$L_p$-Spectrum and Lieb–Thirring Inequalities for Schrödinger Operators on the Hyperbolic Plane

Abstract

This paper deals with the $L_p$-spectrum of Schrödinger operators on the hyperbolic plane. We establish Lieb–Thirring-type inequalities for discrete eigenvalues and study their dependence on p. Some bounds on individual eigenvalues are derived as well.

Appendix

1.1 A.1. Operators, Spectra and Perturbations

We introduce terminology and collect some standard results on operators and spectra. As references see, e.g., [23, 24, 31].

(a) X and Y denote complex Banach spaces, and $\mathcal {B}(X,Y)$ denotes the algebra of all bounded linear operators from X to Y. As usual, we set $\mathcal {B}(X):=\mathcal {B}(X,X)$. The spectrum of a closed operator Z in X will be denoted by $\sigma (Z)$ and $\varrho (Z):={{\mathbb {C}}}{\setminus } \sigma (Z)$ denotes its resolvent set. An isolated eigenvalue $\lambda$ of Z will be called discrete if its algebraic multiplicity $m(\lambda ):=\dim ({\text {Ran}}(P_Z(\lambda ))$ is finite. Here,

$$\begin{aligned} P_Z(\lambda )=\frac{1}{2\pi i} \int _\gamma (\mu -Z)^{-1} \hbox {d}\mu \end{aligned}$$

denotes the Riesz projection of Z with respect to $\lambda$ (and $\gamma$ is a counterclockwise oriented circle centered at $\lambda$, with sufficiently small radius). The set of all discrete eigenvalues is called the discrete spectrum $\sigma _d(Z)$. The essential spectrum $\sigma _\mathrm{ess}(Z)$ is defined as the set of all $\lambda \in {{\mathbb {C}}}$, where $\lambda -Z$ is not a Fredholm operator. We have $\sigma _\mathrm{ess}(Z) \cap \sigma _d(Z) = \emptyset$ and if $\Omega \subset {{\mathbb {C}}}{\setminus } \sigma _\mathrm{ess}(Z)$ is a connected component and $\Omega \cap \varrho (Z) \ne \emptyset$, then $\Omega \cap \sigma (Z) \subset \sigma _d(Z)$. Moreover, each point on the topological boundary of $\sigma (Z)$ either is a discrete eigenvalue or a point in the essential spectrum. The discrete eigenvalues of Z can accumulate at the essential spectrum only. Finally, the spectral mapping theorem for the resolvent says that for $a \in \varrho (Z)$ we have

$$\begin{aligned} \sigma ((Z-a)^{-1}) {\setminus } \{0\} = \{ (\lambda -a)^{-1}: \lambda \in \sigma (Z)\}. \end{aligned}$$

A similar identity holds for the essential and the discrete spectra as well. In the latter case, the algebraic multiplicities of $\lambda \in \sigma _d(Z)$ and $(\lambda -a)^{-1} \in \sigma _d((Z-a)^{-1})$ coincide.

(b) In this paper, the sum $Z+M$ of two closed operators Z, M in X will always denote the usual operator sum defined on ${\text {Dom}}(Z) \cap {\text {Dom}}(M)$ (and the product ZM is defined on $\{ f \in {\text {Dom}}(M): Mf \in {\text {Dom}}(Z)\}$). The operator M is called Z-compact if ${\text {Dom}}(Z)\subset {\text {Dom}}(M)$ and $M(Z-a)^{-1}$ is compact for one (hence all) $a \in \varrho (Z)$. If this is the case, the sum $Z+M$ is closed and for $a \in \varrho (Z+M) \cap \varrho (Z)$ also the resolvent difference

$$\begin{aligned} (Z+M-a)^{-1} - (Z-a)^{-1} = -\,(Z+M-a)^{-1}M(Z-a)^{-1} \end{aligned}$$

is compact. In particular, Weyl’s theorem on the invariance of the essential spectrum under compact perturbations and the spectral mapping theorem imply that $\sigma _\mathrm{ess}(Z)=\sigma _\mathrm{ess}(Z+M)$.

