligned}$$
(27)

Proof

This follows from Lemma 4.6 and the fact that

$$\begin{aligned} \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{p,p} \le \Vert V\Vert _\infty \Vert (-\,\Delta _p-\lambda )^{-1}\Vert _{p,p}. \end{aligned}$$

\(\square \)

Lemma 4.8

Let \(p \ge 2\), \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _p(z)\). If \(V \in L_p({{\mathbb {H}}})\), then \(V(-\,\Delta _p-\lambda )^{-1} \in \Pi _p(L_p({{\mathbb {H}}}))\) and

$$\begin{aligned} \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{\Pi _p}^p \le C_0 \cdot \Vert V\Vert _p^p \cdot \left\{ \begin{array}{ll} \frac{1}{|z+\frac{1}{2}| ({\text {Re}}(z))}, &{}\quad p =2 \\ \left( \frac{1}{{\text {Re}}(z)({\text {Re}}(z) + \frac{1}{2})} \right) ^{p/p'}, &{}\quad p > 2, \end{array}\right. \end{aligned}$$
(28)

where \(C_0\) was defined in (5).

Proof

Since \(V(-\,\Delta _p-\lambda )^{-1}\) is an integral op. with kernel \(k(x,x')=V(x)G_\lambda (x,x')\), its p-summing norm can be computed as follows (see, e.g., [32, Thm.3.a.3 and its proof]):

$$\begin{aligned} \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{\Pi _p}^p&\le \int _{{\mathbb {H}}}\mu (dx) \left( \int _{{\mathbb {H}}}\mu (dx') |V(x)G_\lambda (x,x')|^{p'} \right) ^{p/p'} \\&\le \Vert V\Vert _p^p \sup _{x \in {{\mathbb {H}}}} \Vert G_\lambda (x,.)\Vert _{L_{p'}}^p \\&\le C_0 \Vert V\Vert _p^p \cdot \left\{ \begin{array}{ll} \frac{1}{|z+\frac{1}{2}| ({\text {Re}}(z))}, &{}\quad p =2 \\ \left( \frac{1}{{\text {Re}}(z)({\text {Re}}(z) +\frac{1}{2})} \right) ^{p/p'}, &{}\quad p > 2. \end{array}\right. \end{aligned}$$

Here, in the last inequality, we used (21) and (25), respectively. \(\square \)

Now, we are going to interpolate between the results of the last two lemmas to obtain a result for \(V \in L_r({{\mathbb {H}}}), p< r < \infty .\) We will need the following result of Pietsch and Triebel concerning the complex interpolation spaces of the Schatten–von Neumann and absolutely summing ideals, respectively. We refer again to ‘Appendix A.3’ for the notation and terminology.

Proposition 4.9

([42]). Let \(\mathcal {H}\) and X denote complex Hilbert and Banach spaces, respectively. Moreover, let \(p \ge 1, 0< \theta < 1\) and define \(r>p\) by \( \frac{1}{r} = \frac{\theta }{p}\). Then, the following holds:

  1. (i)

    \([\mathcal {B}(\mathcal {H}),{\mathcal {S}}_p(\mathcal {H})]_\theta = {\mathcal {S}}_{r}(\mathcal {H})\) and \(\Vert T\Vert _{[\mathcal {B},{\mathcal {S}}_p]_\theta }=\Vert T\Vert _{{\mathcal {S}}_r}\) for \(T \in {\mathcal {S}}_r(\mathcal {H})\).

  2. (ii)

    \( [\mathcal {B}(X), \Pi _p(X)]_\theta \subset \Pi _{r,p}(X)\) and

    $$\begin{aligned} \Vert T\Vert _{\Pi _{r,p}} \le \Vert T\Vert _{[\mathcal {B},\Pi _p]_\theta } \text { for } T \in [\mathcal {B}(X),\Pi _p(X)]_\theta . \end{aligned}$$

Remark 4.10

We recall that for \(p=2\) and \(r \ge 2\) we have \(\Pi _{r,2}(\mathcal {H})=S_r(\mathcal {H})\) and \(\Vert .\Vert _{\Pi _{r,2}}=\Vert .\Vert _{\mathcal S_r}\), see [41, Prop. 2.11.28].

Theorem 4.11

Let \(2 \le p \le r < \infty \), \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _p(z)=1/4-(z+\gamma _p)^2\). If \(V \in L_r({{\mathbb {H}}})\), then \(V(-\,\Delta _p-\lambda )^{-1} \in \Pi _{r,p}(L_p({{\mathbb {H}}}))\) and

$$\begin{aligned} \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{\Pi _{r,p}}\le & {} C_0^{1/r} \Vert V\Vert _r \cdot \left\{ \begin{array}{ll} \left( \frac{1}{{\text {dist}}(\lambda ,[1/4,\infty ))}\right) ^{1-2/r} \left( \frac{1}{|z+\frac{1}{2}| ({\text {Re}}(z))} \right) ^{1/r}, &{}\quad p=2, \\ \left( \frac{1}{{\text {Re}}(z)}\right) ^{1+2/p-3/r} \left( \frac{1}{{\text {Re}}(z) + \frac{1}{2}} \right) ^{1-2/p+1/r}, &{}\quad p > 2, \end{array}\right. \end{aligned}$$

where \(C_0\) was defined in (5).

In particular, the theorem shows that for \(V \in L_r({{\mathbb {H}}})\) the operator of multiplication by V is \(-\,\Delta _p\)-compact. This is used in Sect. 3.2.

