*2.2. Study Design*

The study was a single-center, open-label, non-placebo controlled, combined, methodological field trial to determine (a) method(s) for measuring cat allergy symptoms in participants that have immunized their cats with Fel-CuMV.

**Human part:** *Main study*—Screening visits took place at the Clinical Trials Center, University Hospital Zurich (USZ), CH in collaboration with the Department of Dermatology (USZ). After the informed consent was given, the participants underwent two tests to confirm their cat allergy. A skin-prick test was performed using a standard cat allergen and a scratch-test was performed with fur from the own cat. Home-based provocation tests were performed in study weeks 1–3 before vaccination of the cat(s), and in weeks 4, 8, 12, 16, 20 and 24 after vaccination of the cat(s). The provocation test assessed two parameters, the petting time and organ-specific symptoms score (OSSS). The test was only valid if participants had not taken medication containing anti-histamines less than 3 days before the test, otherwise they had to reschedule the test accordingly. If the rescheduling did not happen, the test for that timepoint was invalid and not considered in the statistical analysis. Once a week, the participants filled in a questionnaire retrospectively recording their organ-specific symptoms of the previous week (i.e., general weekly symptom score GWSS). A close-out phone call at the end of the study in week 29 was done to follow up on the well-being and health of the study participant.

*Extension study* - Seven of ten participants of the main study signed the informed consent form and were enrolled in the extension study. They performed a provocation test before the booster injection of their cat(s) in study week 1 (or week 54 in the combined schedule of main and extension study including the intervening time) followed by four additional tests in study weeks 5, 9, 25 and 45 (weeks 58, 62, 78 and 98, respectively). Again, a provocation test was only valid if participants had not taken medication containing anti-histamines less than 3 days before the test. In the event of a breach, the procedure was followed as described in the main study. In addition, participants recorded their general organ-specific symptoms without provocation monthly (i.e., general specific symptoms GSS). The study finished with a close-out phone call by the clinical study team in study week 45 (week 98, respectively).

**Animal part:** All interventions for the cats were performed by study veterinarians at the "Kleintierpraxis Schwäntenmos" in Zumikon, Zurich, "Ennetseeklinik für Kleintiere AG" in Hünenberg, Zug and "Tierklinik Aarau West AG" in Aarau, Aargau.

*Main study*—Thirteen cats were enrolled and received three subcutaneous injections of 100 μg Fel-CuMV formulated in 1 mL in study weeks 4, 7, and 10. Sera were collected in study weeks 4, 10, and 27 and analyzed for specific antibody responses.

*Extension study*—Nine cats received a subcutaneous booster injection of 100 μg Fel-CuMV formulated in 1 mL in week 3 (or week 56 in the combined study schedule). Sera were collected before the boost (week 56), 6 weeks after the boost (week 62) and at the end of the extension cat study (week 78).


**Sequential Management with Gonadotropin-Releasing Hormone Agonist and Dienogest of Endometriosis-Associated Uterine Myoma and Adenomyosis**

Atsushi Imai\*, Hiroshi Takagi, Kazutoshi Matsunami and Satoshi Ichigo *Institute of Endocrine-Related Cancer, Matsunami General Hospital, Japan*


**3.4.2 New anticoagulants**

The limitations of the traditional anticoagulants are particularly obvious in pediatrics; hence, the promotion of the new drugs already approved in adults urgent. Yet there is but individual experience in children with the following substances: the pentasaccharides fondaparinux and idraparinux, and the direct thrombin inhibitors hirudin, bivalirudin, argatroban; ximelagatran has been withdrawn from the market because of hepatic toxicity.(Balsa, 2005; Kuhle et al, 2006).