(c) If Z is closed and densely defined $Z^*$ denotes its adjoint (see [31, Sections III.5.5 and III.6.6]). The spectrum $\sigma (Z^*)$ is the mirror image of $\sigma (Z)$ with respect to the real axis and $[(\lambda -Z)^{-1}]^*=({{\overline{\lambda }}} - Z^*)^{-1}$. Moreover, $\lambda \in \sigma _d(Z)$ iff ${\overline{\lambda }} \in \sigma _d(Z^*)$ and the respective algebraic multiplicities coincide. Finally, we note that if M is another operator in X and ZM is densely defined, then $(ZM)^* \supset M^*Z^*,$ with equality if $Z \in \mathcal {B}(X)$.

1.2 A.2. $l_r$-Ideals and Perturbation Determinants

We recall some results concerning the construction of perturbation determinants on Banach spaces. The main reference is [27], see also [32, 41].

Let X denote a complex Banach space and let $r>0$. A quasi-normed subspace $({\mathcal {I}}, \Vert .\Vert _{{\mathcal {I}}})$ of $\mathcal {B}(X)$ is called an $l_r$-ideal (in $\mathcal {B}(X)$) with eigenvalue constant$\mu _r>0$ if the following holds:

• (1) The finite rank operators, denoted by ${\mathcal {F}}(X)$, are dense in ${\mathcal {I}}$.

• (2) $\Vert L\Vert \le \Vert L\Vert _{{\mathcal {I}}}$ for all $L \in {\mathcal {I}}$.

• (3) If $L \in {\mathcal {I}}$ and $A,B \in \mathcal {B}(X)$, then $ALB \in {\mathcal {I}}$ and

  $$\begin{aligned} \Vert ALB\Vert _{{\mathcal {I}}} \le \Vert A\Vert \Vert L\Vert _{{\mathcal {I}}} \Vert B\Vert . \end{aligned}$$

• (4) For every $L \in {\mathcal {I}}$, one has $\Vert (\lambda _j(L))\Vert _{l_r} \le \mu _r \Vert L\Vert _{{\mathcal {I}}}$. Here, $(\lambda _j(L))$ denotes the sequence of discrete eigenvalues of L, counted according to their algebraic multiplicity (note that by (1) and (2) each $L \in {\mathcal {I}}$ is compact).

In the present paper, we will need only two particular $l_r$-ideals, which we introduce in the following two examples.

Example A.1

Let $\mathcal {H}$ denote a complex Hilbert space and let $r>0$. The Schatten–von Neumann classes${\mathcal {S}}_r(\mathcal {H})$ are defined by

$$\begin{aligned} {\mathcal {S}}_r(\mathcal {H}) = \{ K \in \mathcal {B}(X) : K \text { is compact and } (s_n(K)) \in l_r\}. \end{aligned}$$

Here, $(s_n(K))$ denotes the sequence of singular values of K. Equipped with the (quasi-) norm $\Vert K\Vert _{{\mathcal {S}}_r} := \Vert (s_n(K))\Vert _{l_r}$ this class is an $l_r$-ideal with eigenvalue constant $\mu _r=1$.

Example A.2

Let $1 \le q \le p < \infty$. An operator $L \in \mathcal {B}(X)$ is called (p, q)-summing if there exists $\varrho >0$ such that for all finite systems of elements $x_1,\ldots ,x_n \in X$ one has

$$\begin{aligned} \left( \sum _{k=1}^n \Vert Lx_k\Vert ^p \right) ^{1/p} \le \varrho \sup _{x' \in X', \Vert x'\Vert \le 1} \left( \sum _{k=1}^n |x'(x_k)|^q\right) ^{1/q}. \end{aligned}$$