Proof

A density argument shows that it is sufficient to consider the case where V is a nonnegative simple function. For such a V define

$$\begin{aligned} f: S \rightarrow \mathcal {B}(L_p({{\mathbb {H}}})) + \Pi _p(L_p({{\mathbb {H}}})), \qquad f(w)=V^{\frac{r}{p} w}(-\,\Delta _p-\lambda )^{-1}, \end{aligned}$$

where as above \(S = \{ w : 0 \le {\text {Re}}(w) \le 1\}\). From what we have shown above, we infer that f is continuous and bounded on S and holomorphic in the interior of S. Moreover, since \(\Vert V^{\frac{r}{p} iy}\Vert _{\infty } \le 1\) we obtain from Lemma 4.7 that

$$\begin{aligned} A_0:= \sup _{y \in {{\mathbb {R}}}} \Vert f(iy)\Vert _{p,p} \le \left\{ \begin{array}{ll} \frac{1}{{\text {dist}}(\lambda ,[1/4,\infty ))}, &{}\quad p=2, \\ \frac{1}{({\text {Re}}(z))^{1+\frac{2}{p}}(1+{\text {Re}}(z))^{1-\frac{2}{p}}}, &{}\quad p > 2. \end{array}\right. \end{aligned}$$

Furthermore, Lemma 4.8 implies that

$$\begin{aligned} A_1^p:= & {} \sup _{y \in {{\mathbb {R}}}} \Vert f(1+iy)\Vert _{\Pi _p(L_p)}^p \le C_0 \cdot \Vert V^{\frac{r}{p}}\Vert _p^p \cdot \left\{ \begin{array}{ll} \frac{1}{|z+\frac{1}{2}| ({\text {Re}}(z))}, &{}\quad p =2 \\ \left( \frac{1}{{\text {Re}}(z)({\text {Re}}(z) +\frac{1}{2})} \right) ^{p/p'}, &{}\quad p > 2, \end{array}\right. \end{aligned}$$

and here \(\Vert V^{\frac{r}{p}}\Vert _p^p = \Vert V\Vert _r^r\). But then Propositions A.4 and 4.9 imply that with \(1/r= \theta / p\) (i.e., \(f(\theta )= V(-\,\Delta _p-\lambda )^{-1}\))

$$\begin{aligned}&\Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{\Pi _{r,p}(L_p({{\mathbb {H}}}))} \le \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{[\mathcal {B},\Pi _p]_\theta } \le A_0^{(1-\theta )}A_1^{\theta } \\&\quad \le C_0^{1/r} \Vert V\Vert _r\left\{ \begin{array}{ll} \left( \frac{1}{{\text {dist}}(\lambda ,[1/4,\infty ))}\right) ^{1-2/r} \left( \frac{1}{|z+\frac{1}{2}| ({\text {Re}}(z))} \right) ^{1/r}, &{}\quad p=2, \\ \left( \frac{1}{({\text {Re}}(z))^{1+\frac{2}{p}}(1+{\text {Re}}(z))^{1-\frac{2}{p}}}\right) ^{1-p/r} \left( \frac{1}{{\text {Re}}(z)({\text {Re}}(z) +\frac{1}{2})} \right) ^{p/(rp')}, &{}\quad p > 2. \end{array}\right. \end{aligned}$$

Now, a rearrangement of terms, using the estimate \({\text {Re}}(z)+1 > {\text {Re}}(z) + \frac{1}{2}\), concludes the proof. \(\square \)

The previous theorem will be used to prove the results in Sect. 2.1. To prove the results in Sect. 2.2, we will use the following corollary.

Corollary 4.12

Let \(2 \le p \le r < \infty \), \(z \in {{\mathbb {C}}}_+\) and \(\lambda =\Psi _p(z)\). If \(V \in L_r({{\mathbb {H}}})\), then \(V(-\,\Delta _p-\lambda )^{-1} \in \Pi _{r,p}(L_p({{\mathbb {H}}}))\) and

$$\begin{aligned} \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{\Pi _{r,p}}\le & {} 2^{1-2/p} \Vert V\Vert _r \left\{ \begin{array}{ll} \left( \frac{1}{|z|}\right) ^{1-2/r} \left( \frac{1}{{\text {Re}}(z)} \right) ^{1-\frac{1}{r}}, &{}\quad p=2, \\ \left( \frac{1}{{\text {Re}}(z)}\right) ^{1+2/p-3/r}, &{}\quad p > 2. \end{array}\right. \end{aligned}$$

Proof

The case \(p=2\) follows using Lemma 4.1 to estimate \({\text {dist}}(\lambda ,[1/4,\infty )) \ge |z|{\text {Re}}(z)\), together with the estimate \(|z+1/2| \ge 1/2\) for \(z \in {{\mathbb {C}}}_+,\) and the fact that \(2C_0 \le 1\). The case \(p>2\) follows in the same way using that \({\text {Re}}(z)+\frac{1}{2} \ge \frac{1}{2}\) and \(C_0^{1/r}2^{1-2/p+1/r} \le 2^{1-2/p}\). \(\square \)

5 Proof of Theorems 2.1 and 2.3

Let \(p \ge 2\) and let \(\lambda \in \Sigma _p^c=\varrho (-\,\Delta _p)\). Then, by the Birman–Schwinger principle \(\lambda \) is an eigenvalue of \(H_p=-\,\Delta _p+V\) if and only if \(-1\) is an eigenvalue of \(V(-\,\Delta _p-\lambda )^{-1}\). Hence, in this case we obtain for \(r \ge p\) that

$$\begin{aligned} 1 \le \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert \le \Vert V(-\,\Delta _p-\lambda )^{-1}\Vert _{\Pi _{r,p}}. \end{aligned}$$
(29)

Now, we use Theorem 4.11 to estimate the right-hand side from above and we rearrange terms. We distinguish between two cases:

(i) In case \(p=2\), we obtain with \(z=\Psi _2^{-1}(\lambda )=\sqrt{1/4-\lambda }\)

$$\begin{aligned} ({\text {dist}}(\lambda ,[1/4,\infty )))^{r-2}|z+1/2|{\text {Re}}(z) \le C_0 \Vert V\Vert _r^r. \end{aligned}$$
(30)

A short calculation shows that, since \( z \in {{\mathbb {C}}}_+\), we have \(|z+1/2| \ge 1/\sqrt{2} \cdot (|z|+1/2)\). Hence, using Lemma 4.1 we see that the left-hand side of (30) can be bounded from below as follows:

$$\begin{aligned}&({\text {dist}}(\lambda ,[1/4,\infty )))^{r-2}|z+1/2|{\text {Re}}(z) \nonumber \\&\quad \ge 1/\sqrt{2}\cdot ({\text {dist}}(\lambda ,[1/4,\infty )))^{r-2}(|z|+1/2){\text {Re}}(z) \nonumber \\&\quad = 1/\sqrt{2}\cdot ({\text {dist}}(\lambda ,[1/4,\infty )))^{r-2} (1+1/(2|z|)) |z| {\text {Re}}(z) \nonumber \\&\quad \ge 1/(2\sqrt{2}) \cdot ({\text {dist}}(\lambda ,[1/4,\infty )))^{r-1} |1+1/(2|1/4-\lambda |^{1/2})| . \end{aligned}$$
(31)