**1. Introduction**

Sweet potato (*Ipomoea batatas* (L.) Lam), a member of the bindweed family (*Convolvulaceae* Juss.), is well known for its food, medicinal, and industrial value [1–3]. More than 5000 years ago, this species was first domesticated in the Americas. It is estimated that more than 7000 sweet potato cultivars have been bred to date. Sweet potatoes are considered one

**Citation:** Kirakosyan, R.N.; Kalashnikova, E.A.; Abubakarov, H.G.; Sleptsov, N.N.; Dudina, Y.A.; Temirbekova, S.K.; Khuat, Q.V.; Trukhachev, V.I.; Sumin, A.V. Influence of Mineral Treatment, Plant Growth Regulators and Artificial Light on the Growth of Jewel Sweet Potato (*Ipomoea batatas* Lam. cv. Jewel) In Vitro. *Life* **2023**, *13*, 52. https://doi.org/10.3390/ life13010052

Academic Editors: Hakim Manghwar and Wajid Zaman

Received: 10 November 2022 Revised: 6 December 2022 Accepted: 21 December 2022 Published: 24 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of the most important root crops after potatoes and cassava [4], especially in developing countries in Latin America, Southeast Asia, and Africa [5]. In these countries, sweet potato is a main food crop for the people because of its richness with healthy proteins, vitamins, antioxidants, and minerals [6,7]. Studies have shown that this species possesses a number of pharmaceutical properties, such as antibacterial, antioxidant, anticancer, anti-inflammatory, and antiulcer activities [8–10]. In traditional medicine in many countries, sweet potato is used to treat diabetes, anemia, hypertension, stomach cancer, cardiovascular disease, allergies, constipation, eye disease, arthritis, dengue fever, and nausea [6,10–15]. On the other hand, sweet potato can be used in cooking, as well as in different industrial foods production [16–18].

Traditionally, sweet potato cultivars have been propagated by vegetative propagation using stem cuttings or tubers [7,19]. However, this method has certain limitations, such as slow propagation speed, being time-consuming, and being season-dependent. On the other hand, stem cuttings or tubers used for this method often accumulate pathogens (such as nematodes, insect pests, and pathogens that cause black rot, scurf, and stem rot) and can be spread from one generation to the next, causing great losses in yield and the production of poor-quality tubers [3,7,19]. Therefore, if large-scale, uniform, and disease-free plant material is required for production, this method is not always suitable [3,7]. On the other hand, the seeds of sweet potato are only used to breed and develop new cultivars of sweet potatoes because of their highly heterozygous nature [20]. In order to overcome the limitations, biotechnological methods using plant tissue culture techniques have been used for the commercial scale-up of many cultivars of sweet potato. In previous reports, several types of explants, e.g., nodal segment, axillary shoot, and shoot tip from mature or in vitro plants, have been used as explants for in vitro shoot proliferation of sweet potato cultivars [21–39]. In there, nodal segments have been recognized as the best or most commonly used explants for the micropropagation of various sweet potato cultivars, e.g., 'Carmen Rubin' and 'White Triumph' [32], 'Gaozi No.1' [27], 'Naruto Kintoki' [40], 'KSP 36' and 'KEMB 36' [41], and 'Abees' [4]. However, little is known about the micropropagation of jewel sweet potato, which is dubbed the 'sweet potato queen' due to its high nutritional value and delicious taste with all processing methods. This sweet potato cultivar was bred in 1970 at North Carolina State University. To date, jewel has become one of the most commonly grown sweet potato cultivars, its tubers are characterized by tan skin and orange flesh and that is considered a rich source of provitamin A carotenoid [42,43].

In fact, the efficacy of micropropagation techniques depends on a variety of factors, of which three important factors are culture medium, plant growth regulators (PGRs), and artificial light sources (type and intensity). In most reports, Murashige and Skoog (MS) [44] base medium was reported to be the most suitable medium for shoot initiation, shoot proliferation, and rooting in sweet potato cultivars [21–39]. However, in some sweet potato cultivars, 1/2MS medium has been found to be more suitable for shoot proliferation and rooting, e.g., 'purple flesh sweet potato' [37] and 'red-peeled sweet potato' [4]. Most of the authors have shown that the medium without any PGRs is not suitable for in vitro shoot regeneration. In sweet potato, nutrient media supplemented with cytokinins in combination with auxins were reported to be the best for in vitro shoot proliferation [21–23,26,37]. Besides the above two factors, the type and intensity of artificial light also affect the micropropagation of sweet potatoes. Yang et al. [45] found that the different ratios of photosynthetic photon flux (PPF) of red LED (R) and PPF of blue LED (B) had different effects on the growth of sweet potato plantlets in vitro.

