# Injecting Text and Cross-lingual supervision in few-shot learning from Self-
Supervised Models

###### Abstract

Self-supervised model _pre-training_ has recently garnered significant
interest, but relatively few efforts have explored using additional resources
in _fine-tuning_ these models. We demonstrate how universal phoneset acoustic
models can leverage cross-lingual supervision to improve transfer of
pretrained self-supervised representations to new languages. We also show how
target-language text can be used to enable and improve fine-tuning with the
lattice-free maximum mutual information (LF-MMI) objective. In three low-
resource languages these techniques greatly improved few-shot learning
performance.

Index Terms—  Self-supervised, few-shot learning, lattice-free MMI, cross-
lingual ASR

##  1 Introduction

In the last decade, automatic speech recognition (ASR) systems have benefited
from supervised training of deep neural networks on transcribed speech at
scale. However, since annotated resources for low-resource languages are often
scarce or expensive, considerable efforts have been made towards unsupervised
and semi-supervised learning for ASR, resulting in methods such as
representation learning, pseudo-labeling, local prior matching, and
adversarial training [1, 2, 3, 4]. Recently, self-supervised approaches [5, 6,
7, 8, 9, 10] have become popular for learning general representations of the
input distribution without supervision. In particular, wav2vec2.0 (W2V2) [11]
has been used for downstream tasks such as cross-lingual ASR [12], speech
translation [13], and speaker and language identification [14].

Fig. 1: Our proposed integration of cross-lingual transcribed speech and
unpaired target language text in LF-MMI-based fine-tuning of large self-
supervised models. Cross-lingual supervision is injected using a sequence-
trained universal phone recognizer at the output layer. We use unpaired target
language text to estimate the denominator graph used for LF-MMI computation.

Prior work has investigated improved self-supervised pre-training of models
using side information including (i) using a text-to-speech (TTS) module to
inject text [15], (ii) using teacher models to improve representation learning
[16], and (iii) moving average based learning from a fixed supervised teacher
[17]. However, research on few-shot fine-tuning with these models has received
less attention. In this work, we develop ASR models for new languages using
_very_ small amounts (say, 15 mins.) of transcribed speech. However, we have
access to additional resources, including cross-lingual transcribed and
untranscribed speech, as well as unpaired target language data. We explore the
best strategies to incorporate these in _fine-tuning_ large self-supervised
pre-trained models.

###  1.1 Related Work

Most recent research on fine-tuning of self-supervised models is in the “end-
to-end” framework by learning a linear output layer using the connectionist
temporal classification (CTC) objective [18]. In contrast, “hybrid” ASR
systems which combine acoustic, pronunciation, and language models for
decoding with weighted finite state transducers (WFSTs) [19] are popular in
industry due to their compositionality and efficiency. We focus on such hybrid
ASR models.

Acoustic model transfer in hybrid ASR has used the lattice-free MMI (LF-MMI)
objective [20] — in conjunction with methods such as weight transfer, multi-
task training, and teacher-student learning [21, 22] — for fine-tuning models.
One outstanding problem in LF-MMI based model transfer is how best to
construct the denominator graph when using only small amounts of domain-
matched transcribed speech. This issue was raised in [23], but has not been
explicitly addressed in a multilingual setting.

Recently, Vyas et al. [24] studied the application of supervised fine-tuning
using LF-MMI on self-supervised acoustic models, and showed that
representations learned on English transferred well to low-resource languages
such as Swahili. However, there have been no investigations about the use of
additional resources, such as cross-lingual supervision or unpaired target
language text, in fine-tuning from large self-supervised models.

We leverage ideas from prior work [25, 26, 27] which explored language-
_independent_ approaches for cross-lingual transfer by relying on a universal
phoneset, such as the IPA [28], to train phone models by mixing training data
from all languages. More recently, the sequence training of universal phoneset
models using LF-MMI was explored with various neural architectures [29, 30],
while [31] explored _cross_ -lingual transfer of acoustic models trained using
CTC. Our main contributions in this paper are three-fold:

  1. 1.

We propose using a cross-lingual universal phoneset model for few-shot
learning. This significantly outperforms the conventional method of fine-
tuning by constructing a new output layer.

  2. 2.

We leverage untranscribed speech by exploring the choice of pre-trained, self-
supervised models used in fine-tuning. Experiments show that using matched-
domain and matched-language data in pretraining significantly improves
downstream performance.

  3. 3.

We simultaneously leverage unparied, target language text while addressing the
question of how to construct the denominator graph in LF-MMI based fine-
tuning, using unpaired text to improve the LF-MMI denominator graph phone
language model.

##  2 Technical Preliminaries

The hybrid HMM-DNN framework for speech recognition follows the Bayes
decomposition of the ASR objective. Given a speech input $\textbf{x}^{(u)}$,
the most likely sequence $\hat{\mathbf{w}}^{(u)}$ is given as

$\displaystyle\hat{\mathbf{w}}^{(u)}$
$\displaystyle=\text{arg}\max_{\mathbf{w}}p(\mathbf{w}|\textbf{x}^{(u)})=\text{arg}\max_{\mathbf{w}}p(\mathbf{x}^{(u)}|\mathbf{w})p(\mathbf{w})$
(1)
$\displaystyle=\text{arg}\max_{\mathbf{w},\mathbf{\ell}}\underbrace{p(\mathbf{x}^{(u)}|\mathbf{\ell})}_{H}\underbrace{p(\mathbf{\ell}|\mathbf{w})}_{L}\underbrace{p(\mathbf{w})}_{G},$
(2)

where each term, $H$, $L$, $G$, are estimated independently. $H$ is a
collection of hidden Markov models (HMM), one per (context-dependent) phone,
where a neural network models HMM state emission probabilities. Together, the
HMMs and neural network are termed the acoustic model. The acoustic model
scores can be combined with a pronunciation lexicon, $L$, and language model
(grammar), $G$, to produce scores for hypothesis sequences, w. One advantage
of these models is the ability to optimize each component independently. In
this work, this property enables the re-use of acoustic models trained to
produce units from a universal phoneset.

The acoustic model is often trained using a sequence discriminative objective
such as maximum mutual information (MMI), which aims to maximize the the
point-wise mutual information between the speech signal and reference
transcripts:

$\mathcal{F}_{\text{MMI}}=\sum_{u\in
U}\log\frac{p_{\lambda}(\textbf{x}^{(u)}|\mathbb{M}_{w}^{(u)})}{p_{\lambda}(\textbf{x}^{(u)}|\mathbb{M}_{\text{den}})}.$
(3)

Here, $\mathbb{M}_{w}$, called the numerator graph, is a graph representing
the set of all alignments and pronunciations corresponding to the ground-truth
sequence $w$. $\mathbb{M}_{\text{den}}$, called the denominator graph, is
ideally, an HMM modeling _all_ possible word sequences. The LF-MMI objective
[20] maximizes a lower-bound on the mutual information [32] by using un-
normalized distributions parameterized by a neural network. The denominator
graph is approximated with a 4-gram phone language model. In other words, the
LF-MMI objective is _globally_ normalized, as opposed to CTC, which assumes a
uniform prior between output units and is _locally_ normalized. $\lambda$ is
the set of all HMM parameters, including the DNN that estimates the
probability of HMM states at each time step.

Contrastive self-supervised models aim to learn representations of speech that
maximize a lower bound on the mutual information between a chunk of audio
$\mathbf{x}_{t}$ at time $t$ and its surrounding context $\mathbf{c}_{t}$
using the InfoNCE objective. In wav2vec 2.0 [11], a convolutional feature
extractor generates latent speech representations $\mathbf{z}$ from the
waveform, which are then processed by a Transformer to produce contextualized
representations $\mathbf{c}$. A quantization module converts the continuous
latent vectors $\mathbf{z}$ to discrete representations $\mathbf{q}$. The pre-
training objective predicts the correct quantized vector $\mathbf{q}_{t}$ by
contrasting against distractors. The emission probabilities in DNN-HMMs can be
replaced by a linear transformation of the outputs $\mathbf{c}$, of self-
supervised models. General practice, which we follow, is to freeze the
parameters of the feature extractor during fine-tuning.

##  3 Method

In this section, we will describe our method for injecting each of the 3
different “additional” resources (shown in Fig. 1) — cross-lingual transcribed
speech, untranscribed speech, and unpaired target language text — into few-
shot learning.

###  3.1 Cross-lingual transcribed speech

We use transcribed cross-lingual speech to train a universal phoneme-sequence
model, which we refer to as “multi-phoneset” (as opposed to a “mono-phoneset”
obtained from just the target language). We hypothesize that multi-phoneset
models can be used, to an extent, in new languages without transcribed speech.

To verify this, we conducted some preliminary experiments on hat, where we
analyzed the performance of wide ResNet (WRN) based mono- and multi-phoneset
models when fine-tuned on various amounts of data (Fig. 2). We found that in
the few-shot scenario ($<$1k utterances), the multi-phoneset model
outperformed the conventional approach of retraining a monolingual output
layer. Our goal is to replicate this behavior using larger self-supervised
models capable of producing better pre-trained representations.

$10^{2}$$10^{3}$$10^{4}$$60$$80$$100$# UtterancesWER %Multi-phonesetMono-
phonesetScratch

Fig. 2: Comparison between multi and mono phoneset outputs with increasing
training data on WRN models on hat. In the few-shot scenario ($<$ 1k
utterances), using a shared, universal phoneset provides significant WER
improvements over a monolingual baseline.

###  3.2 Untranscribed speech

We explore the value of using matched-domain and matched-language
untranscribed speech in self-supervised pre-training. [33] demonstrated the
clear value of pre-training on matched acoustic channel untranscribed speech.
Furthermore, some of the improvements of self-supervised contrastive learning
appear to transfer across language [34, 35], and [12] demonstrated that
representations trained on large and diverse cross-lingual speech could
outperform those trained on small amounts of matched-language speech. Using
pre-trained models on speech in new languages results in both mismatch in
language and/or acoustic channel (different auditory environments). It is not
immediately clear which, if any, of these are more harmful for downstream
performance. To investigate these questions, we performed fine-tuning
experiments with publicly available W2V2 models pre-trained on 4 different
kinds of data:

  1. 1.

LV60111https://hf.co/facebook/wav2vec2-large-lv60: Trained on 53.2k hours of
English audiobook recordings from Libri-Light [1]. Few-shot transfer using
this model performs remarkable well on LibriSpeech [36].

  2. 2.

Large-Robust222https://hf.co/facebook/wav2vec2-large-robust: Trained on
English audiobooks (LibriLight [1]), conferences (TED-Lium [37]), and
telephone conversations (Switchboard) [33].

  3. 3.

VoxPopuli-100k333https://hf.co/facebook/wav2vec2-large-100k-voxpopuli: Trained
on 100k hours of speech in 23 languages from the VoxPopuli corpus [38]. The
data is obtained from European parliamentary recordings (and only contains
European languages).