But (30) and (31) show the validity of (4) and conclude the proof of Theorem 2.1

(ii) In case \(p > 2\), we obtain from (29) and Theorem 4.11 that

$$\begin{aligned} ({\text {Re}}(z))^{2r-2}(1+1/(2{\text {Re}}(z)))^{r(1-2/p)+1} \le C_0 \Vert V\Vert _r^r. \end{aligned}$$
(32)

Now, we use Lemma 4.1 and (14) to estimate the left-hand side from below by

$$\begin{aligned} \left( \frac{1}{16} \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|1/4-\lambda |^{1/2}} \right) ^{2r-2} \left( 1 + \frac{|1/4-\lambda |^{1/2}}{8 {\text {dist}}(\lambda ,\Sigma _p)}\right) ^{r(1-2/p)+1}. \end{aligned}$$

This shows the validity of (7) in case \(p > 2\). Finally, the case \(1<p<2\) follows by duality using Proposition 3.3. This concludes the proof of Theorem 2.3.

6 An Abstract Lieb–Thirring Estimate

Theorems 2.5 and 2.7 are proved using the following abstract result. Here, we use terminology from [27], which is reviewed in ‘Appendix A.2.’ Moreover, \(x_+=\max (x,0)\) denotes the positive part of \(x \in {{\mathbb {R}}}\).

Theorem 6.1

Let X denote a complex Banach space, let \(r \ge 1\) and let \(({\mathcal {I}},\Vert .\Vert _{{\mathcal {I}}})\) be an \(l_r\)-ideal in \(\mathcal {B}(X)\). Moreover, let \(Z_0\) and \(Z=Z_0+M\) denote closed operators in X such that

  • for some \(p \in [1,\infty )\) we have \(\sigma (Z_0)=\Sigma _p\) as defined in (1),

  • there exist \(\alpha , \beta ,\gamma \ge 0\) and \(C_1> 0\) such that for all \(z \in {{\mathbb {C}}}_+\) and \(\Psi _p(z)\) as defined in (13):

    $$\begin{aligned} \Vert M(Z_0-\Psi _p(z))^{-1}\Vert _{\mathcal {I}} \le C_1\cdot {\text {Re}}(z)^{-\alpha } \cdot |z|^{-\beta } \end{aligned}$$
    (33)

    and

    $$\begin{aligned} \Vert (Z_0-\Psi _p(a))^{-1}\Vert \le a^{-\gamma }, \qquad a>0. \end{aligned}$$
    (34)

Finally, let \(\tau > 0\) and set

$$\begin{aligned} \delta _1:= & {} r\alpha + 1+ \tau , \\ \delta _2:= & {} (r\beta -1+\tau )_+, \\ \delta _3:= & {} r(1-\alpha -\beta -\gamma ). \end{aligned}$$

Then, there exist constants C and \(C'\), both depending on \(\alpha ,\beta ,\gamma ,r\) and \(\tau \), such that

$$\begin{aligned} \sum _{|\lambda - \frac{1}{pp'}|^{\frac{1}{2}} \le (2C_1)^{\frac{1}{\alpha +\beta }}} \frac{{\text {dist}}(\lambda ,\Sigma _p)^{\delta _1} |\lambda -\frac{1}{pp'}|^{\delta _2}}{(|\lambda - \frac{1}{pp'}|^{\frac{1}{2}}+2\gamma _p)^{\delta _1+\delta _2}} \le C \cdot C_1^{r+\frac{\delta _1+\delta _2+\delta _3}{\alpha +\beta }}(C_1^{\frac{1}{\alpha + \beta }}+\gamma _p)^{r}\nonumber \\ \end{aligned}$$
(35)

and

$$\begin{aligned} \sum _{|\lambda - \frac{1}{pp'}|^{\frac{1}{2}}>(2C_1)^{\frac{1}{\alpha +\beta }}} \frac{{\text {dist}}(\lambda ,\Sigma _p)^{\delta _1} |\lambda -\frac{1}{pp'}|^{\delta _2}}{(|\lambda -\frac{1}{pp'}|^{\frac{1}{2}}+2\gamma _p)^{2\delta _1+2\delta _2 + \delta _3+r+\tau }} {\le } C' \cdot C_1^{r}(C_1^{\frac{1}{\alpha +\beta }}+\gamma _p)^{-\tau } .\nonumber \\ \end{aligned}$$
(36)

Here, in both sums, we are summing over all eigenvalues \(\lambda \in \sigma _d(Z)\cap \Sigma _p^c\) satisfying the stated restrictions, each eigenvalue being counted according to its algebraic multiplicity. Moreover, \(\gamma _p\) is as defined in (6).

In the remainder of this section, we are going to prove the previous theorem. We start with a lemma providing a resolvent norm estimate on \(Z=Z_0+M\).