We denote the infimum of all such $\varrho >0$ by $\Vert L\Vert _{\Pi _{p,q}}$ and the class of all such operators by $\Pi _{p,q}(X)$. In the special case $p=q$, we speak of p-summing operators and write $\Pi _{p}(X)$. We note that for $1\le q_1 \le q_0 \le p_0 \le p_1 < \infty$ we have $\Pi _{p_0,q_0}(X) \subset \Pi _{p_1,q_1}(X)$ and

$$\begin{aligned} \Vert L\Vert _{\Pi _{p_1,q_1}} \le \Vert L\Vert _{\Pi _{p_0,q_0}}, \qquad L \in \Pi _{p_0,q_0}(X). \end{aligned}$$

(61)

Moreover, if $\mathcal {H}$ is a complex Hilbert space, then for $r \ge 2$ we have $\Pi _{r,2}(\mathcal {H})={\mathcal {S}}_r(\mathcal {H})$ and the corresponding norms coincide. Concerning the above properties of an $l_r$-ideal we note that $\Pi _{p,q}(X)$ always satisfies (2) and (3), and it satisfies (1) if $X'$ has the approximation property and is reflexive, see [27, Remark 5.4]. For such X, we can use known information on the eigenvalue distribution of the (p, q)-summing operators to make the following statements:

(a) $(\Pi _p(X),\Vert .\Vert _{\Pi _p})$ is an $l_{\max (p,2)}$-ideal with eigenvalue constant $\mu _{\max (p,2)} = 1$.

(b) If $p>2$, then the eigenvalues of $L \in \Pi _{p,2}(X)$ are in the weak space $l_{p,\infty }({{\mathbb {N}}})$, see [33]. More precisely, if the eigenvalues are denoted decreasingly $|\lambda _1(L)| \ge |\lambda _2(L)| \ge \ldots$ (where each eigenvalue is counted according to its algebraic multiplicity and the sequence is extended by 0 if there are only finitely many eigenvalues), then

$$\begin{aligned} \sup _{j \in {{\mathbb {N}}}} |\lambda _j(L)| j^{1/p} \le 2e \Vert L\Vert _{\Pi _{p,2}}. \end{aligned}$$

In particular, this implies that for $q>p$ and $n \in {{\mathbb {N}}}$

$$\begin{aligned} \sum _{j=1}^n |\lambda _j(L)|^q = \sum _{j=1}^n \left( |\lambda _j(L)| j^{1/p}\right) ^q j^{-q/p} \le \left( 2e \Vert L\Vert _{\Pi _{p,2}}\right) ^q \sum _{j=1}^\infty j^{-q/p}. \end{aligned}$$

Hence, we see that $\Pi _{p,2}(X)$ is an $l_q$-ideal for every $q>p>2$, with eigenvalue constant $\mu _q^q= (2e)^q \sum _{j=1}^\infty j^{-q/p}$. Moreover, by (61) we see that for $p > r \ge 2$ also $\Pi _{p,r}(X)$ is an $l_q$-ideal for every $q>p$, with the same constant $\mu _q$ as before.

The $l_r$-ideals can be used to construct perturbation determinants on Banach spaces: First, for a finite rank operator $F \in \mathcal F(X)$ and $r>0$ we define

$$\begin{aligned} {\det }_r(I-F):= \prod _j \left( (1-\lambda _j(F))\exp \left( \sum _{k=1}^{\lceil r \rceil -1} \frac{\lambda _j^k(F)}{k} \right) \right) . \end{aligned}$$

(62)

Now, one can show that for every $l_r$-ideal $({\mathcal {I}}, \Vert .\Vert _{{\mathcal {I}}})$ there exists a unique continuous function ${\det }_{r,{\mathcal {I}}}(I-.) : ({\mathcal {I}},\Vert .\Vert _{{\mathcal {I}}}) \rightarrow {{\mathbb {C}}}$ that coincides with $\det _r(I-.)$ on the finite rank operators ${\mathcal {F}}(X)$. Moreover, there exists $\Gamma _r > 0$ such that for all $L \in {\mathcal {I}}$ we have