The aim of the present study was to determine the effects of different PGRs, different MS salt concentrations, and different artificial light conditions on the induction, proliferation, and growth of in vitro *Ipomoea batatas* 'Jewel' shoots obtained from the axillary bud and shoot tip explants.

#### **2. Materials and Methods**

#### *2.1. Plant Material Preparation*

The work was carried out at the Department of Biotechnology of the Russian State Agrarian University—Moscow Agricultural Academy named after K. A. Timiryazev (Moscow, Russia). All work on sterilization, introduction into culture in vitro, and further work on the study of callogenesis and morphogenesis were carried out in aseptic conditions of laminar hood flow (Biokom). The object of the study was explants of sweet potato (*Ipomoea batatas* Lam.), cultivar Jewel (USA), provided by the Federal Research Center for Potato named after A. G. Lorch.

Before being placed in vitro, the jewel sweet potato tubers were placed in water to activate the dormant meristem. On the 7th day, the shoots began to form, and on the 21st day, they reached 10–12 cm. The shoot tips of 1.0 cm in length and stem pieces bearing the axillary buds of 1.5 cm in length from plantlets sprouting from tubers were used as explants for experiments. The explants' surface was sterilized with a 0.1% solution of mercuric chloride (HgCl2). They were soaked in the solution for 5 min and then rinsed with sterile distilled water [4].


**4. Conclusion**

452 Selected Topics in DNA Repair

showing false results, that is lower damage. Cebulska et al. (2007) showed negative influence of smoking on the efficiency of DNA repair in lymphocytes. On the other hand, Rzeszowska-Wolny et al. (2005) did not find significant correlation between smoking and DNA damage and DNA repair after the irradiation of human lymphocytes with the dose of 2 Gy. Our results also did not show the influence of smoking on DNA repair of primary damage, except for the tail length after the exposure to 4 Gy dose. The results are similar to

It is still not clear whether the basal damage differ due to gender. Dusinska et al. (2004a) did not find the difference in creation of strand breaks, nor in the amount of DNA damage sensitive to action of specific enzymes such as endonuclease III, or FPG enzyme or alkali labile sites, although male had lower DNA damage after the treatment with hydrogen peroxide (Dusinska et al., 2004a, b). Bajpayee et al. (2002) showed that healthy men have more basal damage in lymphocytes than women. Our results showed that difference was only seen after the 4 Gy dose, where women had higher values for DNA damage in all three parameters observed in comet assay immediately after and 24 hours after the irradiation. There was not similar number of male and female, and that could also influence the results. Trzeciak et al. (2008) showed that there are differences in the amount of damage and the repair rate after the exposure of whole blood sample to 6.3 Gy dose. But most of the experiments did not show the gender influence on DNA repair capacity (Muller et al., 2001,

Age influence was not seen in this experiment. That is in agreement with other authors (Betti et al., 1994; Frenzili et al., 1997). Rzeszowska-Wolny et al. (2005) did not show the influence of sex on DNA damage level and DNA repair by the comet assay after the exposure to 2 Gy dose. Maluf et al. (2001) showed positive correlation of age and DNA repair in the exposed group. Singh et al. (1988) described positive correlation between the

Although former results showed the existence of radio adaptive response, our results did not confirm the relationship between primary DNA damage and adaptive response. Few authors have also shown lower DNA damage levels after the repeated exposure to ionising radiation (Sankaranarayanan et al., 1989; Wang et al., 1991; Ikushima, 1992; Domingues et

APE1 gene is involved in BER repair. Polymorphism in exon 5 is connected with hypersensitivity during the exposure to ionising radiation (Au et al., 2006). Results are in agreement with Au et al. (2006). There was positive correlation of polymorphic variants and