  4. 4.

XLSR-53444https://hf.co/facebook/wav2vec2-large-xlsr-53: Trained on 56k hours
of speech in 53 languages, including CommonVoice and BABEL corpora. It has
previously shown strong cross-lingual transfer performance for ASR and phoneme
recognition tasks [12, 39].

Table 1: Previous SOTA WERs (using FLP training) for the languages used for
fine-tuning experiments in this paper.

Language Model Data (h) WER Pashto (pus) CTC-BLSTM + lang. adversarial [40]
0.2k 43.4 Haitian (hat) Multi-lingual CTC-BLSTM [31] 0.5k 44.3 Georgian (kat)
XLSR-53 w/ CTC fine-tuning [12] 56k 31.1

###  3.3 Unpaired target language text

Conventionally, unpaired target language text is used to train language models
which are then fused with acoustic model outputs during decoding or rescoring.
In contrast, we explore how to use these resources during _training_ of the
acoustic model. Recall from §2 that the LF-MMI objective, contrary to the CTC
objective (often used in fine-tuning self-supervised models), enables us to
incorporate a language model in training through the denominator graph
$\mathbb{M}_{\text{den}}$, which can be supplemented with additional unpaired,
or cross-lingual text. We conjecture that such denominator graph expansion may
lead to better loss computation by estimating a prior which is closer to the
true target distribution (as opposed to CTC’s uniform prior), particularly in
the few-shot scenario.

##  4 Experimental Setup

###  4.1 Data

Prior work in _few-shot_ transfer has mainly used clean, read speech. A
motivation of this work was to study how these representations transfer in
moderately challenging, conversational, and multilingual speech. The BABEL
corpus [41] was developed to support research in ASR and keyword search in
such conditions. The speech data are 8kHz telephone conversations in 25
languages. Each language has two training sets: the full language pack (FLP)
(between 40h and 80h); and the limited language pack (LLP) (10h subset of
FLP). A 10h development split (dev10h) is also defined for each language. We
report performance on dev10h as separate evaluation sets are not publicly
available. To study cross-lingual transfer in the few-shot scenario, we
created 200-utterance subsets from the LLP set for each language (or 15 min.
of speech total from 2-4 speakers).

Pronunciation lexicons containing X-SAMPA [42] pronunciations of all words
seen in the FLP and dev10h sets are provided. X-SAMPA phonemes are an ASCII
representation of the IPA [28], which was constructed to be a cross-lingual
finite symbol set representing all attested phonemes in the world. This label-
set enables training universal phoneset models.

We use 21 languages (XL-21) as our cross-lingual supervision, and report
performance of all our systems on held-out languages, Pashto (pus), Haitian
(hat), and Georgian (kat). Importantly, the FLP audio from these languages are
_included_ in the XLSR-53 training data. In Table 1, we report the state-of-
the-art WERs on these languages using various methods from literature. We used
a 1h subset of the Haitian dev10h for early stopping and hyper-parameter
tuning.

Table 2: Effect of cross-lingual supervision on downstream few-shot WER
performance, using the 15-min train subset. Column 2 shows the amount of
(labeled / unlabeled) data in pre-training.

Model Pre-training (h) pus hat kat XL-21 WRN Mono 0 / 1.2k 86.2 81.5 89.2
XL-21 WRN Multi 0 / 1.2k 75.6 72.9 73.1 XL-21 Wav2Vec 2.0 (random) 0 / 1.2k
77.2 77.9 76.0 XLSR-53 56k / 0 75.9 74.7 81.5 XLSR-53 + XL-21 56k / 1.2k 71.0
67.8 64.5 XLSR-53 (frozen) + XL-21 Multi 56k / 1.2k 63.6 58.9 58.2

Table 3: Effect of pre-training with untranscribed cross-lingual data on the
downstream fine-tuning WER. †The WERs can be reduced to 38.8, 36.9, and 32.2,
respectively, if we decode with an expanded vocabulary with no OOVs.

Model Pre-training (hours) Few-shot ($\sim$15m) LLP ($\sim$10h) FLP
($\sim$80h) Labeled Unlabeled pus hat kat pus hat kat pus hat kat Random WRN 0
0 93.2 95.4 95.4 61.1 57.7 57.6 50.1 46.6 49.5 XL-21 WRN Mono 1.2k 0 86.2 81.5
89.2 56.6 52.3 55.1 44.7 43.3 46.2 LV60 0 60k 81.4 84.2 86.0 50.3 50.3 50.5
42.8 40.8 43.1 Large-Robust 0 63k 81.2 80.1 93.0 49.5 49.1 48.7 41.8 40.4 42.2
VoxPopuli-100k 0 100k 80.3 77.2 85.6 48.9 48.0 47.9 41.1 39.3 41.5 XLSR-53 0
56k 75.9 74.7 81.5 45.5 45.3 45.1 39.4† 37.7† 40.0†

###  4.2 Training details

We used 2 types of acoustic models: 28-10 wide ResNets (WRN) [43], and the
W2V2-based models and architectures. All monolingual and cross-lingual
supervised baselines are reported using the WRN model; for example, XL-21 WRN
refers to the WRN model pre-trained on the 21 BABEL languages with
supervision.

We conducted fine-tuning experiments on the few-shot, LLP, and FLP subsets,
where the training alignments for the few-shot and LLP were obtained by
training GMM-HMMs on the LLP subset, and those for FLP by training on the FLP
data, respectively. We use the cross-lingual GMM-HMMs, occasionally remapping
missing phonemes in the target language to a similar phoneme seen in training
to create the targets for fine-tuning with the ”Multi” models. We trained all
acoustic models with PyTorch [44] and used Kaldi [45] for decoding. We used
Pychain for the LF-MMI implementation [46]. All decoding was performed with a
3-gram language model trained on the FLP transcripts in srilm [47]. This
conforms with a common scenario where larger amounts of unpaired text are more
readily available than transcribed speech and is different from denominator
graph expansion, which we used during _training_. Our implementation and
recipes are publicly available555https://github.com/m-wiesner/nnet_pytorch.
The denominator graph phone-LMs for the “multi” models combined the cross-
lingual and target language training data, weighting the target-language data
by a factor of 10 as in [23].

##  5 Results and Discussion

###  5.1 Effect of phone set expansion

Table 2 shows the few-shot transfer of multilingual baselines in the three
target languages. We find that transferring the supervised cross-lingual
representation from WRN models (row-2) works better than reconstructing the
output layer (row-1). Training the cross-lingual supervised model with the
W2V2 neural architecture (row-3) performed slightly worse than the WRN
baseline. Self-supervised vs. supervised pre-training (rows 3 v. 4) made
little difference in the few-shot scenario. There were likely not enough data
to retrain a output transformations of the self-supervised representations.

We compare 2 approaches that combine the supervised, sequence-level universal
phoneset model, with the pretrained self-supervised models. In the XLSR-53 +
XL-21 (row-5), we performed two sequential fine-tuning steps: first on the
cross-lingual representations, and then on the target language few-shot
supervisions. We also try freezing the XLSR-53 network when fine-tuning on the
XL-21 data (row-6). This was the best-performing approach, almost matching the
_supervised_ models trained on 10h of audio! We suspect that freezing the W2V2
model prevents catastrophic forgetting during cross-lingual fine-tuning, while
simultaneously enabling us to learn a universal phoneset model. Few-shot
learning by transferring a universal output layer worked significantly better
than reconstructing the output layer using small amounts of speech as in [11,
12].

pushatkat$60$$70$$80$WER %RegularExpanded (4-gram)Expanded (2-gram)

Fig. 3: Effect of denominator graph expansion (regular vs. expanded) for LF-
MMI based few-shot cross-lingual transfer of the XLSR-53 model.

###  5.2 Analysis of pre-trained W2V2 models

Self-supervised pre-training outperformed XL-21 based supervised pre-training
in all scenarios, as shown in Table 3, but we found a clear hierarchy among
pretrained models. (i) Matched unlabeled data is king – XLSR-53 was the best
performing base-model and improved over a cross-lingual supervised WRN
baseline by 10-15% relative across the board. It was trained on the smallest
amount of, but _matched_ acoustic and language, data. The LV60 model is
trained on a similar amount of data, but only clean English audio from
LibriLight [1]. It was by far the worst performing model. (ii) Linguistic
diversity, quantity of data, and matched acoustic conditions are important.
For instance, the VoxPopuli-100k model, like the LV60 model, is trained on
clean wide-band speech. However, presumably due to the quantity, and
multilingual (but not matched language) nature, of training data, it was the
second best performing model. Finally, the Robust model, despite training on
much less, and purely _monolingual_ English speech, performed comparably to
the VoxPopuli model, likely thanks to the telephone speech (similar to BABEL)
that it was exposed to in training. Increasing the amount of fine-tuning data,
reduced the difference in performance among models.

###  5.3 Effect of denominator graph expansion

Table 4: Effect of $n$-gram order and unpaired text weight, $\alpha$, on WER in LF-MMI few-shot learning of Haitian ASR models. Weight ($\alpha$) | n=0 | n=1 | n = 2 | n = 3 | n = 4
---|---|---|---|---|---
$\alpha=0.0$ | 81.9 | 66.2 | 65.2 | 68.2 | 74.7
$\alpha=0.1$ | 81.9 | 66.1 | 65.0 | 66.3 | 67.6
$\alpha=0.2$ | 81.9 | 66.3 | 65.1 | 66.1 | 66.8
$\alpha=0.5$ | 81.9 | 65.9 | 65.2 | 66.3 | 67.2

Using additional, unpaired text to train the denominator graph phone-LM
reduces overfitting. Fig. 3, shows that using the resulting ”expanded” LF-MMI
denominator graphs improves the WER across all languages in the few shot
setting. However, using lower-order $n$, in the phone-LM $n$-gram model would
result in less over-fitting. We analyzed the trade-off between modeling-power
and unpaired text weight in the Haitian few-shot scenario with XLSR-53 (Table
4).

The best preforming systems use $n=2$, demonstrating the benefit of using LF-
MMI in fine-tuning as opposed to CTC, which corresponds to the $n=0$ case with
a different HMM topology. We also found that including the unpaired text with
a small weight improved performance across all, but especially higher-order
$n$. Using additional text enables the use of higher-order n-gram models even
in low-resource scenarios. We also see that the improvements from the
universal phoneset model cannot be attributed solely to the inclusion of
cross-lingual phone sequences when training the denominator graph phone-LM.

##  6 Conclusion

In this work, we have shown how cross-lingual supervision can be used during
fine-tuning in few-shot learning to significantly improve transfer of self-
supervised models. We confirmed the importance of using matched-domain speech
in pre-training, and we demonstrated how to incorporate external text in the
phone-LM of the LF-MMI denominator graph in order to help guide fine-tuning.
These techniques resulted in significantly improved few-shot performance in
three low-resource test conditions.