Lemma 6.2

Given assumptions (33) and (34), we have for all

$$\begin{aligned} a \ge (2C_1)^{1/(\alpha + \beta )} \end{aligned}$$
(37)

that \(\Psi _p(a) \in \varrho (Z)\) and

$$\begin{aligned} \Vert (Z-\Psi _p(a))^{-1}\Vert \le 2 a^{-\gamma }. \end{aligned}$$
(38)

Proof

Since for \(a>0\) we have \(\Psi _p(a) \in \Sigma _p^c=\varrho (Z_0)\), we can write

$$\begin{aligned} Z-\Psi _p(a) = Z_0+M-\Psi _p(a) = (I+M(Z_0-\Psi _p(a))^{-1})(Z_0-\Psi _p(a)).\quad \end{aligned}$$
(39)

By assumption (33),

$$\begin{aligned} \Vert M(Z_0-\Psi _p(a))^{-1}\Vert \le \Vert M(Z_0-\Psi _p(a))^{-1}\Vert _{\mathcal I} \le C_1 a^{-\alpha -\beta }. \end{aligned}$$

Hence, we see that for \(a \ge (2C_1)^{1/(\alpha + \beta )}\) the operator \(I+M(Z_0-\Psi _p(a))^{-1}\) is invertible with norm of the inverse being at most 2. But then also \(Z-\Psi _p(a)\) is invertible and using (34) and (39) we obtain

$$\begin{aligned} \Vert (Z-\Psi _p(a))^{-1}\Vert \le 2 \Vert (Z_0-\Psi _p(a))^{-1}\Vert \le 2 a^{-\gamma }. \end{aligned}$$

\(\square \)

Now, for a shorter notation let us set

$$\begin{aligned} b= \Psi _p(a) \end{aligned}$$

with some a satisfying (37). Then, \(b \in \varrho (Z) \cap \varrho (Z_0)\) and

$$\begin{aligned} K:=(Z-b)^{-1} - (Z_0-b)^{-1} = -(Z-b)^{-1}M(Z_0-b)^{-1} \in {\mathcal {I}}. \end{aligned}$$

Let us set \(A=(Z_0-b)^{-1}\) and \(B:=A+K:=(Z-b)^{-1}\). By the spectral mapping theorem

$$\begin{aligned} \lambda \in \varrho (Z_0){\setminus } \{b\} \quad \Leftrightarrow \quad (\lambda -b)^{-1} \in \varrho (A) \end{aligned}$$

(and a similar result holds for Z and B). From [27, Theorem 4.10], see ‘Appendix A.2,’ we know that there exists a holomorphic function \(d : \varrho (A) \rightarrow {{\mathbb {C}}}\) with the following properties:

  1. (p1)

    \(\lim _{|u| \rightarrow \infty } d(u)=1\),

  2. (p2)

    for \(u \in \varrho (A)\) we have

    $$\begin{aligned} |d(u)| \le \exp \left( \mu _r^r \Gamma _r \Vert K(u-A)^{-1}\Vert _{\mathcal I}^r\right) , \end{aligned}$$

    where \(\mu _r\) denotes the eigenvalue constant of \({\mathcal {I}}\) and \(\Gamma _r\) is a universal r-dependent constant, see [28],

  3. (p3)

    \(d(u)=0\) iff \(u \in \sigma (A+K)\),

  4. (p4)

    if \(u \in \varrho (A) \cap \sigma _d(A+K)\), then its algebraic multiplicity (as an eigenvalue) coincides with its order as a zero of d.

Using the spectral mapping theorem, again we see that

$$\begin{aligned} D(\lambda ):=d((\lambda -b)^{-1}) \end{aligned}$$

is well defined and analytic on \(\varrho (Z_0) {\setminus } \{b\}\) and, by (p1), can be analytically extended to \(\varrho (Z_0)=\Sigma _p^c\) by setting \(D(b)=1\). Moreover, by spectral mapping and (p3) and (p4) we know that \(D(\lambda )=0\) iff \(\lambda \in \sigma (Z)\) and if \(\lambda \in \varrho (Z_0) \cap \sigma _d(Z)\), then its algebraic multiplicity coincides with its order as a zero of D. Finally, since

$$\begin{aligned} ((\lambda -b)^{-1}-A)^{-1}=((\lambda -b)^{-1}-(Z_0-b)^{-1})^{-1}=(\lambda -b)(Z_0-b)(Z_0-\lambda )^{-1} \end{aligned}$$

we see that

$$\begin{aligned} K((\lambda -b)^{-1}-A)^{-1}= & {} ((Z-b)^{-1}-(Z_0-b)^{-1})(\lambda -b)(Z_0-b)(Z_0-\lambda )^{-1} \\= & {} (b-\lambda )(Z-b)^{-1}M(Z_0-\lambda )^{-1} \end{aligned}$$

and hence for \(\lambda \in \varrho (Z_0)=\Sigma _p^c\) we have by (p2)

$$\begin{aligned} |D(\lambda )|\le & {} \exp \left( \mu _r^r \Gamma _r |\lambda -b|^r \Vert (Z-b)^{-1}\Vert ^r \Vert M(Z_0-\lambda )^{-1}\Vert _{{\mathcal {I}}}^r\right) . \end{aligned}$$

Writing \(b=\Psi _p(a)\) and \(\lambda =\Psi _p(z)\), with \(z \in {{\mathbb {C}}}_+\), the assumption (33) and Lemma 6.2 hence imply that

$$\begin{aligned} |D(\Psi _p(z))| \le \exp \left( \mu _r^r \Gamma _r |\Psi _p(z)-\Psi _p(a)|^r 2^ra^{-r\gamma } C_1^r {\text {Re}}(z)^{-r\alpha } |z|^{-r\beta }\right) . \end{aligned}$$
(40)

Here, the holomorphic function \(D \circ \Psi _p\) is defined on the right half-plane \({{\mathbb {C}}}_+\). In the following, it will be necessary to transfer this function to the unit disk \({{\mathbb {D}}}\) using the conformal map

$$\begin{aligned} \Phi _a: {{\mathbb {D}}}\rightarrow {{\mathbb {C}}}_+, \quad \Phi _a(w) = a \frac{1-w}{1+w} \end{aligned}$$

with inverse

$$\begin{aligned} \Phi _a^{-1}(z)=\frac{a-z}{a+z}. \end{aligned}$$

Lemma 6.3

Let \(w \in {{\mathbb {D}}}, z=\Phi _a(w) \in {{\mathbb {C}}}_+\) and \(\lambda = \Psi _p(z) \in \Sigma _p^c\). Then, the following holds:

$$\begin{aligned} 1+w = \frac{2a}{a+z},&\quad 1-w=\frac{2z}{a+z} \end{aligned}$$
(41)
$$\begin{aligned} |\Psi _p(z)-\Psi _p(a)|&\le \frac{4a(a+2\gamma _p)}{|1+w|^2} \end{aligned}$$
(42)
$$\begin{aligned} a \frac{1-|w|}{|1+w|^2}&\le {\text {Re}}(z) \le 2a \frac{1-|w|}{|1+w|^2} \end{aligned}$$
(43)
$$\begin{aligned} \frac{a\cdot {\text {dist}}(\lambda ,\Sigma _p)}{8|a+z|^2|1/4-\lambda |^{1/2}}&\le 1-|w| \le \frac{16a\cdot {\text {dist}}(\lambda ,\Sigma _p)}{|a+z|^2|1/4-\lambda |^{1/2}}. \end{aligned}$$
(44)