$$\begin{aligned} |{\det }_{r,{\mathcal {I}}}(I-L)| \le \exp \left( \mu _r^r \Gamma _r \Vert L\Vert _{{\mathcal {I}}}^r\right) , \end{aligned}$$

where $\mu _r$ denotes the eigenvalue constant of ${\mathcal {I}}$. Finally, if $A \in \mathcal {B}(X)$ and $K \in {\mathcal {I}}$, we define the r-regularized perturbation determinantd of A by K (with respect to ${\mathcal {I}}$) as follows:

$$\begin{aligned} d: \varrho (A) \ni \lambda \mapsto {\det }_{r,{\mathcal {I}}}(I-K(\lambda -A)^{-1}). \end{aligned}$$

Then, the following holds: (i) d is analytic on $\varrho (A)$. (ii) $\lim _{|\lambda | \rightarrow \infty } d(\lambda )=1.$ (iii) For $\lambda \in \varrho (A)$, we have

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

(iv) $d(\lambda )=0$ iff $\lambda \in \sigma (A+K)$. (v) If $\lambda \in \varrho (A) \cap \sigma _d(A+K)$, then its algebraic multiplicity as an eigenvalue of $A+K$ coincides with its order as a zero of d.

1.3 A.3. Complex Interpolation

We review some aspects of Calderon’s method of complex interpolation, see [7] or [3].

Let $S := \{ z \in {{\mathbb {C}}}: 0 \le {\text {Re}}(z) \le 1\}$ and let (X, Y) denote an interpolation couple of complex Banach spaces (i.e., X and Y are complex Banach spaces continuously embedded in a topological vector space V). Then, $X \cap Y$ and $X+ Y$ become Banach spaces when equipped with the norms $\Vert z\Vert _{X \cap Y} = \max (\Vert z\Vert _X,\Vert z\Vert _Y)$ and $\Vert z\Vert _{X+Y}=\inf \{ \Vert x\Vert _X + \Vert y\Vert _Y : z = x+y, x \in X, y \in Y\}$, respectively. We denote by $\mathcal G(X,Y)$ the vector space of all functions $f: S \rightarrow X+Y$ which satisfy the following properties:

• f is holomorphic in the interior of S,

• $f \in C_b(S;X+Y)$, i.e., f is continuous and bounded on S,

• $t \mapsto f(it) \in C_b({{\mathbb {R}}};X)$ and $t \mapsto f(1+it) \in C_b({{\mathbb {R}}};Y)$.

Then, ${\mathcal {G}}(X,Y)$ becomes a Banach space with the norm

$$\begin{aligned} \Vert f\Vert _{{\mathcal {G}}(X,Y)}:= \max \left( \sup _{t \in {{\mathbb {R}}}} \Vert f(it)\Vert _X, \sup _{t \in {{\mathbb {R}}}} \Vert f(1+it)\Vert _Y\right) . \end{aligned}$$

For $0< \theta < 1$, the complex interpolation spaces$[X,Y]_\theta$ are introduced as follows:

$$\begin{aligned}{}[X,Y]_\theta = \{ f(\theta ) : f \in {\mathcal {G}}(X,Y)\}, \quad \Vert z\Vert _{[X,Y]_\theta } = \inf _{f \in {\mathcal {G}}(X,Y), f(\theta )=z} \Vert f\Vert _{{\mathcal {G}}(X,Y)}. \end{aligned}$$

One can show that

$$\begin{aligned} X \cap Y \subset [X,Y]_\theta \subset X+Y, \end{aligned}$$

both embeddings being continuous.

Example A.3

For a $\sigma$-finite measure space $(M,\mu )$ and $p_0,p_1 \in [1,\infty ]$, we have

$$\begin{aligned}{}[L_{p_0}(M),L_{p_1}(M)]_\theta = L_p(M), \quad \text {where} \quad \frac{1}{p} = \frac{1-\theta }{p_0} + \frac{\theta }{p_1}. \end{aligned}$$

Moreover, the corresponding norms coincide.