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# Degeneration of the spectral gap with negative Robin parameter

Derek Kielty<EMAIL_ADDRESS>
###### Abstract.

The spectral gap of the Neumann and Dirichlet Laplacians are each known to
have a sharp positive lower bound among convex domains of a given diameter.
Between these cases, for each positive value of the Robin parameter an
analogous sharp lower bound on the spectral gap is conjectured. In this paper
we show the extension of this conjecture to negative Robin parameters fails
completely, by proving the spectral gap of an explicit family of domains can
be exponentially small.

##  1\. Introduction

The Robin Laplacian eigenvalue problem with parameter $\alpha$ on a bounded
Lipschitz domain $\mathcal{D}$ is

$\begin{cases}-\Delta u=\lambda u,\quad&\text{on}\leavevmode\nobreak\
\mathcal{D}\\\ \partial_{\nu}u+\alpha u=0,\quad&\text{on}\leavevmode\nobreak\
\partial\mathcal{D}.\end{cases}$

It has a discrete spectrum of eigenvalues for each
$\alpha\in(-\infty,\infty)$:

$\lambda_{1}(\mathcal{D})<\lambda_{2}(\mathcal{D})\leq\lambda_{3}(\mathcal{D})\leq\dots\to\infty.$

Recall that when $\alpha=0$ this is the Neumann eigenvalue problem. The
limiting case $\alpha=\infty$ corresponds to Dirichlet boundary conditions.
See [8, Chapter 4] for foundational material on the Robin problem.

In this paper we prove that when $\alpha<0$ the spectral gap

$(\lambda_{2}-\lambda_{1})(\mathcal{D})$

can be arbitrarily small among domains $\mathcal{D}$ of a given diameter.
Investigation of this result for $\alpha<0$ was motivated by the following
conjecture of Andrews, Clutterbuck, and Hauer [3, §10] for $\alpha\geq 0$:

######  Conjecture 1 (Robin gap).

If $\alpha\geq 0$ and $\mathcal{D}\subset\mathbb{R}^{n}$ is a bounded convex
domain then

$(\lambda_{2}-\lambda_{1})(\mathcal{D})\geq(\lambda_{2}-\lambda_{1})(I),$

where $I\subset\mathbb{R}$ is an open interval of length equal to the diameter
of $\mathcal{D}$.

The Robin gap conjecture is known to hold in the Neumann case ($\alpha=0$) due
to Payne and Weinberger [16]. A new proof was given by Andrews and Clutterbuck
[2, pp. 901]. In the Dirichlet case ($\alpha=\infty$) it is known, not only
for the Laplacian, but also for Schrödinger operators with a convex potential,
thanks to Andrews and Clutterbuck [2, Corollary 1.4].

Figure 1. The _double cone_ domain $\mathcal{D}_{\theta}$ in 3-dimensions with
$\theta=\pi/5$.

To state our main result, define the family of convex _double cone_ domains of
angle $\theta$ by

$\mathcal{D}_{\theta}=\\{(x,y)\in(-1,1)\times\mathbb{R}^{n-1}:|y|<\tan(\theta/2)(1-|x|)\\},\quad\text{for
each}\leavevmode\nobreak\ \theta\in(0,\pi),$

as shown in Figure 1.

######  Theorem 2 (Degeneration of the Robin gap).

If $\alpha<0$ then $(\lambda_{2}-\lambda_{1})(\mathcal{D}_{\theta})\to 0$ as
$\theta\to 0$. In fact, if $\alpha<0$ then for each $\epsilon>0$ there exists
a constant $C>0$ such that

$(\lambda_{2}-\lambda_{1})(\mathcal{D}_{\theta})\leq
C\exp\\{-4(1-\epsilon)|\alpha|/\theta\\},\quad\text{for all }\theta\text{
sufficiently small}.$ (1)

The scaling relation
$\lambda_{j}(t\mathcal{D};\alpha)=t^{-2}\lambda_{j}(\mathcal{D};t\alpha)$ then
shows that $(\lambda_{2}-\lambda_{1})(t\mathcal{D}_{\theta})\to 0$ as
$\theta\to 0$, for each $t>0$. Since $\mathcal{D}_{\theta}$ has diameter $2$
for each $\theta\in(0,\pi/2]$, Theorem 2 implies that among convex
$\mathcal{D}$ of any given diameter, the spectral gap
$(\lambda_{2}-\lambda_{1})(\mathcal{D})$ can be made arbitrarily small when
$\alpha<0$. In particular, since $(\lambda_{2}-\lambda_{1})(I)>0$ there exist
convex domains $\mathcal{D}$ such that
$(\lambda_{2}-\lambda_{1})(\mathcal{D})<(\lambda_{2}-\lambda_{1})(I)$, and so
the Robin gap conjecture fails to extend to $\alpha<0$.

We believe the upper bound in Theorem 2 is sharp in the sense that the
estimate (1) fails when $\epsilon=0$ as $\theta\to 0$. Although we do not
prove this the proof of our upper bound on $\lambda_{2}(\mathcal{D}_{\theta})$
and Lemma 3 suggests it is true.

The remainder of this paper is structured as follows: in Section 2 we discuss
a heuristic explaining Theorem 2, in Section 3 we present a brief overview of
relevant literature, in Section 4 we prove a lemma about convergence of
Schrödinger eigenvalues, and in Section 5 we prove Theorem 2. Finally, in
Section 6 we suggest a variant of Theorem 2 to a modification of the domains
$\mathcal{D}_{\theta}$ and discuss the possibility of extending Conjecture 1
for $\alpha<0$ to a restricted class of domains.

##  2\. A heuristic for small gaps in Theorem 2

The Robin eigenvalue problem with $\alpha<0$ on the infinite cone of opening
angle $\theta<\pi$

$\mathcal{C}_{\theta}=\\{(x,y)\in(0,\infty)\times\mathbb{R}^{n-1}:|y|<\tan(\theta/2)x\\},$
(2)

has at least one negative eigenvalue and a corresponding ground state (see
equation (16)) that concentrates exponentially fast (in $L^{2}$-norm), as
$\theta\to 0$. In the 2-dimensional case, it is known that _all_ the
eigenfunctions concentrate in this fashion (see [11, Theorem 5.1]).

Because of this concentration we expect the eigenfunctions of
$\mathcal{D}_{\theta}$ concentrate at each vertex $(\pm 1,0)$ like the ground
state of $\mathcal{C}_{\theta}$ concentrates at the origin. Consequently, the
Robin problem on $\mathcal{D}_{\theta}$ should be approximated by the Robin
problem on two copies of $\mathcal{C}_{\theta}$ as $\theta\to 0$. This
approximation suggests $(\lambda_{2}-\lambda_{1})(\mathcal{D}_{\theta})\to 0$
as $\theta\to 0$ since the Robin problem on
$\mathcal{C}_{\theta}\sqcup\mathcal{C}_{\theta}$ has a spectral gap of zero
because the first eigenvalue has multiplicity two.

##  3\. Literature

For an overview of the Robin Laplacian and related shape optimization problems
see the article by Bucur, Freitas, and Kennedy [8, Chapter 4]. In the study of
super conductors, the Robin Laplacian with negative parameter is also known
for its connection to Ginsburg-Landau theory [9, §4].

Recently, Ashbaugh and the current author [5], as well as Andrews,
Clutterbuck, and Hauer [4] proved sharp lower bounds on the Robin spectral gap
for 1-dimensional Schrödinger operators with various classes of single-well
potentials. In contrast with Theorem 2, some of these lower bounds continue to
hold for some $\alpha<0$.

The Robin Laplacian on the infinite cone with small aperture will play a
crucial role throughout this paper. In work of Khalile and Pankrashkin [11]
they computed eigenvalue asymptotics for the small aperture limit of
2-dimensional infinite cones. This geometric limit is related to the
$\alpha\to-\infty$ limit, which was studied by Khalile [10] who proved that
the first $N=N(\mathcal{D})$ Robin eigenvalues of a fixed curvilinear polygon
$\mathcal{D}$ are determined by the angles of the vertices as
$\alpha\to-\infty$ (to leading order). For smooth domains the second term in
the $\alpha\to-\infty$ asymptotic is determined by the mean curvature; see
work of Pankrashkin and Popoff [15]. In related work, Bögli, Kennedy, and Lang
[6] proved convergence of certain eigenvalues of the Robin Laplacian to those
of the Dirichlet Laplacian for certain limits $\alpha\to\infty$, for complex
$\alpha$.

##  4\. Convergence of Schrödinger eigenvalues on expanding balls

In this section we prove the convergence and calculate the convergence rate of
the first eigenvalue $E=E(R)$ of the Schrödinger eigenvalue problem

$\begin{cases}(-\Delta-\frac{\kappa}{r})\varphi=E\varphi,\quad&\text{on}\leavevmode\nobreak\
B(R)\\\ (\partial_{r}+\gamma)\varphi=0,\quad&\text{on}\leavevmode\nobreak\
\partial B(R),\end{cases}$ (3)

as $R\to\infty$, where $B(R)$ is the ball of radius $R$ centered at the
origin, $\kappa>0$ is constant, and $\gamma\in(-\kappa/(n-1),\infty]$. When
$\gamma=\infty$ we impose Dirichlet boundary conditions on $\partial B(R)$.
Since problem (3) is radial it reduces to an ODE eigenvalue problem that we
analyze using special functions. In the proof of Theorem 2 we will get a lower
bound on $\lambda_{1}(\mathcal{D}_{\theta})$ by proving a lower bound on
$\lambda_{1}(\mathcal{D}_{\theta})-\lambda_{1}(\mathcal{C}_{\theta})$ in terms
of $E(R)$ with $R=\cot(\theta/2)$, as $\theta\to 0$. Combining this lower
bound with an upper bound on
$\lambda_{2}(\mathcal{D}_{\theta})-\lambda_{1}(\mathcal{C}_{\theta})$ will
prove Theorem 2.

The aforementioned ODE eigenvalue problem gives rise to the _confluent
hypergeometric equation_

$\rho\frac{d^{2}F}{d\rho^{2}}+(b-\rho)\frac{dF}{d\rho}-aF=0,$ (4)

for certain $a\in\mathbb{R}$ and $b\notin\\{0,-1,-2,\dots\\}$. The solution
$F=F(a,b;\rho)$ is a linear combination of the confluent hypergeometric
functions of the first and second kind, $M(a,b;\rho)$ and $U(a,b;\rho)$,
respectively. For more information on this ODE see Chapter 13 by Slater of the
book edited by Abramowitz and Stegun [1].