Polymorphic variants of hOGG1 gene showed positive correlation with TI and TM measured immediately after the exposure to the dose of 2 Gy. The results are in agreement with Aka et al. (2004) who showed the connection between polymorphic variants of hOGG1 Ser326Cys with lower capability of repair of oxidative DNA damage in the exposed, but not in the control group measured 60 and 120 minutes after the exposure to 2 Gy. After 120 minutes from the exposure, 71 % of individuals with high level of DNA damage had polymorphic variant of hOGG1 gene. Cornetta et al. (2006) did not show the influence of

Goode et al. (2002)) showed the connection between polymorphic variants of hOGG1 with higher risk of lung, head and neck cancer. hOGG1 and XRCC1 are involved in the same

Aka et al. (2004) on nuclear power plant workers.

2002; Rajaee-Behbahani et al., 2001; Marcon et al., 2003).

DNA damage level 120 minutes after the exposure to 2 Gy.

age and DNA damage in non-smokers.

hOGG1 polymorphism on DNA damage.

al., 1993).

The results indicate that in addition to individual gene polymorphisms, the influence of combinations of polymorphic genes to DNA damage and repair should be tested.


*Edited by Sher Bahadar Khan and Kalsoom Akhtar*

This book focuses on the chemical structure and applications of CeO2. It covers the recent developments in a wide range of CeO2 applications, particularly catalysis corrosion protection, fuel cells, sensors, and UV-blocking. It also provides a concise but thorough coverage of the chemical structure and applications of CeO2. Thus, this book provides an overview of chemical structure, applications, and recent attributes of CeO2 for a broad audience, including beginners, graduate students, and specialists in both academic and industrial sectors.

Published in London, UK © 2019 IntechOpen © Mcastrom / iStock

Cerium Oxide - Applications and Attributes

Cerium Oxide

Applications and Attributes

*Edited by Sher Bahadar Khan and Kalsoom Akhtar*


**3. Discussion**

## *3.1. The Characteristics of NAC Gene Family in L. indica*

In this study, we identified 21 complete *LiNAC* genes with Open Reading Frames based on *L. indica* transcriptome data and analyzed their basic information. We predicted and analyzed 21 LiNAC proteins, such as isoelectric power, relative molecular weight, and transmembrane structure, LiNAC7, and LiNAC16 have a transmembrane structure at the C-terminal, and it is speculated that they are likely to be NAC-membrane-binding transcription factors. Almost all *LiNAC* genes are predicted to be localized in the nucleus (Table 1); *LiNACs* are probably mostly functional in the nucleus. The conserved structure and motif of *LiNACs* were also examined (Figure 4B) and it was shown that all of them possessed the normal NAM structure (Figure 4C). Nearly all *LiNAC* genes have motif 3, motif 4, and motif 1, although motif 2 is missing from Group II, which may be related to the loss of assembled data. The motifs found in the gene members of various evolutionary branches vary and motif 3, motif 4, motif 2, motif 1, and motif 5 may be the conserved motifs in the NAC family (Figure S2). Phylogenetic trees of *L. indica* and *Arabidopsis* divide into 2 groups with 14 subgroups (Figure 1) and *LiNAC* genes were not found in many subgroups (e.g., ANAC011, ATAF, and AtNAC3/NAP). Overexpression of *AtNAP* causes early senescence in *A. thaliana* and members of the ATAF subgroup have a significant role in leaf senescence and other characteristics of plants, indicating that these subgroups may be involved in the process of plant senescence. *LiNAC8* is closely connected to AT1G32770.1, AT3G61910.1, and AT2G46770, for instance, leading one to hypothesize about the function of the *LiNAC* gene based on genetic similarities. AT1G32770.1 (*SND1*) is a key gene that regulates the synthesis of cell secondary walls. In *Arabidopsis*, *SND1* is specifically expressed in xylem fibers, and functional silencing led to secondary wall thinning and the inability of pedicel fiber cells to form secondary walls and drooping. The other subgroups of the *LiNAC* gene may play different biological functions.