Proof of the lemma

The identities in (41) are immediate consequences of the definitions. To see (42), we compute, using (13),

$$\begin{aligned} |\Psi _p(z)-\Psi _p(a)|= & {} | (a+\gamma _p)^2-(z+\gamma _p)^2| = |a^2-z^2 + 2\gamma _p(a-z)|. \end{aligned}$$

Hence, since

$$\begin{aligned} a-z = a\left( 1-\frac{1-w}{1+w}\right) = \frac{2aw}{1+w}, \qquad a+z = \frac{2a}{1+w}, \end{aligned}$$

we obtain

$$\begin{aligned} |\Psi _p(z)-\Psi _p(a)|= & {} \left| \frac{4wa^2}{(1+w)^2} +\frac{4wa\gamma _p}{1+w} \right| \\= & {} \frac{4|w|a}{|1+w|^2} |a + \gamma _p(1+w)| \le \frac{4a(a+2\gamma _p)}{|1+w|^2}. \end{aligned}$$

The estimates in (43) follow from

$$\begin{aligned} {\text {Re}}(z)= a \frac{1-|w|^2}{|1+w|^2}. \end{aligned}$$
(45)

Finally, in order to show (44) we first use Lemma 4.1 to obtain

$$\begin{aligned} \frac{|z+\gamma _p| {\text {Re}}(z)}{4} \le {\text {dist}}(\lambda ,\Sigma _p) \le 16 |z+\gamma _p| {\text {Re}}(z). \end{aligned}$$
(46)

(Here, we ignore the fact that a better estimate is valid if \(p=2\).) Since \(z+\gamma _p=\sqrt{1/4-\lambda }\), we obtain, also using (41) and (45), that

$$\begin{aligned} |z+\gamma _p| {\text {Re}}(z) = \frac{a |1/4-\lambda |^{1/2}(1-|w|^2)}{|1+w|^2} = (1-|w|^2) \frac{|1/4-\lambda |^{1/2}|a+z|^2}{4a}.\nonumber \\ \end{aligned}$$
(47)

But (47) and (46) imply (44). \(\square \)

Now, let us introduce the holomorphic function

$$\begin{aligned} h: {{\mathbb {D}}}\rightarrow {{\mathbb {C}}}, \quad h(w)= D(\Psi _p(\Phi _a(w))). \end{aligned}$$

Then, \(h(w)= 0\) if and only if \(\lambda = \Psi _p(\Phi _a(w)) \in \sigma _d(Z) \cap \Sigma _p^c\) (and order and multiplicity coincide) and \(h(0)=D(\Psi _p(a))=D(b)=1\). Moreover, using the previous lemma and (40) a short computation shows that

$$\begin{aligned} |h(w)|\le & {} \exp \left( 8^r\mu _r^r \Gamma _rC_1^r a^{r(1-\alpha -\beta -\gamma )}(a+2\gamma _p)^r \frac{1}{|1+w|^{r(2-2\alpha -\beta )}|1-w|^{r\beta }(1-|w|)^{r\alpha }}\right) . \end{aligned}$$

So we see that h grows at most exponentially for w approaching the unit circle, with the rate of explosion depending on whether w approaches 1 or \(-1\) or a generic point of the boundary, respectively. A theorem of Borichev, Golinskii and Kupin [5, Theorem 0.3] allows us to transform this information on the growth of h into the following information on its zero set: The theorem says that for all \(\tau > 0\) there exists a constant \(C=C(\alpha ,\beta ,\gamma ,r, \tau ) > 0\) such that

$$\begin{aligned}&\sum _{h(w)=0, w \in {{\mathbb {D}}}} {(1-|w|)^{r\alpha +1+\tau }|1-w|^{(r\beta -1+\tau )_+}|1+w|^{(r(2-2\alpha -\beta ) -1 +\tau )_+}} \nonumber \\&\quad \le C \cdot C_1^r \cdot a^{r(1-\alpha -\beta -\gamma )}(a+2\gamma _p)^r, \end{aligned}$$
(48)

where each zero of h is counted according to its order. Using Lemma 6.3, we see that the summands on the LHS are bounded below by

$$\begin{aligned} \left( \frac{a}{8} \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|a+z|^2|1/4 -\lambda |^{1/2}}\right) ^{r\alpha +1+\tau } \left| \frac{2z}{a+z} \right| ^{(r\beta -1+\tau )_+}\left| \frac{2a}{a+z}\right| ^{(r(2-2\alpha -\beta )-1+\tau )_+}. \end{aligned}$$

Hence, we have proved the following lemma.

Lemma 6.4

Assume (33) and (34). Let \(\tau > 0\) and set

$$\begin{aligned} \delta _1:= & {} r\alpha + 1+ \tau , \\ \delta _2:= & {} (r\beta -1+\tau )_+, \\ \delta _3:= & {} r(1-\alpha -\beta -\gamma ),\\ \delta _4:= & {} (r(2-2\alpha -\beta )-1+\tau )_+. \end{aligned}$$

Then, there exists \(C=C(\alpha ,\beta ,\gamma ,r,\tau )\) such that for all \(a \ge (2C_1)^{1/(\alpha + \beta )}\) we have

$$\begin{aligned} \sum _{\lambda \in \sigma _d(Z)\cap \Sigma _p^c} \left( \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|\frac{1}{4} -\lambda |^{1/2}}\right) ^{\delta _1} \frac{|z|^{\delta _2}}{|a+z|^{2\delta _1+\delta _2+\delta _4}} \le C \cdot C_1^r \cdot a^{-\,\Delta _1{-}\delta _4{+}\delta _3}(a{+}2\gamma _p)^r.\nonumber \\ \end{aligned}$$
(49)

Here, each eigenvalue is counted according to its alg. multiplicity and \(z=\Psi _p^{-1}(\lambda )\).