Proposition A.4

Let $f \in {\mathcal {G}}(X,Y)$ and set

$$\begin{aligned} A_0= \sup _{t \in {{\mathbb {R}}}} \Vert f(it)\Vert _X \quad \text {and} \quad A_1=\sup _{t \in {{\mathbb {R}}}} \Vert f(1+it)\Vert _Y. \end{aligned}$$

Then, $\Vert f(\theta )\Vert _{[X,Y]_\theta } \le A_0^{1-\theta } A_1^\theta .$

Proof

If $A_0, A_1 \ne 0$, the function $g(w):= ({A_0}/{A_1})^{w-\theta } f(w), w \in S,$ is in ${\mathcal {G}}(X,Y)$ with $g(\theta )=f(\theta )$ and

$$\begin{aligned} \sup _{t \in {{\mathbb {R}}}} \Vert g(it)\Vert _X \le A_0^{1-\theta } A_1^\theta \quad \text {and} \quad \sup _{t \in {{\mathbb {R}}}} \Vert g(1+it)\Vert _Y \le A_0^{1-\theta } A_1^\theta . \end{aligned}$$

Hence, $\Vert f(\theta )\Vert _{[X,Y]_\theta } \le \Vert g(\theta )\Vert _{\mathcal G(X,Y)} \le A_0^{1-\theta } A_1^\theta .$ If one of $A_0,A_1$ vanishes, we can replace it by $\varepsilon > 0$ in the definition of g and then send $\varepsilon \rightarrow 0$. If $A_0=A_1=0$, we can choose $g(w)=0$ to obtain the result. $\square$

In this paper, we will not need interpolation results for operators between abstract interpolation spaces. However, we will need the following more concrete result known as the Stein interpolation theorem [47] (see also [48]).

Remark A.5

Let us recall that a simple function on a measure space $(M,\mu )$ is a finite linear combination of characteristic functions of measurable sets of finite measure.

In the following, we denote the norm of $L_p(M)$ by $\Vert .\Vert _p$ and the operator norm of $T:L_p \rightarrow L_q$ by $\Vert T\Vert _{p,q}$.

Theorem A.6

Let $(M,\mu )$ and $(N,\nu )$ be $\sigma$-finite measure spaces and assume that for every $w \in S$, $T_w$ is a linear operator mapping the space of simple functions on M into measurable functions on N. Moreover, suppose that for all simple functions $f: M \rightarrow {{\mathbb {C}}}$ and $g : N \rightarrow {{\mathbb {C}}}$, the product $T_wf \cdot g$ is integrable and that

$$\begin{aligned} S \ni w \mapsto \int _N (T_wf)(x)g(x) \nu (dx) \end{aligned}$$

is continuous and bounded on S and holomorphic in the interior of S. Finally, suppose that for some $p_j,q_j \in [1,\infty ], j=0,1,$ and $A_0,A_1 \ge 0$ we have

$$\begin{aligned} \Vert T_{it}f\Vert _{{q_0}} \le A_0 \Vert f\Vert _{{p_0}}, \quad \Vert T_{1+it}f\Vert _{{q_1}} \le A_1 \Vert f\Vert _{{p_1}} \end{aligned}$$

for all $t \in {{\mathbb {R}}}$ and all simple functions $f: M \rightarrow {{\mathbb {C}}}$. Then, for each $\theta \in (0,1)$ and

$$\begin{aligned} 1 / {p_\theta } = {(1-\theta )}/ {p_0} + {\theta } /{p_1}, \quad 1/ {q_\theta } = {(1-\theta )}/ {q_0} + {\theta }/ {q_1}, \end{aligned}$$

the operator $T_\theta$ can be extended to a bounded operator in $\mathcal {B}(L_{p_\theta }(M),L_{q_\theta }(N))$ and

$$\begin{aligned} \Vert T_\theta \Vert _{{p_\theta },{q_\theta }} \le A_0^{1-\theta } A_1^\theta . \end{aligned}$$