Further, 18 of the 21 *LiNAC* genes were shown to be engaged in the two biological processes of GO, leaf growth and branch creation, according to KEGG and GO enrichment analyses (Figure 5). While lignin, a phenylpropane-derived polymer, together with cellulose and hemicellulose, forms the cell wall of plant vascular tissue and provides mechanical support for plant upright growth, KEGG enrichment results revealed that *LiNAC* genes were involved in the phenylpropanoid biosynthesis pathway. As a result, it is hypothesized that the *LiNAC* genes may influence the production of secondary cell walls in plants to regulate the weeping trait.


*2.2. Data Collection*

Clinical data were collected by chart abstraction and automatic export from the electronic health records and included socio-demographics, comorbidities, and pre-existing risk factors for a severe COVID-19 course. Comorbidities were classified according to International Statistical Classification of Diseases and Related Health Problems codes (ICD10). For all patients, age-adjusted Charlson Comorbidity Index [26] and Clinical Frailty Scale [27] were calculated. Patient outcomes, including in-hospital mortality, admission to ICU, need for invasive ventilation, and length of hospital stay (LOS), were collected by chart review. To assess nutritional status, we calculated nutritional risk screening (NRS) 2002 score [28] and body mass index (BMI).

#### *2.3. Laboratory Analysis*

Laboratory values correspond to blood samples obtained within the first four days of hospitalization. Serum or plasma levels for vitamins A, B12, D, and E, as well as folic acid, zinc, selenium, and copper, were measured. As a surrogate for vitamin A and for vitamin

E we measured total retinol and total α-tocopherol concentrations, respectively, by high performance liquid chromatography (HPLC). The method was modified for small sample volume and based on best practice guideline ([29], modified). Vitamin B12 and folates were measured by chemiluminescence microparticle immunoassays on an Abbott Alinity i system. Total vitamin D (cholecalciferol) concentrations were measured by chemiluminescence immunoassay on a DiaSorin LIAISON XL system. Trace elements (copper, selenium, zinc) were measured by inductively coupled plasma mass spectrometry in collision mode (helium). The method was set up without digestion and simple dilution with an alkaline solution containing an internal standard element (rhodium). Table S1 shows cut-off values for deficiencies.


Preface

This book deals with some of the most advanced research observations and their applications in electronic device manufacturing in a very concise and structured way. It will help readers to gain in-depth knowledge about lead-free solder and its significance. Generally, the electronics world has been shifting towards lead-free electronic devices because of strict rules, government legislation, and serious health and environmental issues arising out of the use of lead in electronic devices.

It is now a great challenge for academia and industry to study and analyze lead-free solder joints and to recommend the manufacturing of reliable lead-free electronic devices. Moreover, the task becomes more complicated when miniaturization hit the electronics world recently.

All five chapters were written by acclaimed researchers from all over the world describing detailed and necessary facts and figures on all aspects and scopes of lead-free soldering, its application, and the immense possibilities of advanced materials development.

*Chapter 1* gives a general overview and discusses the need for lead-free solder research. It also proposes some of the very recent observations towards achieving reliable lead-free solder joints. Kar et al. observe very interesting effects of using multilayers of Sn for controlling intermetallic compounds growth at the lead-free solder join interface. *Chapter 2* describes the importance of intermetallics in leadbased and Pb-free solder alloys starting from their formation, characteristics, and effects on material joints. Since the growth of intermetallics over time is determined by solid-state diffusion rules and varies from system to system, readers will get a broad overview of the interaction of intermetallics with substrates and will be able to determine the exact effect on solder properties. The chapter explains how the mechanical properties and growth kinetics of intermetallics at the solder joint interface can be altered by the addition of rare earth elements. *Chapter 3* discusses the soldering of metallic and ceramic materials by lead-free active Sn and Bi-Inbased solders. Readers will gain detailed knowledge of the possibilities of soldering ceramic materials, which is limited due to the poor wettability of ceramic substrates with commercial solders at classical soldering technologies because of the different thermal expansions of soldered materials. This chapter describes how the technology of soldering with active solders is selected for joining ceramic materials along with applications of ultrasound for mechanical activation of solders, which disrupts the surface oxides, changes the surface energy of ceramic materials, and supports the diffusion processes in the interface. *Chapter 4* explains the room temperature formation of intermixing layers between a Cu/glass stack. For fabricating highly reliable Cu/glass structures, atomically scaled interface bonding is currently necessary. This chapter describes how these nanoscale ZnO adhesion layers are prepared and used for reducing the processing temperature so that it does not affect the packaging technologies. *Chapter 5* discusses ways of improving lead-free micronsized solder joint cracks. These can be used in flex-on-board interconnections, which have become very popular in mobile electronics applications. However, crack formation within these micron-sized solder joints remains a major problem, which