In order to finish the proof of Theorem 6.1, we need to distinguish between ‘small’ and ‘large’ eigenvalues. Namely, introducing

$$\begin{aligned} 0< \eta := (2C_1)^{1/(\alpha + \beta )} \end{aligned}$$

we consider the cases

$$\begin{aligned} \text{(i) } \quad |\lambda -1/(pp')|^{1/2} \le \eta \qquad \text {and} \qquad \text{(ii) } \quad |\lambda -1/(pp')|^{1/2} > \eta , \end{aligned}$$

respectively. Note that by (12)

$$\begin{aligned} \lambda - 1/(pp')=-\,z(z+2\gamma _p). \end{aligned}$$
(50)

Case (i): Since for \(z \in {{\mathbb {C}}}_+\) we have \(|z+2\gamma _p| \ge |z|\), we obtain

$$\begin{aligned} \eta \ge |\lambda -1/(pp')|^{1/2}= |z(z+2\gamma _p)|^{1/2} \ge |z|. \end{aligned}$$

Now, we apply (49) with \(a=\eta \), the sum being restricted to those \(\lambda \) satisfying the first case, and use the estimate \(|z+\eta | \le |z| + \eta \le 2\eta \). We obtain

$$\begin{aligned} \sum _{|\lambda -1/(pp')|^{1/2} \le \eta } \left( \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|\frac{1}{4} -\lambda |^{1/2}}\right) ^{\delta _1} |z|^{\delta _2}\le & {} C \cdot C_1^r \cdot \eta ^{\delta _1+\delta _2+\delta _3}(\eta +\gamma _p)^{r} \nonumber \\\le & {} C \cdot C_1^{r+\frac{\delta _1+\delta _2+\delta _3}{\alpha +\beta }}(C_1^{\frac{1}{\alpha + \beta }}+\gamma _p)^{r}. \end{aligned}$$
(51)

Remark 6.5

Note that here the constants C are different from each other and from the one in (49), but they depend on the same parameters. Also, in the following this constant may change from line to line.

It remains to estimate the sum on the left-hand side of the previous inequality from below in a suitable manner. To this end, we note that since \(1/4-1/(pp')=\gamma _p^2\) we have

$$\begin{aligned} |\lambda -1/4|^{1/2} \le \left( |\lambda -1/(pp')|+ \gamma _p^2 \right) ^{1/2} \le |\lambda -1/(pp')|^{1/2}+ \gamma _p. \end{aligned}$$
(52)

Moreover, this estimate implies that, with \(z=\Psi _p^{-1}(\lambda )=-\,\gamma _p + \sqrt{1/4-\lambda }\),

$$\begin{aligned} |z+2\gamma _p| \le |z+\gamma _p| + \gamma _p = |1/4 - \lambda |^{1/2} + \gamma _p \le |\lambda -1/(pp')|^{1/2} + 2 \gamma _p. \end{aligned}$$
(53)

Finally, the previous inequality and (50) show that

$$\begin{aligned} |z| = \frac{|\lambda -1/(pp')|}{|z+2\gamma _p|} \ge \frac{|\lambda -1/(pp')|}{|\lambda -1/(pp')|^{1/2} + 2 \gamma _p}. \end{aligned}$$
(54)

Remark 6.6

It is important to note that (52)–(54) are valid for all \(\lambda \in \Sigma _p^c\).

Now, we can use (54) and (52) to estimate the sum in (51) from below by

$$\begin{aligned} \sum _{|\lambda -1/(pp')|^{1/2} \le (2C_1)^{1/(\alpha +\beta )}} {\text {dist}}(\lambda ,\Sigma _p)^{\delta _1} \frac{|\lambda -\frac{1}{pp'}|^{\delta _2}}{(|\lambda - \frac{1}{pp'}|^{1/2}+2\gamma _p)^{\delta _1+\delta _2}}. \end{aligned}$$

This completes the proof of inequality (35).

Case (ii): For those \(\lambda \) satisfying the second case, we have

$$\begin{aligned} \eta < |\lambda -1/(pp')|^{1/2} = |z(z+2\gamma _p)|^{1/2} \le |z+2\gamma _p|. \end{aligned}$$

Now, we restrict the sum in (49) to those \(\lambda \) satisfying the second case, multiply left- and right-hand side of (49) by \(a^{\delta _1+\delta _4-\,\Delta _3}(a+2\gamma _p)^{-r-1-\tau }\) and integrate a from \(\eta \) to \(\infty \).

Then, as a result for the RHS we obtain

$$\begin{aligned} C \cdot C_1^r \cdot \int _\eta ^\infty da \, (a+2\gamma _p)^{-1-\tau } = C \cdot C_1^r \cdot \frac{1}{\tau }(\eta +2\gamma _p)^{-\tau }. \end{aligned}$$
(55)

Moreover, for the LHS we obtain

$$\begin{aligned}&\int _{\eta }^\infty da \, a^{\delta _1+\delta _4-\,\Delta _3}(a+2\gamma _p)^{-r-1-\tau } \sum _{|\lambda -1/(pp')|^{1/2}>\eta } \left( \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|\frac{1}{4} -\lambda |^{1/2}}\right) ^{\delta _1} \frac{|z|^{\delta _2}}{|a+z|^{2\delta _1+\delta _2+\delta _4}} \nonumber \\&\quad \ge \sum _{|\lambda -1/(pp')|^{1/2}>\eta } \left( \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|\frac{1}{4} -\lambda |^{1/2}}\right) ^{\delta _1} |z|^{\delta _2} \int _{\eta }^\infty da \frac{a^{\delta _1+\delta _4-\,\Delta _3}}{(a+2\gamma _p)^{r+1+\tau }(a+|z|)^{2\delta _1+\delta _2+\delta _4}}. \nonumber \\ \end{aligned}$$
(56)