In Lemma 3 we show that the eigenvalues of (3) with radial eigenfunctions are
determined by the positive solutions $z$ of the transcendental equation

$2\frac{a_{0}}{b_{0}}M(a_{0}+1,b_{0}+1;\rho_{0})=(1-2\gamma
z/\kappa)M(a_{0},b_{0};\rho_{0}),$ (5)

where $m=(n-3)/2$, $a_{0}=a_{0}(z)=m+1-z$, $b_{0}=2m+2$, and
$\rho_{0}=\rho_{0}(z)=\kappa R/z$. When $\gamma=\infty$ the transcendental
equation is $M(a_{0},b_{0};\rho_{0})=0$. To see this case formally, divide by
$\gamma$ in (5) and let $\gamma\to\infty$.

Part (ii) of the following lemma shows that the first eigenvalue of (3)
converges to
$-\kappa^{2}/(n-1)^{2}$ as $R\to\infty$, which is the first eigenvalue of
$-\Delta-\kappa/r$ on $\mathbb{R}^{n}$.

######  Lemma 3.

Assume that $\gamma\in(-\infty,\infty]$ and $\kappa>0$ are fixed.

(i) If $R<\infty$ and $E\neq 0$ then the radial eigenfunctions of (3) are
multiples of

$M(a_{0},b_{0};2\sqrt{|E|}r)e^{-\sqrt{|E|}r}$ (6)

and have eigenvalues $E=\operatorname{sgn}(E)\kappa^{2}/4z^{2}$, where $z$ is
a positive solution of (5). If $R=\infty$ (i.e. the domain is
$\mathbb{R}^{n}$) then the radial eigenfunctions (that decay as $r\to\infty$)
of (3) are multiples of (6) with eigenvalues $E=-\kappa^{2}/4(j+(n-3)/2)^{2}$
for $j\in\mathbb{N}$. Furthermore, $M(a_{0},b_{0};2\sqrt{|E|}r)$ is a
polynomial of degree $j-1$.

(ii) Assume further that $\gamma\in(-\kappa/(n-1),\infty]$. For each
$\epsilon>0$, the smallest positive solution $z$ of (5) satisfies
$z=(n-1)/2+O(\exp\\{2(1-\epsilon)\kappa R/(n-1)\\})$, as $R\to\infty$.
Consequently, the first eigenvalue $E$ of (3) satisfies

$E(R)=-\kappa^{2}/(n-1)^{2}+O\bigg{(}\exp\bigg{\\{}-(1-\epsilon)\frac{2\kappa}{n-1}R\bigg{\\}}\bigg{)},\quad\text{as
}R\to\infty.$

Computing the spectrum of the Schrödinger operator $-\Delta-\kappa/r$ on
$B(R)$ and $\mathbb{R}^{n}$ is closely related to computing the spectrum of
the hydrogen atom from quantum mechanics. In fact, on $\mathbb{R}^{3}$ this
calculation is precisely the hydrogen atom calculation (see the book by Landau
and Lifschitz [12, Chapter V], for example). For general $n\geq 3$, the ODE in
(7) below is the radial part of the Schrödinger operator for the 3-D hydrogen
atom with angular moment $m$ (after changing variables). This physical
interpretation breaks down when $n=2$ since in this case $m<0$. The operator
in (7) also appears in work of Khalile and Pankrashkin [11] as an effective
operator for the Robin Laplacian with negative Robin parameter on
$\mathcal{C}_{\theta}$ as $\theta\to 0$, when $n=2$.

###### Proof of Lemma 3.

Part (i): Let $\varphi=\varphi(r,\xi)$ be an arbitrary eigenfunction of (3)
with eigenvalue $E=E(R)$ written in $(r,\xi)$-spherical coordinates. Since
problem (3) is radially symmetric, $\varphi$ factors as
$\varphi(r,\xi)=u(r)Y_{\ell}(\xi)$, where $Y_{\ell}$ is a spherical harmonic
(eigenfunction of the Laplace–Beltrami operator on $\mathbb{S}^{n-1}$) with
eigenvalue $\ell(\ell+n-2)$ for $\ell\in\mathbb{N}\cup\\{0\\}$.

Assume that $\ell=0$ so that $\varphi(r,\xi)$ is a constant times $u(r)$. In
this case, $-\Delta-\kappa/r$ reduces to the radial part of the Laplacian plus
the potential $-\kappa/r$ so that

$\begin{cases}-u^{\prime\prime}-\frac{n-1}{r}u^{\prime}-\frac{\kappa}{r}u=Eu,\quad&\text{on}\leavevmode\nobreak\
(0,R)\\\ (u^{\prime}+\gamma u)(R)=0.\end{cases}$

In the above problem and the ones below we impose Dirichlet boundary
conditions when $\gamma=\infty$. Let $m=(n-3)/2$ and $v(r)=r^{m+1}u(r)$ so
that

$\begin{cases}-v^{\prime\prime}+\frac{m(m+1)}{r^{2}}v-\frac{\kappa}{r}v=Ev,\quad&\text{on}\leavevmode\nobreak\
(0,R)\\\ (v^{\prime}+(\gamma-\frac{m+1}{R})v)(R)=0.\end{cases}$ (7)

Since $E\neq 0$ define $z=\kappa(2\sqrt{|E|})^{-1}>0$ and rescale the radial
coordinate by defining $\rho=\kappa r/z$ and let $w(\rho)=v(z\rho/\kappa)$. It
follows that $w$ satisfies

$\begin{cases}-w^{\prime\prime}+\frac{m(m+1)}{\rho^{2}}w-(\frac{z}{\rho}-\frac{1}{4})w=0,\quad&\text{on}\leavevmode\nobreak\
(0,\rho_{0})\\\
(w^{\prime}+(\frac{z\gamma}{\kappa}-\frac{m+1}{\rho_{0}})w)(\rho_{0})=0,\end{cases}$

where $\rho_{0}=\kappa R/z$.

Guessing a solution of the form $w(\rho)=\rho^{m+1}e^{-\rho/2}F(\rho)$ we find
that $F(\rho)=F(a_{0},b_{0};\rho)$ is a solution of (4) on the interval
$(0,\rho_{0})$ with $a_{0}(z)=m+1-z$, $b=b_{0}=2m+2$, and boundary condition
$2F^{\prime}(a_{0},b_{0};\rho_{0})=(1-z\gamma/\kappa)F(a_{0},b_{0};\rho_{0})$.
When $\gamma=\infty$ the boundary condition $F(a_{0},b_{0};\rho_{0})=0$ is
immediate from the form of the ansatz. Thus, the original radial
eigenfunctions $\varphi$ are multiples of
$F(a_{0},b_{0};2\sqrt{|E|}r)e^{-\sqrt{|E|}r}$.

The general solution of (4) is a linear combination of the confluent
hypergeometric functions of the first and second kind,
$M(a_{0},b_{0};\rho)$and $U(a_{0},b_{0};\rho)$, respectively. It follows from
formulas 13.4.21, 13.5.5, and 13.5.7 in [1] that

$U^{\prime}(a_{0},b_{0};\rho)=-a_{0}\frac{\Gamma(b_{0})}{\Gamma(a_{0}+1)}\rho^{-b_{0}}+o(\rho^{-b_{0}}),\quad\text{as
}\rho\to 0,$

so that $U^{\prime}(a_{0},b_{0};\rho)\not\in L^{2}(B(R))$ since $b_{0}=n-1$.
Thus, $F(\rho)$ must have no $U(a_{0},b_{0};\rho)$ component to guarantee that
$\varphi\in H^{1}(B(R))$. This shows that $F(\rho)=M(a_{0},b_{0};\rho)$.

Finally, using the boundary condition we computed for $F$ and the formula
$M^{\prime}(a,b;\rho)=\frac{a}{b}M(a+1,b+1;\rho)$ [1, formula 13.4.8] we
deduce that $z$ must satisfy the transcendental equation (5). Conversely, each
positive solution $z$ to the transcendental equation defines an eigenvalue
$E=\operatorname{sgn}(E)\kappa^{2}/4z^{2}$ corresponding to a radial
eigenfunction of (3). Thus, the eigenvalues $E$ corresponding to radial
eigenfunctions of (3) are in one-to-one correspondence with solutions $z$ of
(5), and part (i) of the lemma on $B(R)$ follows from this correspondence.

The same calculation shows that on $\mathbb{R}^{n}$ the eigenfunctions are
multiples of (6) with eigenvalue $-\kappa^{2}/4z^{2}$ and that
$w(\rho)=\rho^{m+1}e^{-\rho/2}M(a_{0},b_{0};\rho)$. Since we are assuming that
the original eigenfunction $\varphi$ tends to zero as $r\to\infty$, the
function $w$ grows at most polynomially. If $a$ and $b$ are fixed and
$a\notin\\{0,-1,-2,\dots\\}$ then $M(a,b;\rho)\sim\rho^{a-b}e^{\rho}$ as
$\rho\to\infty$ (see [1, 13.1.4]). If $a\in\\{0,-1,-2,\dots\\}$ then
$M(a,b;\rho)$ is a polynomial of degree $-a$ (see (9)). Thus, we must have
$a_{0}\in\\{0,-1,-2,\dots\\}$ to ensure $\varphi$ tends to zero as
$r\to\infty$. It follows from the definition of $a_{0}=m+1-z$ that
$z=\kappa(2\sqrt{|E|})^{-1}$ and $E=-\kappa^{2}/4(j+(n-3)/2)^{2}$ for
$j\in\mathbb{N}$.

Part (ii): We split the the proof into two cases depending on the size of
$|\gamma|$. Recall that $\kappa>0$ and $m=(n-3)/2$.

Case 1: $|\gamma|<\kappa/(2(m+1))$: We show that the smallest positive
$z=z(\rho_{0})$ that solves (5) exists and tends to $m+1$ from below
exponentially fast, as $\rho_{0}\to\infty$ (equivalently as $R\to\infty$ in
our original variables). We do this by inverting this problem to show that
there is a $\rho_{0}=\rho_{0}(z)$ that solves (5) exists and tends to
infinity, as $z\nearrow m+1$. First we fix $c>1$ and $0<z<m+1$ close to $m+1$
and apply the intermediate value theorem to the function

$\mathcal{M}(\rho)=2\frac{a_{0}}{b_{0}}M(a_{0}+1,b_{0}+1;\rho)-(1-2\gamma
z/\kappa)M(a_{0},b_{0};\rho).$

Since $\mathcal{M}(\rho)=0$ if and only if $\rho$ satisfies the transcendental
equation, we show that for $c$ and $z$ as above there exists a $\rho_{0}(z)$
such that $\mathcal{M}(\rho_{0}(z))=0$ and

$\rho_{-}(z)\leq\rho_{0}(z)\leq\rho_{+}(z),$ (8)

where

$\rho_{-}(z)=\log(1-2\gamma
z/\kappa)+\log\bigg{(}\frac{m+1}{m+1-z}\bigg{)}\quad\text{and}\quad\rho_{+}(z)=c\log\bigg{(}\frac{m+1}{m+1-z}\bigg{)}.$

(Note we need $z$ large enough so that $0<\rho_{-}(z)<\rho_{+}(z)$). The
desired convergence for $E(R)$ as $R\to\infty$ follows from solving these
inequalities for $z$ and using the definition of $z$ and $\rho_{0}$.