occurs in low-temperature curable acrylic polymer resins after bonding processes. The authors describe how the elastic modulus of an anisotropic conductive film resin assembly can be altered for low-melting solder materials and electronic device packaging so that these cracks can be controlled and reliability maintained. In this study, the mechanism of an Sn-58Bi solder joint crack with low-temperature curable acrylic adhesive is investigated.

The main focus of this book is to make scientists/researchers aware of the most advanced developments and applications in lead-free solders. We hope this book will be helpful not only for those who work in the field of solder jointing and its applications but also for researchers and scientists working on photonics and semiconductor devices. Finally, we would like to thank all the contributing authors for their hard work in the preparation of this book.

> **Abhijit Kar** Jagadis Bose National Science Talent Search, Kolkata, India

> > **1**

**Chapter 1**

Interface

**1. Introduction**

has come into the limelight.

conventionally used Sn37Pb solder.

*Monalisa Char and Abhijit Kar*

Introductory Chapter: Overview

of Multilayered Thin Film of Sn

All electronic devices starting from a small remote-controlled car to large aerospace vehicles require numerous interconnects within the circuit to complete the electrical pathways for its smooth functionality. For making these contact points, soldering plays the most important role. Until recently, Sn63-Pb37 (Sn-Pb eutectic) solders were used widely for making these contact points. The whole electronic industry was depending on the Sn-Pb alloy due to their low cost, good solder ability, low melting temperature, and satisfactory mechanical and functional properties. However, it has been observed that there arise serious environmental and health hazards caused due to the extensive use of electronic gadgets containing lead. So, there came strict restrictions imposed by the Restriction of Hazardous Substance (2002/95/EC (RoHS 1)) and Waste Electrical and Electronic (WEEE) directives and other similar bodies in use of lead in the electronic gadgets. Hence, the need for the development of lead-free solder alloy for electronics and microelectronic devices

However, the miniaturization of electronic devices aiming at high performance has made the task very complex for all the researchers to develop reliable and costeffective electronic joining materials/technologies. Scientists have developed different lead-free soldering alloys for making consumer electronic devices; however, after more than couples of decades, it is yet to conclude the best possible solder that can withstand different harsh environmental conditions and provide good reliability and improve the service life of the device/joints (compared to the Sn-Pb eutectic solder). In recent days, we have observed that the service life of any electronic gadgets have been reduced drastically. These early failures of various consumer electronic gadgets have been often linked with the absence of a good lead-free solder material, which can withstand various service-exposed conditions [1–8]. After many years of research, different lead-free solder alloys like SAC105, SAC205, SAC305, SAC405, SN100C, Sn-Zn-, Sn-Cu-, and Sn-Bi-based solders have been explored by the industry and academia [9–11]. However, SAC305 (Sn-3.0Ag-0.5Cu), a lead-free solder alloy, has been identified as a nearest substitute of the

of Pb-Free Solders and Effect

on the Lead-Free Solder Joint


2. The model

The system is presented in Figure 1. The system consists of an infinite nanowire (in the longitudinal x-direction), whose width (in the transverse y-direction) is w, a point defect, whose distance from the boundary of the nanowire is wε, and an external capacitor, whose voltage and charge can be controlled by a power source.

### Figure 1.

Model schematic. The capacitor creates the transverse electric field, and the point defect simulates the single atom.