Now, we change variables in the integral in (56), obtaining that

$$\begin{aligned}&\int _{\eta }^\infty da \frac{a^{\delta _1+\delta _4-\,\Delta _3}}{(a+2\gamma _p)^{r+1+\tau }(a+|z|)^{2\delta _1+\delta _2+\delta _4}}\nonumber \\&\quad = |z+2\gamma _p|^{\delta _1+\delta _4-\,\Delta _3+1} \nonumber \\&\qquad \int _{\eta /|z+2\gamma _p|}^\infty db \frac{b^{\delta _1+\delta _4-\,\Delta _3}}{(b|z+2\gamma _p|+2\gamma _p)^{r+1+\tau }(b|z+2\gamma _p|+|z|)^{2\delta _1+\delta _2+\delta _4}} \nonumber \\&\quad \ge |z+2\gamma _p|^{-\,\Delta _1-\delta _2-\delta _3-r-\tau } \int _{1}^\infty db \frac{b^{\delta _1+\delta _4-\delta _3}}{(b+1)^{r+1+\tau +2\delta _1+\delta _2+\delta _4}}, \end{aligned}$$
(57)

where in the last step we used that \(|z| \le |z+2\gamma _p|\) and \(2\gamma _p \le |z+2\gamma _p|\) for \(z \in {{\mathbb {C}}}_+\), and that \(\eta < |z+2\gamma _p|\) as had been shown above. From (57), (56) and (55), we obtain that

$$\begin{aligned}&\sum _{|\lambda -1/(pp')|^{1/2}>\eta } \left( \frac{{\text {dist}}(\lambda ,\Sigma _p)}{|\frac{1}{4} -\lambda |^{1/2}}\right) ^{\delta _1} \frac{|z|^{\delta _2}}{|z+2\gamma _p|^{\delta _1+\delta _2+\delta _3+r+\tau }}\nonumber \\&\quad \qquad \le C \cdot C_1^{r}(C_1^{1/(\alpha +\beta )}+\gamma _p)^{-\tau }. \end{aligned}$$
(58)

Finally, we use (52)–(54) to estimate the left-hand side of (58) from below by

$$\begin{aligned} \sum _{|\lambda -1/(pp')|^{1/2}>(2C_1)^{1/(\alpha +\beta )}} \frac{{\text {dist}}(\lambda ,\Sigma _p)^{\delta _1} |\lambda -1/(pp')|^{\delta _2}}{(|\lambda -1/(pp')|^{1/2}+2\gamma _p)^{2\delta _1+2\delta _2 + \delta _3+r+\tau }}. \end{aligned}$$

This shows that also inequality (36) is valid and concludes the proof of Theorem 6.1.

7 Proof of Theorems 2.5 and 2.7

In this final section, we use Theorem 6.1 to prove Theorems 2.5 and 2.7, starting with the former. We set \(H_0=-\,\Delta _p\) and \(H_p=-\,\Delta _p+V\) acting in \(L_p({{\mathbb {H}}}), 1< p < \infty \).

7.1 Proof of Theorem 2.5

Let \(r \ge 2\). Since Theorem 2.5 is obviously true if \(\Vert V\Vert _r=0\), we can assume that this is not the case. Now, we apply Theorem 6.1 with the \(l_r\)-ideal \({\mathcal {S}}_r(L_2({{\mathbb {H}}}))\) (see ‘Appendix A.2’ and Example A.1). By Corollary 4.12, we have

$$\begin{aligned} \Vert V(H_0-\Psi _2(z))^{-1}\Vert _{{\mathcal {S}}_r} \le \Vert V\Vert _r {\text {Re}}(z)^{-(1-1/r)}|z|^{-(1-2/r)}, \quad z \in {{\mathbb {C}}}_+. \end{aligned}$$

Moreover, Lemma 4.6 shows that for \(a>0\)

$$\begin{aligned} \Vert (H_0-\Psi _2(a))^{-1}\Vert _{2,2} \le a^{-2}. \end{aligned}$$

Hence, we can apply Theorem 6.1 with \(Z_0=H_0, M=V, p=2\) and

$$\begin{aligned} C_1 = \Vert V\Vert _r, \quad \alpha = 1-1/r, \quad \beta = 1-2/r, \quad \gamma =2, \end{aligned}$$

and so \(1/(\alpha +\beta )= r/(2r-3)\) and

$$\begin{aligned} \delta _1 = r+\tau , \qquad \delta _2 = (r-3+\tau )_+, \qquad \delta _3 = 3-3r. \end{aligned}$$

Then, (35) implies, using that \(\gamma _2=0\),

$$\begin{aligned} \sum _{|\lambda -1/4|^{r-3/2} \le (2\Vert V\Vert _r)^{r}} \frac{{\text {dist}}(\lambda ,[1/4,\infty ))^{r+\tau }}{|\frac{1}{4} -\lambda |^{\frac{r+\tau -(r-3+\tau )_+}{2}}} \le C \cdot \Vert V\Vert _r^{\frac{r}{2r-3} ( r+\tau +(r-3+\tau )_+)}. \end{aligned}$$

In particular, if we restrict to \(\tau \in (0,1)\) and consider the cases \(2 \le r \le 3 -\tau \) and \(r>3-\tau \) separately, the validity of Theorem 2.5, part (ia) and (ib), is easily derived.

Similarly, (36) implies that

$$\begin{aligned} \sum _{|\lambda -1/4|^{r-3/2} > (2\Vert V\Vert _r)^{r}} \frac{{\text {dist}}(\lambda ,[1/4,\infty ))^{r+\tau }}{|\frac{1}{4} -\lambda |^{(3+3\tau )/2}} \le C' \Vert V\Vert _r^{\frac{r}{2r-3}(2r-3-\tau )}. \end{aligned}$$

This shows the validity of Theorem 2.5, part (ii), and concludes the proof of the theorem.