We prove (8) by showing that $\mathcal{M}(\rho)<0$ for $\rho<\rho_{-}(z)$ and
that $\mathcal{M}(\rho)>0$ for $\rho>\rho_{+}(z)$ in order to apply the
intermediate value theorem. To prove these estimates on $\mathcal{M}$ we use
the power series representation

$M(a,b;\rho)=\sum_{k=0}^{\infty}\frac{a^{(k)}}{b^{(k)}}\frac{\rho^{k}}{k!},\quad\text{for
}a\in\mathbb{R},\leavevmode\nobreak\ b\notin\\{0,-1,-2,\dots\\},$ (9)

where $a^{(k)}$ is the rising factorial. That is, $a^{(0)}=1$ and
$a^{(k)}=a(a+1)\cdots(a+k-1)$ for $k\geq 1$. Observe that the intermediate
value theorem applies because $\rho\mapsto M(a,b;\rho)$ is analytic.

Since $z<m+1$ we know $M(a_{0},b_{0};\leavevmode\nobreak\
\cdot\leavevmode\nobreak\ )$ is positive due to (9). Thus, as soon as we prove
that $\rho_{0}(z)$ exists for some $0<z<m+1$ we know that it corresponds to
the first eigenvalue $E$ of (3) since the ground state is the unique positive
eigenfunction.

First we prove $\mathcal{M}(\rho)<0$ when $0<\rho<\rho_{-}(z)$. Using that
$a_{0}+1=m+2-z<2m+3=b_{0}+1$ we have

$M(a_{0}+1,b_{0}+1;\rho)<\sum_{k=0}^{\infty}\frac{\rho^{k}}{k!}=e^{\rho}.$

From this estimate and that $\rho<\rho_{-}(z)$ we have

$2\frac{a_{0}}{b_{0}}M(a_{0}+1,b_{0}+1;\rho)<\frac{m+1-z}{m+1}e^{\rho}<(1-2\gamma
z/\kappa)<(1-2\gamma z/\kappa)M(a_{0},b_{0};\rho),$

since $M(a_{0},b_{0};0)=1$ and $\rho\mapsto M(a_{0},b_{0};\rho)$ is
increasing. Thus, $\mathcal{M}(\rho)<0$ when $\rho<\rho_{-}(z)$.

Now we prove $\mathcal{M}(\rho)>0$ when $\rho>\rho_{+}(z)$ and $z<m+1$ is
sufficiently close to $m+1$. First observe that for each $\delta\in(0,1)$
there is a $K$ independent of $z$ such that

$2\frac{k+m+1-z}{k+2m+2}\geq 1+\delta,\quad\text{for each }k\geq K.$ (10)

The $k^{\text{th}}$ coefficient of the power series of the first term of
$\mathcal{M}$ satisfies the following identity by considering the first and
last factors in the rising factorials

$2\frac{a_{0}}{b_{0}}\frac{(a_{0}+1)^{(k)}}{(b_{0}+1)^{(k)}}=2\frac{m+1-z}{2m+2}\frac{(m+2-z)^{(k)}}{(2m+3)^{(k)}}=2\frac{k+m+1-z}{k+2m+2}\frac{(m+1-z)^{(k)}}{(2m+2)^{(k)}},$

for each $k\geq 0$. Applying the lower bound (10) we have

$2\frac{a_{0}}{b_{0}}\frac{(a_{0}+1)^{(k)}}{(b_{0}+1)^{(k)}}\geq(1+\delta)\frac{a_{0}^{(k)}}{b_{0}^{(k)}},\quad\text{for
each }k\geq K.$ (11)

Dropping the first $K$ terms in the power series of $M(a_{0}+1,b_{0}+1;\rho)$,
applying (11), and adding and subtracting $1+\delta$ times the first $K$ terms
in the series for $M(a_{0},b_{0};\rho)$ shows

$\displaystyle
2\frac{a_{0}}{b_{0}}M(a_{0}+1,b_{0}+1;\rho)\geq(1+\delta)\sum_{k=K}^{\infty}\frac{a_{0}^{(k)}}{b_{0}^{(k)}}\frac{\rho^{k}}{k!}=(1+\delta)M(a_{0},b_{0};\rho)-(1+\delta)\sum_{k=0}^{K-1}\frac{a_{0}^{(k)}}{b_{0}^{(k)}}\frac{\rho^{k}}{k!}.$

Subtracting and adding $(2\gamma z/\kappa)M(a_{0},b_{0};\rho)$ to the right
side shows

$2\frac{a_{0}}{b_{0}}M(a_{0}+1,b_{0}+1;\rho)\geq(1-2\gamma
z/\kappa)M(a_{0},b_{0};\rho)+\mathcal{E}(\rho),$ (12)

where

$\mathcal{E}(\rho)=(\delta+2\gamma
z/\kappa)M(a_{0},b_{0};\rho)-(1+\delta)\sum_{k=0}^{K-1}\frac{(m+1-z)^{(k)}}{(2m+2)^{(k)}}\frac{\rho^{k}}{k!}.$

To conclude that $\mathcal{M}(\rho)>0$ from (12) we will show that
$\mathcal{E}(\rho)>0$ for $\rho\geq\rho_{+}(z)$, as $z\nearrow m+1$, by
reducing to showing that $\mathcal{E}(\rho_{+}(z))>0$. First, notice that for
$\delta$ sufficiently close to 1 the factor $\delta+2\gamma z/\kappa$ is
positive since $\gamma>-\kappa/(2(m+1))$. Consequently,
$\rho\mapsto\mathcal{E}(\rho)$ is an increasing function for fixed $z$ and
large $\rho$. In particular, the first term of $\mathcal{E}$ is bounded from
below by $C\rho^{K}$ for large $\rho$, where $C>0$ is independent of $\rho$,
and the second term is only a polynomial of degree $K-1$. Thus, it suffices to
prove $\mathcal{E}(\rho_{+}(z))>0$, as $z\nearrow m+1$. Since the second term
of $\mathcal{E}(\rho_{+}(z))$ behaves like $\log(1/(m+1-z))^{K-1}$, as
$z\nearrow m+1$, it is enough to show that $M(a_{0},b_{0};\rho_{+}(z))$ grows
like a negative power of $1+m-z$, as $z\nearrow m+1$.

We estimate the factor $a_{0}^{(k)}/b_{0}^{(k)}$ from below using that
$m+1-z>0$ and converting the rising factorials to ordinary factorials so that
for $k\geq 1$

$\frac{a_{0}^{(k)}}{b_{0}^{(k)}}=\frac{(m+1-z)^{(k)}}{(2m+2)^{(k)}}\geq(m+1-z)\frac{(k-1)!}{(2m+2)^{(k)}}=(m+1-z)\frac{(k-1)!(2m+1)!}{(2m+2+(k-1))!}.$

In what follows $C$ is a positive constant that is independent of both $k$ and
$z$ that may change from line to line. Noting that $(2m+1)!$ only depends on
the dimension $n$ and making cancellations leads to the lower bound

$\frac{a_{0}^{(k)}}{b_{0}^{(k)}}\geq(m+1-z)\frac{C}{(k+2m+1)^{2m+2}},\quad\text{for
}k\geq 0.$

Using the above lower bound, factoring $c=c^{1/2}c^{1/2}>1$, and using the
Taylor series for $e^{\rho}$ we have

$\displaystyle M(a_{0},b_{0};\rho_{+}(z))$ $\displaystyle\geq$ $\displaystyle
C(m+1-z)\sum_{k=0}^{\infty}\bigg{(}\frac{c^{k/2}}{(k+2m+1)^{2m+2}}\bigg{)}\frac{1}{k!}[c^{1/2}\log((m+1)/(m+1-z))]^{k}$
$\displaystyle\geq$ $\displaystyle
C(m+1-z)\sum_{k=0}^{\infty}\frac{1}{k!}[c^{1/2}\log((m+1)/(m+1-z))]^{k}$
$\displaystyle\geq$ $\displaystyle\frac{C}{(m+1-z)^{c^{1/2}-1}}.$

Since $c>1$ we know $M(a_{0},b_{0};\rho_{+}(z))$ blows up algebraically so
that $\mathcal{E}(\rho_{+}(z))$ is positive as $z\nearrow m+1$. Thus,
$\mathcal{M}(\rho)>0$ for all $\rho\geq\rho_{+}(z)$ and $z<m+1$ sufficiently
large.

Since $\rho\mapsto\mathcal{M}(a,b;\rho)$ is continuous there exists a solution
$\rho_{0}=\rho_{0}(z)$ to (5) that satisfies the inequalities (8) by the
intermediate value theorem. Exponentiating these inequalities, using that
$1-2\gamma z/\kappa$ is uniformly positive in $z$, solving for $z$, and using
the definition $\rho_{0}=\kappa R/z$ we find that for $R$ sufficiently large
there are constants $C_{1},C_{2}>0$ such that

$m+1-C_{1}e^{-\kappa R/cz}\leq z\leq m+1-C_{2}e^{-\kappa R/z}.$

Using $z<m+1=(n-1)/2$ and that $c>1$ is arbitrary we that

$\frac{n-1}{2}-C_{1}\exp\bigg{\\{}-(1-\epsilon)\frac{2\kappa}{n-1}R\bigg{\\}}\leq
z\leq\frac{n-1}{2}+C_{2}\exp\bigg{\\{}-(1-\epsilon)\frac{2\kappa}{n-1}R\bigg{\\}},$

for all $R$ sufficiently large and for each $\epsilon>0$. This shows the first
estimate in part (ii) of the lemma for $|\gamma|<\kappa/(2(m+1))$.

From the definition $z=\kappa(2|E|^{1/2})^{-1}$ and that $E<0$ we find that

$E(R)=-\kappa^{2}/(n-1)^{2}+O\bigg{(}\exp\bigg{\\{}-(1-\epsilon)\frac{2\kappa}{n-1}R\bigg{\\}}\bigg{)},\quad\text{as
}R\to\infty,\quad\text{for each }\epsilon>0,$ (13)

by linearizing the reciprocal function. This completes the proof of part (ii)
of the lemma for $|\gamma|<\kappa/(2(m+1))$.