Single-Atom Field-Effect Transistor DOI: http://dx.doi.org/10.5772/intechopen.81526

Mathematically, the model can be described by the 2D stationary Schrödinger equation

$$\frac{\partial^2 \Psi}{\partial \mathbf{x}^2} + \frac{\partial^2 \Psi}{\partial \mathbf{y}^2} + (E - U(\mathbf{y}) + F\mathbf{y})\Psi = -D(\mathbf{r} - \mathbf{r\_0})\Psi \tag{1}$$

in which normalized units (where Planck constant is ℏ ¼ 1 and the electron's mass is m ¼ 1=2) were used. In this equation

$$U(\boldsymbol{\nu}) = \begin{cases} \mathbf{0} & \mathbf{0} \prec \boldsymbol{\mathcal{Y}} \prec \boldsymbol{w} \\ \infty & \text{else} \end{cases} \tag{2}$$

is the boundaries' potential, which confines the dynamics to the wire's geometry. F is the electric field.

The point defect, which models the single atom, is presented by the asymmetric impurity D function (IDF) [19–21]

$$D(\mathbf{r}) \equiv \lim\_{\rho \to 0} \frac{2\sqrt{\pi}\delta(y)\exp\left(-\mathbf{x}^2/\rho^2\right)}{\rho\ln\left(\rho\_0/\rho\right)}\tag{3}$$

which is located at r0 ¼ ^yε, and ρ0 is related to the impurity's bound eigenenergy by

$$E\_0 = -\frac{16\exp\left(-\gamma\right)}{\rho\_0^2} \cong -\frac{8.98}{\rho\_0^2} \tag{4}$$

where γ ffi 0:577 is Euler constant [22]. On the other hand, the relation between ρ0 and the physical properties of a real physical impurity (which has a finite radius a and a finite local potential V0) is

$$\rho\_0 = 2a \exp\left(\frac{2}{V\_0 a^2} + \frac{\chi}{2}\right). \tag{5}$$

The homogeneous eigenstates solutions of the wire, i.e., solutions without the point defect, are

$$
\Psi\_{n,E}(\mathbf{x}, \boldsymbol{\uprho}) = \exp\left(i k\_n \boldsymbol{\uprho}\right) \chi\_n(\boldsymbol{\uprho}) \tag{6}
$$

where χnð Þy are the eigenstates of the one-dimensional differential equation

$$\frac{\partial^2 \chi\_n(\mathbf{y})}{\partial \mathbf{y}^2} + (E\_n - U(\mathbf{y}) + F\mathbf{y})\chi\_n(\mathbf{y}) = \mathbf{0} \tag{7}$$

with the corresponding eigenvalues

$$k\_n^2 = E - E\_n.\tag{8}$$

These eigenstates can be written using the Airy functions Ai and Bi [22] and the normalized parameter

$$\xi \equiv \mathcal{y}F^{1/3} + \left(E - k\_n^2\right)/F^{2/3} = F^{1/3}\left\{\mathcal{y} + \left(E - k\_n^2\right)/F\right\},$$

as

$$\chi\_n(\xi) = N\{Ai(\xi)Bi(\xi\_0) - Ai(\xi\_0)Bi(\xi)\}\tag{9}$$

where N is the normalization constant, ξ0 � F1=3 fEn=Fg, where the eigenvalues En are determined by the infinite solutions of the transcendental equation.

χnðξwÞ ¼ 0, when ξw � F1=3 fw þ En=Fg.

In the case of a weak electric field, the eigenstates can be written to a first order in the electric field as a superposition of the free (zero electric field) eigenstates

$$\Phi\_m(y) = \sqrt{\frac{2}{w}} \sin \left( m \pi y / w \right) \tag{10}$$

namely,

$$\chi\_m(\mathbf{y}) = \phi\_m(\mathbf{y}) + F\nu^3 \sum\_q \phi\_q(\mathbf{y}) \frac{(-1)^{m+q} - 1}{2} \frac{mq}{(m+q)^3 (m-q)^3} \frac{8}{\pi^4} \tag{11}$$

with the corresponding eigenenergies (again to the first order in F)

$$E\_n \cong \left(n\pi/w\right)^2 + \frac{1}{2}Fw.\tag{12}$$

Clearly, in the absence of the point defect (the atom), there is no coupling between the transverse direction and the longitudinal one, i.e., the capacitor cannot affect the longitudinal conductance.