7.2 Proof of Theorem 2.7

In view of Proposition 3.3, it is sufficient to prove the theorem in case \(p>2\). Let \(r \ge p > 2\) and \(\Vert V\Vert _r \ne 0\) (otherwise the theorem is trivially satisfied). As remarked in ‘Appendix A.2,’ Example A.2, if \(r \ge p > 2\) the (r, p)-summing ideal \(\Pi _{r,p}(L_p({{\mathbb {H}}}))\) is an \(l_R\)-ideal, where

$$\begin{aligned} R = r + \varepsilon (r) \quad \text { and } \quad \varepsilon (r)= \left\{ \begin{array}{ll} 0, &{}\quad \text {if } r=p \\ \varepsilon _0, &{}\quad \text {if } r>p . \end{array}\right. \end{aligned}$$
(59)

Here, \(\varepsilon _0 > 0\) can be chosen arbitrarily small. By Corollary 4.12, we have

$$\begin{aligned} \Vert V(H_0-\Psi _p(z))^{-1}\Vert _{\Pi _{r,p}} \le 2^{1-2/p}\Vert V\Vert _r {\text {Re}}(z)^{-(1+2/p-3/r)}, \quad z \in {{\mathbb {C}}}_+, \end{aligned}$$

and Lemma 4.6 shows that for \(a>0\)

$$\begin{aligned} \Vert (H_0-\Psi _p(a))^{-1}\Vert _{p,p} \le a^{-2}. \end{aligned}$$

Hence, we can apply Theorem 6.1 with the \(l_R\)-ideal \(\Pi _{r,p}, Z_0=H_0, M=V\) and

$$\begin{aligned} C_1 = 2^{1-2/p}\Vert V\Vert _r, \quad \alpha = 1+2/p-3/r, \quad \beta = 0, \quad \gamma = 2, \end{aligned}$$

so that \(1/(\alpha +\beta )= r/(r(1+2/p)-3)\) and for \(\tau \in (0,1)\)

$$\begin{aligned} \delta _1= & {} R(1+2/p-3/r)+1+\tau \\= & {} r(1+2/p)-2+\tau + \varepsilon (r)(1+2/p-3/r), \\ \delta _2= & {} (-1+\tau )_+=0, \\ \delta _3= & {} R(1-(1+2/p-3/r)-2)=R(-2-2/p+3/r) \\= & {} 3-r(2+2/p)-\varepsilon (r)(2+2/p-3/r). \end{aligned}$$

Before applying (35) in the present situation, we note that for \(|\lambda -\frac{1}{pp'}|^{\frac{1}{2}} \le (2C_1)^{1/(\alpha +\beta )}\) we trivially have \( (|\lambda - 1/(pp')|^{\frac{1}{2}}+2\gamma _p) \le (2C_1)^{1/(\alpha +\beta )} + 2\gamma _p\) and hence (35) implies that

$$\begin{aligned}&\sum _{|\lambda - \frac{1}{pp'}|^{\frac{1}{2}} \le (2C_1)^{\frac{1}{\alpha +\beta }}} {\text {dist}}(\lambda ,\Sigma _p)^{\delta _1} \cdot |\lambda -\frac{1}{pp'}|^{\delta _2} \\&\quad \le C \cdot C_1^{r+\frac{\delta _1+\delta _2+\delta _3}{\alpha +\beta }}(C_1^{\frac{1}{\alpha + \beta }}+\gamma _p)^{r+\delta _1+\delta _2}\\&\quad \le C \cdot C_1^{r}(C_1^{\frac{1}{\alpha + \beta }}+\gamma _p)^{r+2\delta _1+2\delta _2+\delta _3}. \end{aligned}$$

Inserting the parameters computed above, the previous estimate and a short computation shows that with

$$\begin{aligned} \varepsilon _1:= \tau +\varepsilon (r)(1+2/p-3/r), \qquad \varepsilon _2:= 2\tau +\varepsilon (r)(2/p-3/r), \end{aligned}$$

we have

$$\begin{aligned}&\sum _{|\lambda - \frac{1}{pp'}|^{\frac{1}{2}} \le (2\Vert V\Vert _r)^{r/(r(1+2/p)-3)}} {\text {dist}}(\lambda ,\Sigma _p)^{r(1+2/p)-2+\varepsilon _1} \\&\quad \le C \cdot \Vert V\Vert _r^{r} (\Vert V\Vert _r^{r/(r(1+2/p)-3)}+\gamma _p)^{r(2/p+1)-1+\varepsilon _2}. \end{aligned}$$

Note that choosing \(\varepsilon (r)\) sufficiently small we can achieve that \(0< \varepsilon _1, \varepsilon _2 < 4\tau \). Since

$$\begin{aligned} k=r(2-2\gamma _p)-2=r\left( 1+\frac{2}{p}\right) -2 \quad \text {if} \quad p > 2, \end{aligned}$$
(60)

this concludes the proof of Theorem 2.7, part (i).

Similarly, considering ‘large’ eigenvalues we first note that from (36) we obtain, using that here \(\delta _2=0\),

$$\begin{aligned} \sum _{|\lambda - \frac{1}{pp'}|^{\frac{1}{2}}>(2C_1)^{\frac{1}{\alpha +\beta }}} \frac{{\text {dist}}(\lambda ,\Sigma _p)^{\delta _1} }{(|\lambda -\frac{1}{pp'}|^{\frac{1}{2}}+2\gamma _p)^{2\delta _1+ \delta _3+r+\tau }} \le C' \cdot C_1^{r}\left( C_1^{\frac{1}{\alpha +\beta }}+\gamma _p\right) ^{-\tau }. \end{aligned}$$

Inserting the parameters this shows that, with

$$\begin{aligned} \varepsilon _3:= 3\tau + \varepsilon (r)(2/p-3/r) \end{aligned}$$

and k as in (60) we have

$$\begin{aligned}&\sum _{|\lambda - \frac{1}{pp'}|^{\frac{1}{2}} > (2\Vert V\Vert _r)^{r/(k-1)}} \frac{{\text {dist}}(\lambda ,\Sigma _p)^{k+\varepsilon _1} }{(|\lambda -\frac{1}{pp'}|^{\frac{1}{2}}+2\gamma _p)^{k+1+\varepsilon _3}} \\&\quad \le C' \cdot \Vert V\Vert _r^{r}(\Vert V\Vert _r^{r/(k-1)}+\gamma _p)^{-\tau }. \end{aligned}$$

Since we can choose \(\varepsilon (r)\) sufficiently small such that \(0<\varepsilon _1,\varepsilon _3 < 4 \tau \), this shows the validity of part (ii) of Theorem 2.7 and concludes the proof of the theorem.