Case 2: $\gamma\in[\kappa/(2(m+1)),\infty]$: In the remainder of the proof let
$E(R,\gamma)$ denote the first eigenvalue of the Schrödinger problem (3). To
prove the asymptotic for $E(R,\gamma)$ when
$\gamma\in[\kappa/(2(m+1)),\infty]$ notice that $\gamma\mapsto E(R,\gamma)$ is
increasing for $\gamma\in(-\infty,\infty]$ by the min-max principle. In
particular, $E(R,0)\leq E(R,\gamma)\leq E(R,\infty)$ for each $\gamma>0$.
Thus, the right side of (13) is a lower bound for $E(R,\gamma)$ and it
suffices to prove it is also an upper bound for the first Dirichlet eigenvalue
$E(R,\infty)$.

By part (i) of the lemma, the ground state of the $-\Delta-\kappa/r$ on
$\mathbb{R}^{n}$ is $\exp\\{-\sqrt{|E(\infty)|}r\\}$ with eigenvalue
$E(\infty)=-\kappa^{2}/(n-1)^{2}$. Let $\chi$ be a smooth radial cutoff
function that is 1 for $r<1-\epsilon$, decreases to zero for $1-\epsilon\leq
r\leq 1$, and is zero for $r>1$. The function
$f(r)=\chi(r/R)\exp\\{-\sqrt{|E(\infty)|}r\\}$ restricted to the ball $B(R)$
is a valid trial function for the first eigenvalue $E(R,\infty)$ of (3) so
that

$\displaystyle E(R,\infty)$
$\displaystyle\leq\frac{\int_{B(R)}f(-\Delta-\kappa/r)f\,dV}{\int_{B(R)}f^{2}\,dV}$
$\displaystyle=\frac{E(\infty)\int_{B((1-\epsilon)R)}\exp\\{-2\sqrt{|E(\infty)|}r\\}\,dV+\int_{B(R)\setminus
B((1-\epsilon)R)}f(-\Delta-\kappa/r)f\,dV}{\int_{B((1-\epsilon)R)}\exp\\{-2\sqrt{|E(\infty)|}r\\}\,dV+\int_{B(R)\setminus
B((1-\epsilon)R)}f^{2}\,dV}$
$\displaystyle=E(\infty)+O\bigg{(}\exp\\{-(1-\epsilon)\frac{2\kappa}{n-1}R\\}\bigg{)},\quad\text{as
}R\to\infty,\quad\text{for each }\epsilon>0,$

The last equality follows from using the decay of
$\exp\\{-\sqrt{|E(\infty)|}r\\}$ and that $\chi(r/R)$ has bounded
$r$-derivatives. These facts show that the second terms in the numerator and
denominator are the same size as the error term in (13) since
$E(\infty)=-\kappa^{2}/(n-1)^{2}$.

This completes the proof of the asymptotic for $E(R,\gamma)$ for all
$\gamma\in(-\kappa/(2(m+1)),\infty]$. Note that the asymptotic for $z$ in part
(ii) of the lemma follows from the formula
$z=\kappa(2\sqrt{|E(R,\gamma)|})^{-1}$. ∎

##  5\. Proof of Theorem 2

We prove Theorem 2 by getting an upper bound on
$\lambda_{2}(\mathcal{D}_{\theta})$ and a lower bound on
$\lambda_{1}(\mathcal{D}_{\theta})$ in terms of the first eigenvalue of the
infinite cone $\mathcal{C}_{\theta}$ (defined in (2)). Specifically, for each
$\epsilon>0$ we show that there exists a $C>0$ such that

$\lambda_{2}(\mathcal{D}_{\theta})\leq\lambda_{1}(\mathcal{C}_{\theta})+C\exp\\{2(1-\epsilon)\alpha/\sin(\theta/2)\\},\quad\text{as
}\theta\to 0,$ (14)

by a trial function argument using (16) and

$\lambda_{1}(\mathcal{D}_{\theta})=\lambda_{1}(\mathcal{C}_{\theta})+O(\exp\\{2(1-\epsilon)\alpha/\sin(\theta/2)\\}),\quad\text{as
}\theta\to 0,$ (15)

by using a combination of trial function arguments and Lemma 3.

To prove both inequality (14) and (15) we will need to use the ground state of
the infinite cone $\mathcal{C}_{\theta}$ to construct trial functions. A
direct calculation shows the $L^{2}$-normalized ground state of the Robin
Laplacian on $\mathcal{C}_{\theta}$ is

$\phi_{\theta}(x,y)=A_{\theta}e^{\alpha x/\sin(\theta/2)},\quad\text{when
}\alpha<0\text{ and }\theta<\pi,$ (16)

with eigenvalue
$\lambda_{1}(\mathcal{C}_{\theta})=-\alpha^{2}/\sin^{2}(\theta/2)$, where the
constant satisfies

$A_{\theta}^{-2}=|\alpha|^{-n}\omega_{n-1}\Gamma(n)\tan^{n-1}(\theta/2)\sin^{n}(\theta/2).$
(17)

Here $\omega_{n-1}$ is the volume of the $(n-1)$-dimensional unit sphere and
$\Gamma$ is the Gamma function.

Upper bound on $\lambda_{2}(\mathcal{D}_{\theta})$: In order to get an upper
bound on $\lambda_{2}(\mathcal{D}_{\theta})$ we transplant two copies of
$\phi_{\theta}$ to construct a trial function. Fix an $\epsilon>0$ and let
$\chi=\chi(x,y)$ denote a smooth cutoff function on $\mathbb{R}^{n}$ that is
constant in $y$, equals 1 for $x\leq 1-\epsilon$, 0 for $x\geq 1$, and is less
than 1 everywhere. Let $\psi_{\theta}=\chi\phi_{\theta}$ be a cut-off cone
ground state and

$F(x,y)=(x+1,y)\quad\text{and}\quad G(x,y)=(1-x,y)$

be rigid motions. Define the trial function
$\tilde{\phi}_{\theta}:\mathcal{D}_{\theta}\to\mathbb{R}$ by putting a cut-off
cone ground state at each vertex of $\mathcal{D}_{\theta}$:

$\tilde{\phi}_{\theta}=\frac{(\psi_{\theta}\circ F)-(\psi_{\theta}\circ
G)}{\sqrt{2}}.$

Notice that $\tilde{\phi}_{\theta}$ is odd with respect to $x$, by
construction.

Since $\mathcal{D}_{\theta}$ is reflection symmetric in $x$ each eigenfunction
with simple eigenvalue is either even or odd in $x$. The ground state is even
since it is non-negative. Thus, the odd function $\tilde{\phi}_{\theta}$ is
orthogonal to the ground state, and so it is a valid trial function for
$\lambda_{2}(\mathcal{D}_{\theta})$. By the min-max principle,

$\lambda_{2}(\mathcal{D}_{\theta})\leq\frac{\int_{\mathcal{D}_{\theta}}|\nabla\tilde{\phi}_{\theta}|^{2}\,dV+\alpha\int_{\partial\mathcal{D}_{\theta}}\tilde{\phi}_{\theta}^{2}\,dS}{\int_{\mathcal{D}_{\theta}}\tilde{\phi}_{\theta}^{2}\,dV}=\frac{\int_{\mathcal{T}_{\theta}}|\nabla\psi_{\theta}|^{2}\,dV+\alpha\int_{\Sigma_{\theta}}\psi_{\theta}^{2}\,dS}{\int_{\mathcal{T}_{\theta}}\psi_{\theta}^{2}\,dV},$
(18)

where we define

$\mathcal{T}_{\theta}=\mathcal{C}_{\theta}\cap\\{x<1\\}\quad\text{and}\quad\Sigma_{\theta}=\partial\mathcal{T}_{\theta}\cap\\{0\leq
x<1\\},$

to be the truncated cone and the curved part of its boundary.

A straightforward calculation will show there are remainder terms

$\operatorname{Rem}_{i}(\theta)=O(\exp\\{2(1-2\epsilon)\alpha/\sin(\theta/2)\\}),\quad\text{as
}\theta\to 0,\quad\text{for }i=1,2,3,$

such that

$\int_{\mathcal{T}_{\theta}}|\nabla\psi_{\theta}|^{2}\,dV=\int_{\mathcal{C}_{\theta}}|\nabla\phi_{\theta}|^{2}\,dV+\operatorname{Rem}_{1}(\theta),\quad\int_{\Sigma_{\theta}}\psi_{\theta}^{2}\,dV=\int_{\partial\mathcal{C}_{\theta}}\phi_{\theta}^{2}\,dV+\operatorname{Rem}_{2}(\theta),$

and

$\int_{\mathcal{T}_{\theta}}\psi_{\theta}^{2}\,dV=\int_{\mathcal{C}_{\theta}}\phi_{\theta}^{2}\,dV+\operatorname{Rem}_{3}(\theta)=1+\operatorname{Rem}_{3}(\theta),$

as $\theta\to 0$. Combining these estimates with the upper bound (18) shows we
have a remainder term
$\operatorname{Rem}(\theta)=(\operatorname{Rem}_{1}(\theta)+\alpha\operatorname{Rem}_{2}(\theta))/(1+\operatorname{Rem}_{3}(\theta))=O(\exp\\{2(1-2\epsilon)\alpha/\sin(\theta/2)\\})$
as $\theta\to 0$ such that

$\lambda_{2}(\mathcal{D}_{\theta})\leq\frac{\int_{\mathcal{C}_{\theta}}|\nabla\phi_{\theta}|^{2}\,dV+\alpha\int_{\partial\mathcal{C}_{\theta}}\phi_{\theta}^{2}\,dS}{\int_{\mathcal{T}_{\theta}}\psi_{\theta}^{2}\,dV}+\operatorname{Rem}(\theta)=\frac{\lambda_{1}(\mathcal{C}_{\theta})}{\int_{\mathcal{T}_{\theta}}\psi_{\theta}^{2}\,dV}+\operatorname{Rem}(\theta)\leq\lambda_{1}(\mathcal{C}_{\theta})+\operatorname{Rem}(\theta),$

as $\theta\to 0$ since $\int_{\mathcal{T}_{\theta}}\psi_{\theta}^{2}\,dV\leq
1$ and $\lambda_{1}(\mathcal{C}_{\theta})<0$. This proves (14) since
$\epsilon$ is arbitrary.