There is an exception, of course, if the capacitor occupies a finite region in space, in which case the electric field does create a coupling between the modes. But in the regime of a weak electric field, this coupling amq is also very weak

$$a\_{mq} = Fw^3 \frac{(-1)^{m+q} - 1}{2} \frac{mq}{\left(m+q\right)^3 \left(m-q\right)^3 \pi^4} \,\mathrm{s}.\tag{13}$$

Even to adjacent modes (where most of the energy is transferred), the coupling is very weak

$$a\_{m,m\pm1} = \mp F \nu^3 \frac{m(m\pm1)}{(2m\pm1)^3} \frac{8}{\pi^4} \tag{14}$$

For example, the coupling between the first and the second modes is as small as a1, 2 ffi 0:0061Fw3 .

However, the presence of the point defect breaks the Cartesian symmetry and increases the coupling between the modes.

The general solution, which takes the point defect into account, is

$$\Psi(\mathbf{r}) = \Psi\_{\rm inc}(\mathbf{r}) - \frac{G^+(\mathbf{r}, \mathbf{r}\_0)\Psi\_{\rm inc}(\mathbf{r}\_0)}{1 + \int d\mathbf{r}' G^+(\mathbf{r}', \mathbf{r}\_0)D(\mathbf{r}' - \mathbf{r}\_0)} \left[d\mathbf{r}'D(\mathbf{r}' - \mathbf{r}\_0)\tag{15}\right]$$

where ψincð Þr is the incoming wavefunction, which can be chosen as one of the eigenmodes χnð Þy , which in the case of a weak electric field can be approximated by Eq. (11) (or Eq. (10), i.e., by ϕmð Þy ); Gþðr; r0Þ is the outgoing 2D Green function, i.e., Gþðr; r0Þ is the solution of the partial differential equation

$$-\nabla^2 G^+(\mathbf{r}, \mathbf{r}\_0) + [U(\mathbf{y}) - E - F\mathbf{y}]G^+(\mathbf{r}, \mathbf{r}\_0) = -\delta(\mathbf{r} - \mathbf{r}\_0) \tag{16}$$

which can be written in terms of the 1D eigenstates χmð Þy as

$$G(\mathbf{r}, \mathbf{r}') = \sum\_{n=1}^{\infty} \frac{\chi\_n(\mathbf{y}) \chi\_n^\*(\mathbf{y}')}{2i\sqrt{E - E\_n}} \exp\left(i\sqrt{E - E\_n}|\mathbf{x} - \mathbf{x}'|\right). \tag{17}$$

The scattered solution is therefore

$$\Psi(\mathbf{r}) = \exp\left(ik\_n \mathbf{x}\right) \chi\_n(\boldsymbol{y}) - \sum\_{m=1}^{\infty} A\_{n,m} \chi\_m(\boldsymbol{y}) \exp\left(ik\_m |\mathbf{x}|\right) \tag{18}$$

with the coefficients

$$A\_{n,m} \equiv \lim\_{\rho \to 0} \frac{\frac{\chi\_{\mathfrak{a}}(\epsilon)\chi\_{m}^{\*}(\epsilon)}{2ik\_{m}}}{\frac{\ln \left(\rho\_{0}/\rho\right)}{2\pi} + \sum\_{q=1}^{\infty} \frac{\chi\_{q}(\epsilon)\chi\_{q}^{\*}(\epsilon)}{2ik\_{q}} \exp\left(-\left|E - E\_{q}\right|\rho^{2}/4\right)}.\tag{19}$$

The transmitted solution (x > 0) is thus

$$\Psi(\mathbf{r}) = \sum\_{m=1}^{\infty} \exp\left(ik\_m \mathbf{x}\right) \chi\_m(\mathbf{y}) \{\delta\_{n,m} - A\_{n,m}\}.\tag{20}$$

Eq. (18) is a generic solution; however, there are two types of energies, for which the solution reveals a universal pattern.