For the sake of completeness we compute $\operatorname{Rem}_{3}(\theta)$; the
other remainder terms are similar. The inequality
$\int_{\mathcal{T}_{\theta}}\psi_{\theta}^{2}\,dV\leq\int_{\mathcal{C}_{\theta}}\phi_{\theta}^{2}\,dV$
follows from $\chi$ being less than one. For the opposite inequality, since

$\int_{T_{\theta}}\psi_{\theta}^{2}\,dV\geq\int_{\mathcal{C}_{\theta}\cap\\{x<1-\epsilon\\}}\phi_{\theta}^{2}\,dV=\int_{\mathcal{C}_{\theta}}\phi_{\theta}^{2}\,dV-\int_{\mathcal{C}_{\theta}\cap\\{x\geq
1-\epsilon\\}}\phi_{\theta}^{2}\,dV,$

it suffices to show the second term on the right is exponentially small. Since
$\phi_{\theta}$ is constant in $y$ we have that

$\displaystyle\int_{\mathcal{C}_{\theta}\cap\\{x\geq
1-\epsilon\\}}\phi_{\theta}^{2}\,dV$
$\displaystyle=A_{\theta}^{2}\int_{1-\epsilon}^{\infty}\exp\\{2\alpha
x/\sin(\theta/2)\\}\text{Vol}(B(\tan(\theta/2)x))\,dx$
$\displaystyle=\Gamma(n)^{-1}\int_{2(1-\epsilon)|\alpha|/\sin(\theta/2)}^{\infty}\exp\\{-y\\}y^{n-1}\,dy,$

where we have made the change of variables $y=|\alpha|x/\sin(\theta/2)$ and
used the definition of $A_{\theta}$ in (17). Since
$\exp\\{-y\\}y^{n-1}\lesssim\exp\\{-\frac{1-2\epsilon}{1-\epsilon}y\\}$ for
all sufficiently large $y$, by integrating we have

$\int_{\mathcal{C}_{\theta}\cap\\{x\geq
1-\epsilon\\}}\phi_{\theta}^{2}\,dV\lesssim\exp\\{2(1-2\epsilon)\alpha/\sin(\theta/2)\\},\quad\text{as
}\theta\to 0,$

which shows that
$\operatorname{Rem}_{3}(\theta)=O(\exp\\{2(1-2\epsilon)\alpha/\sin(\theta/2)\\})$,
as $\theta\to 0$.

Lower bound on $\lambda_{1}(\mathcal{D}_{\theta})$: We prove the lower bound
by viewing
$\lambda_{1}(\mathcal{D}_{\theta})-\lambda_{1}(\mathcal{C}_{\theta})$ as the
first eigenvalue of a problem on the truncated cone $\mathcal{T}_{\theta}$.
Then we “push out” the problem on $\mathcal{T}_{\theta}$ to a radial problem
on a spherical sector $\mathcal{S}_{\theta}$. Finally, we transform the
problem on $\mathcal{S}_{\theta}$ into the Schrödinger eigenvalue problem (3)
whose solutions were analyzed in Lemma 3. We break up the proof into four
steps.

Step 1: Let $u$ denote the $L^{2}$-normalized ground state of
$\mathcal{D}_{\theta}$ restricted and translated to
$\mathcal{T}_{\theta}=\mathcal{C}_{\theta}\cap\\{x<1\\}$. Since the ground
state of $\mathcal{D}_{\theta}$ is even in $x$, we know $u$ has Neumann
boundary conditions on the flat part of the boundary defined by

$\Gamma_{\theta}=\overline{\mathcal{T}_{\theta}}\cap\\{x=1\\}$

and retains its Robin boundary conditions on the complement
$\Sigma_{\theta}=\partial\mathcal{T}_{\theta}\cap\\{0\leq x<1\\}$.

For notational convenience write $\phi$ for the Robin ground state of
$\mathcal{C}_{\theta}$, instead of $\phi_{\theta}$. A direct calculation shows
that the ratio $v=u/\phi$ is the first eigenfunction of the $\tau$-Laplacian
operator $\Delta_{\tau}(\cdot)=\tau^{-1}\text{div}(\tau\nabla(\cdot))$ with
$\tau=\phi^{2}$. More specifically, $v$ satisfies

$\begin{cases}-\Delta_{\tau}v=\mu_{1}v,&\text{on }\mathcal{T}_{\theta}\\\
\partial_{\nu}v+(\alpha/\sin(\theta/2))v=0,&\text{on }\Gamma_{\theta}\\\
\partial_{\nu}v=0,&\text{on }\Sigma_{\theta},\end{cases}$ (19)

where
$\mu_{1}=\mu_{1}(\mathcal{T}_{\theta})=\lambda_{1}(\mathcal{D}_{\theta})-\lambda_{1}(\mathcal{C}_{\theta})$
is the first eigenvalue of (19).

Since $\phi^{2}$ is uniformly positive on $\overline{\mathcal{T}_{\theta}}$,
the above eigenvalue problem has a discrete spectrum with Rayleigh quotient

$\frac{\int_{\mathcal{T}_{\theta}}|\nabla
f|\phi^{2}\,dx+(\alpha/\sin(\theta/2))\int_{\Gamma_{\theta}}f^{2}\phi^{2}\,dS}{\int_{\mathcal{T}_{\theta}}f^{2}\phi^{2}\,dx},\quad\text{for
}f\in H^{1}(\mathcal{T}_{\theta})\setminus\\{0\\}.$

Note that $\mu_{1}(\mathcal{T}_{\theta})<0$ by using the min-map principle
with a constant trial function.

Step 2: Consider the spherical coordinates $(r,\xi)$, where
$\xi=(\xi_{1},\dots,\xi_{n-1})$ is the vector of angle coordinates and
$\xi_{1}$ is the angle between the point $(r,\xi)$ and the $x$-axis. We will
show $\mu_{1}(\mathcal{S}_{\theta})\leq(1+o(1))\mu_{1}(\mathcal{T}_{\theta})$,
as $\theta\to 0$, where $\mu_{1}(\mathcal{S}_{\theta})$ is the first
eigenvalue of a radial problem on the spherical sector

$\mathcal{S}_{\theta}=\\{(r,\xi):0<r<\cos(\theta/2)^{-1},\leavevmode\nobreak\
|\xi_{1}|<\theta/2\\}.$

To define this problem let the diffeomorphism
$P:\mathcal{S}_{\theta}\to\mathcal{T}_{\theta}$ be defined in spherical
coordinates by

$P(r,\xi)=(pr,\xi),$

where $p=p(\xi_{1})=\cos(\theta/2)\sec(\xi_{1})$ and the Rayleigh quotient by

$\frac{\int_{\mathcal{S}_{\theta}}|\nabla f|^{2}(\phi^{2}\circ
P)\,dV+(\beta/\sin(\theta/2))\int_{P^{-1}(\Gamma_{\theta})}f^{2}(\phi^{2}\circ
P)\,dS}{\int_{\mathcal{S}_{\theta}}f^{2}(\phi^{2}\circ P)\,dV},\quad\text{for
}f\in H^{1}(\mathcal{S}_{\theta})\setminus\\{0\\}.$

The parameter $\beta=\beta(\theta)$ is equal to $(1+o(1))\alpha$, as
$\theta\to 0$, and will be defined precisely at the end of Step 2. The above
Rayleigh quotient corresponds to the $\sigma$-Laplacian eigenvalue problem on
$\mathcal{S}_{\theta}$ with $\sigma=\phi^{2}\circ P$, Robin conditions on the
spherical part of the boundary
$P^{-1}(\Gamma_{\theta})=\partial\mathcal{S}_{\theta}\cap\partial
B(\cos(\theta/2)^{-1})$, and Neumann conditions on $\Sigma_{\theta}$.

We take $f=v\circ P$ as a trial function for $\mu_{1}(\mathcal{S}_{\theta})$
so that

$\mu_{1}(\mathcal{S}_{\theta})\leq\frac{\int_{\mathcal{S}_{\theta}}|\nabla[v\circ
P]|^{2}(\phi^{2}\circ
P)\,dV+(\beta/\sin(\theta/2))\int_{P^{-1}(\Gamma_{\theta})}[v\circ
P]^{2}(\phi^{2}\circ P)\,dS}{\int_{\mathcal{S}_{\theta}}[v\circ
P]^{2}(\phi^{2}\circ P)\,dV}.$ (20)

Let $J_{P}$ and $J_{P^{-1}}$ be the Jacobian matrices of $P$ and $P^{-1}$, and
$I_{\mathcal{S}_{\theta}}$ and $I_{\mathcal{T}_{\theta}}$ be the identities on
$\mathcal{S}_{\theta}$ and $\mathcal{T}_{\theta}$. After writing $P$ and
$P^{-1}$ in rectangular coordinates a calculation shows that

$\lVert P-I_{\mathcal{S}_{\theta}}\rVert_{L^{\infty}},\leavevmode\nobreak\
\lVert J_{P}-I_{\mathcal{S}_{\theta}}\rVert_{L^{\infty}},\lVert
P^{-1}-I_{\mathcal{T}_{\theta}}\rVert_{L^{\infty}},\lVert
J_{P^{-1}}-I_{\mathcal{T}_{\theta}}\rVert_{L^{\infty}}\to 0,\quad\text{as
}\theta\to 0.$ (21)

The norm of the Jacobian matrices is the $L^{\infty}$-norm of the operator
norm.

We calculate the denominator of (20) by changing variables and using (21) so
that

$\int_{\mathcal{S}_{\theta}}[v\circ P]^{2}(\phi^{2}\circ
P)\,dV=\int_{\mathcal{T}_{\theta}}v^{2}\phi^{2}|\det(J_{P^{-1}})|\,dV=(1+o(1))\int_{\mathcal{T}_{\theta}}v^{2}\phi^{2}\,dV=1+o(1),\quad\text{as
}\theta\to 0,$

since $v=u/\phi$ and $u$ is $L^{2}$-normalized.

Now we prove an upper bound on the gradient term

$\int_{\mathcal{S}_{\theta}}|\nabla[v\circ P]|^{2}(\phi^{2}\circ
P)\,dV\leq\lVert
J_{P}\rVert_{L^{\infty}}^{2}\int_{\mathcal{S}_{\theta}}|(\nabla v)\circ
P|^{2}(\phi^{2}\circ P)\,dV\leq M(\theta)\int_{\mathcal{T}_{\theta}}|\nabla
v|^{2}\phi^{2}\,dV,$

where $M(\theta)=\lVert
J_{P}\rVert_{L^{\infty}}^{2}\lVert\det(J_{P^{-1}})\rVert_{L^{\infty}}=1+o(1)$,
as $\theta\to 0$ by applying the estimates in (21).

Finally, to handle the boundary term we prove the lower bound

$\int_{P^{-1}(\Gamma_{\theta})}[v\circ P]^{2}(\phi^{2}\circ
P)\,dS\geq\int_{\Gamma_{\theta}}v^{2}\phi^{2}\,dS.$ (22)

By parametrizing the spherical cap $P^{-1}(\Gamma_{\theta})$ with the map
$P^{-1}|\Gamma_{\theta}$ we have

$\displaystyle\int_{P^{-1}(\Gamma_{\theta})}[v\circ P]^{2}(\phi^{2}\circ
P)\,dS=\int_{\Gamma_{\theta}}v^{2}\phi^{2}\sqrt{\det(G)}\,dy,$

where $G=G(y)$ is the Gramian matrix with entries
$G_{ij}=\langle\partial_{i}P^{-1}|_{\Gamma_{\theta}},\partial_{j}P^{-1}|_{\Gamma_{\theta}}\rangle$
with $P^{-1}|_{\Gamma_{\theta}}$ written in rectangular coordinates $y_{i}$
for $i=1,\dots,n-1$. Observe that $\sqrt{\det(G)}\,dy$ is the infinitesimal
$(n-1)$-dimensional volume element associated with the map $P^{-1}$ that maps
the flat disk $\Gamma_{\theta}$ onto the spherical cap
$P^{-1}(\Gamma_{\theta})$. Thus, $\sqrt{\det(G)}$ is larger than 1 pointwise
so (22) holds.

Applying the estimates on each part of the Rayleigh quotient to (20) shows
that

$\mu_{1}(\mathcal{S}_{\theta})\leq(1+o(1))\bigg{[}M(\theta)\int_{\mathcal{T}_{\theta}}|\nabla
v|^{2}\phi^{2}\,dV+(\beta/\sin(\theta/2))\int_{\Gamma_{\theta}}v^{2}\phi^{2}\,dS\bigg{]},\quad\text{as
}\theta\to 0.$

Choosing $\beta=M(\theta)\alpha$ and factoring out the $M(\theta)$ from the
gradient term and $\beta$ shows that
$\mu_{1}(\mathcal{S}_{\theta})\leq(1+o(1))\mu_{1}(\mathcal{T}_{\theta})$, as
$\theta\to 0$.

Step 3: Observer that $\mu_{1}(\mathcal{S}_{\theta})$ is the first eigenvalue
$\mu_{1}(B(1))$ of the problem

$\begin{cases}-\Delta_{\sigma}w=\mu_{1}w,\quad&\text{on}\leavevmode\nobreak\
B(1)\\\
\partial_{r}w+(\beta/\sin(\theta/2))w=0,\quad&\text{on}\leavevmode\nobreak\
\partial B(1),\end{cases}$

where $\sigma=\phi^{2}\circ P$ and $\partial_{r}$ is the radial derivative
(i.e. the normal derivative). Indeed, since
$\sigma=\exp\\{2\alpha\cot(\theta/2)r\\}$ is radial, the first eigenfunction
$w$ is also radial, and therefore the restriction $w|_{\mathcal{S}_{\theta}}$
has Neumann boundary conditions on $\Sigma_{\theta}$. The restriction is still
an eigenfunction, and since it has the correct boundary conditions and is non-
negative it must be a ground state of the problem on $\mathcal{S}_{\theta}$
and so $\mu_{1}(\mathcal{S}_{\theta})=\mu_{1}(B(1))$.

Now we transfer all of the $\theta$ dependence to the domain by rescaling the
radial coordinate by $r\to\cot(\theta/2)r$ and letting $R=\cot(\theta/2)$ and
$\tilde{w}(r)=w(\tan(\theta/2)r)$ so that

$\begin{cases}-\Delta_{\tilde{\sigma}}\tilde{w}=\nu_{1}\tilde{w},\quad&\text{on}\leavevmode\nobreak\
B(R)\\\
\partial_{r}\tilde{w}+\beta\tilde{w}=0,\quad&\text{on}\leavevmode\nobreak\
\partial B(R),\end{cases}$ (23)

where $\tilde{\sigma}(r)=\sigma(\tan(\theta/2)r)=e^{2\alpha r}$ and
$\nu_{1}(R)=\tan(\theta/2)^{2}\mu_{1}(\mathcal{S}_{\theta})$.

Step 4: We make a final change of variables to eliminate the weight
$\tilde{\sigma}$ and Robin parameter $\alpha$ so that (23) becomes the
Schrödinger eigenvalue (3). A direct calculation shows that the radial
function $\varphi(r)=\tilde{w}(r)/e^{-\alpha r}$ satisfies the Schrödinger
eigenvalue problem

$\begin{cases}(-\Delta-\frac{(n-1)|\alpha|}{r}+\alpha^{2})\varphi=\nu_{1}\varphi,\quad&\text{on}\leavevmode\nobreak\
B(R)\\\
(\partial_{r}+(\beta-\alpha))\varphi=0,\quad&\text{on}\leavevmode\nobreak\
\partial B(R).\end{cases}$

Since $\varphi\geq 0$ we know that $\nu_{1}$ is the first eigenvalue of the
Schrödinger eigenvalue problem.

Recall from the definition of $\beta$ (at the end of Step 2) that
$\beta-\alpha=o(1)$ as $\theta\to 0$. Thus, for fixed $\gamma<0$ we have that
$E(R)+\alpha^{2}\leq\nu_{1}(R)$, where $E(R)$ is the first eigenvalue of (3)
with Robin parameter $\gamma$. Using that $R=\cot(\theta/2)$ and combining
this inequality with those from Steps 1, 2, and 3, there is a constant $C>0$
such that

$CR^{2}(E(R)+\alpha^{2})\leq
CR^{2}\nu_{1}(R)=C\mu_{1}(\mathcal{S}_{\theta})\leq\mu_{1}(\mathcal{T}_{\theta})=\lambda_{1}(\mathcal{D}_{\theta})-\lambda_{1}(\mathcal{C}_{\theta})<0.$

Using part (ii) of Lemma 3 with $\kappa=(n-1)|\alpha|$ and
$\gamma\in(-|\alpha|,0)$ we have that
$E(R)+\alpha^{2}=O(\exp\\{-2(1-\epsilon)|\alpha|R\\})$ as $R\to\infty$, for
each $\epsilon>0$, so that

$\lambda_{1}(\mathcal{D}_{\theta})-\lambda_{1}(\mathcal{C}_{\theta})=O(\exp\\{-2(1-\epsilon)|\alpha|/\sin(\theta/2)\\}),\quad\text{as
}\theta\to 0,\text{ for each }\epsilon>0.$

This last estimate follows by using that $R=\cot(\theta/2)$, that the extra
factor of $R^{2}$ can be absorbed into the error term since $\epsilon$ is
arbitrary, and that the factor of $\cos(\theta/2)$ in the exponent can be
absorbed by the factor of $(1-\epsilon)$ for the same reason.

Combining the upper and lower bounds: Combining this lower bound with the
upper bound on $\lambda_{2}(\mathcal{D}_{\theta})$ and using that
$\sin(\theta/2)\leq\theta/2$ we conclude that there is a $C>0$ such that
$(\lambda_{2}-\lambda_{1})(\mathcal{D}_{\theta})\leq
C\exp\\{-4(1-\epsilon)|\alpha|/\theta\\}$ for all $\theta$ sufficiently small
since $\alpha<0$. This concludes the proof of Theorem 2.

######  Remark.

To prove the lower bound for $\lambda_{1}(\mathcal{D}_{\theta})$ we used the
map $P^{-1}$ to “push out” the truncated cone $\mathcal{T}_{\theta}$ to the
spherical sector $\mathcal{S}_{\theta}$. This idea was used to compute bounds
on eigenvalues of thin isosceles triangles first by Freitas in [7, §3] for
Dirichlet boundary conditions and later by Laugesen and Suideja in [14, proof
of Lemma 5.2] for Neumann boundary conditions.

##  6\. Questions

Theorem 2 shows that Conjecture 1 fails to extend to $\alpha<0$ for general
convex $\mathcal{D}$. In a positive direction Laugesen recently showed that
Conjecture 1 holds and is sharp for all $\alpha\in(-\infty,\infty]$ for
rectangular boxes (i.e. products of intervals) [13, Theorem III.8]. This
inspires the question: _Is there a general class of convex domains such that
the gap inequality in Conjecture 1 holds when $\alpha<0$?_ Since the proof of
Theorem 2 appears to rely on the fact that the domain $\mathcal{D}_{\theta}$
has vertices with small opening angle, we might consider minimizing
$(\lambda_{2}-\lambda_{1})(\cdot)$ over convex domains with a uniformly
bounded Lipschitz constant.

Surprisingly, it seems that the gap still fails to have a positive lower bound
over this restricted class of domains. To see this we sketch an approximation
argument. Define the _truncated double cone_ domains

$\mathcal{D}_{\theta,\epsilon}=t(\epsilon)\\{(x,y)\in\mathcal{D}_{\theta}:|x|<1-\epsilon\\},\quad\text{for
}\epsilon\in[0,1),$

where the rescaling $t(\epsilon)$ is such that $\mathcal{D}_{\theta,\epsilon}$
has diameter 2 for each $\epsilon$ sufficiently small. (Note that
$t(\epsilon)=1+o(1)$, as $\epsilon\to 0$.) For $\theta$ fixed,
$\\{\mathcal{D}_{\theta,\epsilon}\\}_{\epsilon\in[0,1)}$ has a Lipschitz
constant that is uniform in $\epsilon\in[0,1)$ since
$\partial\mathcal{D}_{\theta,\epsilon}$ is locally the graph of $y\mapsto
t(\epsilon)\max\\{\tan(\theta/2)|y|,\epsilon\\}$ near $x=\pm
t(\epsilon)(1-\epsilon)$. In addition,
$\mathcal{D}_{\theta,\epsilon}\to\mathcal{D}_{\theta}$ in the Hausdorff
distance and
$|\partial\mathcal{D}_{\theta,\epsilon}|\to|\partial\mathcal{D}_{\theta}|$ as
$\epsilon\to 0$ so we expect that

$\lambda_{j}(\mathcal{D}_{\theta,\epsilon})\to\lambda_{j}(\mathcal{D}_{\theta}),\quad\text{as
}\epsilon\to 0,$

for each $j\in\mathbb{N}$; see [8, Remark 4.33] for a discussion of Robin
continuity results.

Assuming the above limit, we can use Theorem 2 to find a “diagonal sequence”
$\epsilon(\theta)$ converging to zero such that

$(\lambda_{2}-\lambda_{1})(\mathcal{D}_{\theta,\epsilon(\theta)})\to
0,\quad\text{as }\theta\to\infty,$

and $\mathcal{D}_{\theta,\epsilon(\theta)}$ has diameter 2 for each $\theta$.
Observe that the truncated domains
$\\{\mathcal{D}_{\theta,\epsilon(\theta)}\\}_{\theta\in(0,\pi/2)}$ also have
uniformly bounded Lipschitz constants. This shows that
$\lambda_{2}-\lambda_{1}$ can be made arbitrarily small among convex domains
of a given diameter and with a uniformly bounded Lipschitz constant. This
example demonstrates the difficulty of determining a general class of domains
where Conjecture 1 holds when $\alpha<0$, even with a weaker lower bound.

## Acknowledgements

The author would like to thank Richard Laugesen for his encouragement and many
helpful conversations on this project, as well as Mark Ashbaugh for some
enlightening conversations about special functions. The author gratefully
acknowledges support from the University of Illinois Campus Research Board
award RB19045 (to Richard Laugesen).

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